Metaphysics 101 by Rev Jonathan Barlow Gee

“Metaphysics 101” 1st edition is hereby copyright (©) by: Jonathan Barlow Gee, its author, on this: May 5, 2017. a publication of:www.benpadiah.com ------------------------------------------------------------------------------

:: Table of Contents ::

Intro Metaphysics = pg. 4 the hypercube = pg. 6 applications of the hypercube = pg. 10 the hypercross = pg. 12 Phi/Pi = pg. 16 the Torus = pg. 17 Physics Diagrams Original Misc. = pg. 24 on QED = pg. 40 further musings on QED = pg. 44 possible QED experiment musings on the photo-electric effect = pg. 47 advanced QED = pg. 48 Modeling elipses using cones and photo-optics = pg. 49 modeling photo-electric QED with lenses = pg. 51 the double-slit experiment = pg. 55 Chaos from Nothingness = pg. 56 shadow time = pg. 58 Modeling zero-point with gyroscopic motion in a fluid dynamic medium more info on the Lorenz strange attractor = pg. 61 some diagrams depicting its construction = pg. 62 on cosmology = pg. 66 ANGLE OF TIME = pg. 70 GRAVITY = TIME = pg. 71 Solar Precession Relative to the Zodiac = pg. 74 Topology the 3-star = pg. 76 Modeling the fourth dimension with light and motion = pg. 77 some topology = pg. 81 Formal Propositions Equations = pg. 83 Metaform = pg. 84 IMG 113 = pg. 86 Time Diagrams = pg. 87 Super-Reimann metric tensor & static DeSitter tensor combined. = pg. 88 Squaring the Circle using the Pythagorean Triangle = pg. 90 the Pythagorean Spiral = pg. 102 Infinite division of congruent similarities = pg. 112 Number Theory = pg. 117 on the tables of the eighth and the ninth = pg. 119 2

Basic Metaphysics part 1: the Torus = pg. 123 part 2: the Tesseract = pg. 132 part 3: the Hypercross = pg. 139 part 4: Hypershapes = pg. 143 Advanced Metaphysics Electrons = pg. 153 Photons = pg. 156 the Photo-Electric Effect = pg. 160 a Light Cone = pg. 167 Cosmological Model = pg. 172 the 4 Forces = pg. 178 Black Holes and Worm Holes = pg. 181 ------------------------------------------------------------------------------

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“Basic Metaphysics” 4

by:

Jon Gee

“Basic Metaphysics” by:

Jonathan Barlow Gee originally published on:

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www.benpadiah.com/basic_intro.html circa

MMIII A.D.

(2003 ce).

this e-book edition was published: 2/10/2009 (+YP) front and back cover art by: Gustave Doré. the information herein is hereby released by the author into the Public Domain. please distribute thie e-book freely. :: Table of Contents ::

the hypercube pg. 6

applications of the hypercube pg. 10

the hypercross pg. 12

PHI/PI pg. 16

the Torus pg. 17

the hypercube A point is one dimensional. A line (the side of a plane) is two dimensional. A cube is three dimensional. A "hypercube" is four dimensional. The shadow of a line (seen from above one end) is a one-dimensional point. The shadow of a cube (seen from above one side) is a two-dimensional plane. So, too, the shadow of a hypercube is three dimensional. If one were to turn it around in three dimensions, a cube can cast two-dimensional shadows of different shapes. For example, the shadow of a cube seen from above the midpoint of one of its faces is a square. The shadow of a cube seen from above one of its corners is a hexagon. Etc. So, too, as one rotates and reorients a hypercube in four dimensions, it casts three dimensional shadows of different shapes. It is by the shapes of these shadows alone that we can rightly describe the movements of the hypercube. Just as there are certain angles from which a cube can be viewed that cast regular shaped shadows - the hexagon, the square, etc. - so also are there certain angles and positions at which the hypercube will cast regular shaped three dimensional shadows. One of these is, of course, the cube itself. The angle at which the hypercube casts a cubic shadow I call "standard position." A hypercube also casts a regular solid shape at "nested position," and the shape it casts then is a cube-within-a-cube. This shape is the one most commonly associated with the hypercube. Here is a picture of the hypercube at "nested position":

Another position at which the hypercube casts a regular shadow is what I call the "conjoined position." The shadow it casts at this position is a shape geometres know as a "tesseract." The word tesseract is another name for the hypercube. Here is a picture of the "conjoined" hypercube's shadow, the tesseract: 6

The last position at which the hypercube casts a regular shaped shadow which we will be dealing with here is called "antipode" position. Like the cube-within-a-cube and the tesseract, the hypercube at antipode also consists of two cubes. The "nested" cube-within-a-cube shows one cube within the other from a view above one of the faces of the hypercube. The "conjoined" tesseract shows one cube off-set from the other (by diaganols) and this is the view from above one of the hypercube's corners. The "antipode" position is a view from above one of the hypercube's edges. Here is a picture of the "antipode" position of the hypercube:

Remember, all of these regular solid shapes are only different shadows cast by the same hypercube. The only difference is that the hypercube is being seen from different angles. Just as a cube casts different shaped two dimensional shadows as it is rotated in three dimensional space, so, too, does the hypercube cast different shaped three dimensional shadows as it rotates in four-space. The rotation of a hypercube differs from the rotation of a cube, however. Because a hypercube is comprised of two cubes, each with the same volume, its rotation moves one of these cubes through the other. Here is a picture showing a few stages of this process:

It should be remembered, however, that, just as the cube can be rotated around three axes of 3space, so does the hypercube rotate through itself along a 4-d axis. This means that, while the simple rotation depicted above is occuring between two cubes on opposite sides, the same rotation is actually occuring for the hypercube through 6 cubes, 3 opposite 3.

Here is a computer rendering of the rotation of the hypercube:

Here is a depiction of a "slice" of a hypercube taken from one corner to the opposite corner:

Notice that the corner of the hypercube is a tetrahedron, that 1/2 through the hypercube is an octahedron, and that 1/4 and 3/4 through the hypercube is a zonehedral.

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applications of the hypercube

There are applications of the hypercube in many fields of study. Consider the following lattice which explains a complex series of relationships in quantum mechanics:

Also, ancient Hebrew mystics, when looking for an alternative to the tetractys of Pythagoras, stumbled upon the hypercube at antipode position for their "tree of life" diagram. Here, we see that each corner, or node, of the tree describes an attribute of YHVH, and that the twenty-two "paths" connecting these "sefirot" emanations are assigned to the twenty-two letters of the Hebrew alef-bet. This arrangement is called the Gra.

The tree of life diagram may be better recognised by its later depiction, rendered by the Safed school of Ha QBLH, known as the Ari arrangement.

the hypercross When an ordinary 3-cube is unfolded, it forms a cross of six unit squares:

So it has been reckoned that when the hypercube is unfolded, it forms a cross of eight unit cubes. Here, we see that the central cube is surrounded by six cubes, one for each side, plus a subtended eighth cube.

Here is a depiction of this type of hypercross by Salvador DalĂ:

However, this type of hypercross is comprised of eight unit cubes, while the flat cross formed by the unfolded 3-cube is only comprised of six unit squares. 13

Another tye of hypercross can be formed without the subtended eighth cube. Like the unfolded 3cube, it has six cube sides around each side. Here is a picture of it:

This type of hypercross should not be misunderstood as lacking the eighth subtended cube, however. The eighth cube is simply hidden within this form of the hypercross, between the six surrounding cubes and the central seventh. It is what is known as an "impossible" cube. Here is a picture of the "impossible" cube in the hypercross:

This type of impossible cube was discovered, along with a similar impossible triangle, in the 20th century by mathematician Roger Penrose. Such impossible shapes were then incorporated into the architectures depicted by Dutch artist Maurit Cornelius Escher:

This type of cube is called "impossible" because it cannot exist in three space, although it can be depicted two dimensionally. Here is a wooden scultpure of the impossible cube. It is comprised of two separate sculptures, one above and one below, and then photographed at an angle which allows them to appear as if they were a single cohesive whole. 14

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PHI/PI

This diagram depicts a large part of what I work with on a daily basis. It shows the orientation between what I consider, for shorthand notation, a pi and a phi spiral. It should be noted that arithmetic expansion occurs unilinearly, as a diaganol vector on a cartesian coordinate graph. 16

Now, exponential expansion occurs when the exponent of the integer is increased. For example, if one takes the exponential expansion of the numberline of arithmetic expansion, such that, 1^2, 2^2, 3^2, etc... where it then forms the summed numberline, such that, 1, 4, 9, etc.... A third kind of expansion rate exists, however it has been considered more or less esoteric, because it forms a very specific inherent pattern when graphed. This kind of expansion rate is associated with the Fibonnaci sequence of numbers, such that, 1, 1, 2, 3, 5, 8, 13, etc... where the rate of expansion is determined by the addition of only the preceding two numbers to form the sum digit third in the sequence. When this sequence is graphed as points around an origin in a cartesian grid, it forms a spiral, and this spiral has been found everywhere throughout nature, from the branching patterns on plants to the proportions of the human body. This is the pattern I call, for shorthand, phi. Returning to exponential expansion, we find that, when graphed around the origin point in a cartesian graph, this type of expansion also forms a spiral pattern. The difference between this, exponential, and the Fibonnaci spiral is that the exponential spiral is rectalinear, whereas the phi spiral progresses triangularly. This type of spiral, which is rectalinear and exponential, is what I call, for notation, pi. The difference between them is the solution to an equation I refer to as phi/pi. Above the origin point, phi appears as the upper central spiral, and pi as the lower central spiral. Discuss this section on the forums

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the Torus Now, I would like to take a moment to compare some self-evident facts that we can observe in nature. I am not going to claim to have invented any of these things, as I hear doing so results in bad karma. It is a self-evident fact, for example, that the seven basic colours of the spectrum of light (red, orange, yellow, green, blue, indigo, violet in order) can be mapped onto the surface of a torus, or hypersphere, in only one way, such that each of the seven colours occupies the same area on the surface of the shape.

Now, once these colours have been mapped onto the surface of the torus, we see that the outline between each of the mapped areas forms a spiral that wraps around the surface of the hypersphere. 17

The spiral that outlines the seven coulour spectrum is, and this also is a completely self-evident fact and was not "invented" by any human hands, a "phi" spiral.

Now, this "phi" spiral revolves AROUND the circumference of the torus (clockwise or counterclockwise) depending upon the rotation THROUGH the centre of the torus (outward from centre or inward toward centre, respectively) of the seven coloured areas mapped onto its surface. 20

In other words, the "phi" spiral revolves around the circmference as a MEASUREMENT of the surface of the torus. It MEASURES the fourth-dimensionality of this shape by MOVING, that is, it changes over time, and is therefore, like the clock, a means of measuring the passage of the fourth dimension. However, this MEASUREMENT is only of the SURFACE of the torus, measuring the revolution from the circumference (seen from above) to the centre. This is a measurement of AREA, that is, of the combined seven areas of the mapped colour spectrum.

However, what if we measure the VOLUME of the torus, that is, the interior of the rotating radii of the hyersphere? Just as the "phi" spiral REVOLVES as it measures the SURFACE AREA, so too do we need a measurement device for the ROTATION of the INTERIOR VOLUME. We know that, as the "phi" spiral "revolves" around the "top" and "bottom" of the hypersphere, so too, when we look at the torus from the "side" we see there is "rotation" of each radius, "right" or "left." Thus, just as the "revolution" of the "phi" spiral tells us about the external surface area, so too can the rotation of these radii tell us about the internal volume. Now, when the "phi" spiral is revolving "clockwise" it means the torus is rotating "outward." When the "phi" spiral is revolving "counterclockwise" it means the torus is rotating "inward." But, just as the "top" rotates "inward" while the "bottom" rotates "otward," yet there are not two phi spirals, only one continuous spiral measuring the external area, so, even though there are two radii "sides" that rotate "clockwise" or "counterclockwise" respectively, there are not two different spirals, one for each radius, one moving "clockwise" while the other moves "counterclockwise," but only one continuous spiral measuring the internal volume. So, we have "phi" measuring the outside, and another, single and continuous, spiral measuring the inside. I call this spiral "pi" for short hand, but the spiral I mean when I refer to this spiral as "pi" is really the "spiral mirabilis" derived from "e," the so-called "natural number." I have found that by dividing the recirpocal of "phi" (1/1.618) by "pi" (3.14) (thus adding one due to the rules for “phi” as a transcendental #), you arrive at 1.37, which is the same as the so-called "natural" number. Therefore "e," also called the “Fine Structure Constant,” would equal the combined "phi" (exterior) and "pi" (interior) spirals. This would mean that, if "phi" was both clockwise and counterclockwise, and "pi" was both counterclockwise and clockwise, as in the torus, then the "natural" spiral would be equal to the combination of their motions, that is, would be the sum of their spin, expressable mathematically as phi^2 / pi = e. All of these observations arise from self-evident facts of nature. None of them is my own personal invention, nor, I would posit, the "invention" of anyone other than the Creator of this universe, God. Therefore, please feel at utter liberty to discuss these ideas with no worry that I will oppose your applications of them. We are free here, and we are equal. My own personal applications of this model, the "phi/pi" spiral model for the measurement of the fourth dimension and the hypershapes that exist therein, form the basis for my cosmological diagrams that I included with my published book, "the Metaphysicians’ Desk Reference." discuss this section on the forums

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Original MISC http://www.benpadiah.com/MISC_diagrams/pages/originalMISC.html

1A) this is the standard graph of above and below the limit of photic light which defines time as the standard arrow of entropy. Below, time progresses left to right, and above, opposite this. NAM (destiny) and NAM.TAR (fate or free will) increase and decrease in direct relationship to one another. The past is increasingly fixed (permanance) while the future is ever-more in flux (change). At either "end" of the timeline is a singularity connecting the above and below. â&#x20AC;&#x2DC;

1B) when the standard graph is mapped onto a sphere, the relationships between above and below photic entropy become even clearer, as the apparent "twin" singularities are revealed as only a single polar axis, and the future and past below connect to their counterparts above around the equator, which is NAM between the two hemispheres of NAM.TAR. Change or permanence are then seen as the direction of rotation.

1C) In the second iteration of the standard diagram, the apparent "twin" singularities are both mapped as time-strems themselves, perpendicular to the standard time streams above and below the rate of photic entropy. The reason for doing this is to signify the multiverse and nulliverse effects of the aleph sub n toroid. Here we see that the same quanta are measured (photons on the left and below and tachyons above and to the right). Also we can more clearly the permanence and change become diaganol axes of effect by the interaction of these quanta.

1D) When the second iteration is spherically mounted the reason for the double diaganol axes can be seen. The opposite corners of the transform map to each other and the double axis is revealed to be merely the precession, or rotational tilt, of the axis of the singularity. At ~ 45 degrees to each other, the equators (still NAM.TAR) of the apex of the rotational tilt of the polar axis are at ~ 90 degrees to one another, or perfect perpendicularity.

2A) This model represents the static DeSitter metric, which, as Hawking points out, describes a cosmology which is similar to a black hole. In DeSitter, it is empty and will expand exponentially at a fixed temperature. While Hawking points out the difference between this and the observed universe (which is relatively full and a non-fixed temperature) the DeSitter metric may yet describe the properties of underlying super-symmetric strings or branes which comprise the fabric of the space-time continuum's geometric surface.

2B) What the DeSitter static metric describes is a unification by super-symmetry of the closed geometry of Euclidian space (the middle sphere, right) and the open geometry of LorentzianDeSitter space (the space between the outer spheres, right), where the radius = 0 on either side of the static metric is where the polar axis is a singularity. Since observers can be in varying places in the "space-like" past or future (infinity), their views represent diaganols which would be equivalent to the precession of the polar axis of the central sphere (right), when the singularity is replaced with a duration and becomes an event horizon.

2C) The model to the left represents the super-Reimann metric tensor. Originally the Reimann metric tensor was a collection of 10 numbers, represented as 16 variables, describing the curvature of four-dimensional space. When extended to n-dimensions, the Reimann metric tensor also provides for sections of curvature that describe Einstein's equations for gravity and, through super-symmetry, those of Maxwell for photic radiation, Yang and Mills for the nuclear forces, and, theoretically, for matters particles such as quarks and leptons.

2D) The super-Reimann and DeSitter metrics are essentially in agreement, that is, they are complementary. The empty corner spaces in the super-Reimann are rightly filled by the static DeSitter. The reason for this is that the empty spaces in the super-Reimann at first appear to resemble a series of squares, that is, exponential expansion, however their scale relative to one another is arithmetic. Thus, we see they are all complementary to, and, actually, overlap one another. The three spheres (right) are only different positions of one sphere, whose polar axis precesses.

3A) Here we see the combination of the DeSitter and super-Reimann metrics, with the elemntal forces duly marked, and with the second iteration of the standard graph acting as a short-hand for the elements within 4-d spacetime, and the DeSitter shown properly for sub-gravitic and super "cirtical mass" spatial curvatures, which are really complementary, and one and the same. The two, sub/super spacetime static metrics act as two parallel branes which attract toward one another, until the diaganol event horizon is warped into the ploar axis singularity of one or the other, forming a new universe through blackhole-like conditions.

3B) The geometry of this mechanism (described by the formula: manifestation = the precession of dimension, or, by the shorthand phi/pi) has already ben thouroughly described elsewhere (see tau sub tau). Here, we see the diagram for this basic geometry seen at 45 degrees and at 4-d antipode. The seven spheres involved can be likened to the seven historically known planets, as well as to the seven color spectrum. In the center is the second iteration of the 4-space static DeSitter short-hand.

3C) When enlarged and rotated from 4-d antipode to apogee the second iteration of the static deSitter short-hand assumes a nested appearance. The spiralling involution / evolution which will become the universal light cone should be apparent by the light of chockmah. Here we see the equation expressed in the ancient names and the primordial satvas. Another expression for this is the serpent of kundalini wrapped around the Akashic egg.

3D) If we take the 4-d apogee short-hand second iteration of the DeSitter metric and drop one dimension, we obtain a cube containing a diaganol. This differs from the standard diagram describing chaos theory only in that here we see the diaganol is a spiral, representing entropy over time which serves to further limit the chaos factor.

4A) Here we see the previous diagram from the side with the top and bottom of the spiral "unfolded." We see the difference between the standard chaos factor (the dotted diaganol) and the spiral in that the x & y variables of the spiral may repeat, while those of the standard chaos factor do not. To map these we use a dual/diaganol Cartesian quadrant.

4B) The labelling of this quadrant, since it measures the spiral of entropy over time, is based on the four elemental forces of 4-spacetime. These are represented as the western alchemical, eastern trigram and satva symbols, and positioned around and within the quadrant geometry of the spiral, such that, as the world line orbits along it, the different combinations occur in a counter-rotational sequence.

4C) The combinations of elemental forces can be plotted on a periodic chart. The numbers in each square's lower right corner represent the degree of difference from the square below, where the lowest of each column connects to the same column's top. The attribution of letters derives from the Enochian, and the other traits are self-explanatory.

4D) When we return to the model of chaos theory we can further map the demi-elements onto the exterior (or as a median, interior) of the cube. Here we see, by the relationships between these attributes, that the elongation of the singularity into an event horizon by the extension of duration establishes the standard diaganol measure of chaos as a fixed diaganol sqr.rt. of 3 (from corner to corner). The relationship between the demi-elemental attributes and the corner-tocorner sqr.rt. of 3 chaos factor is that of a lorentz transform, to wit refer back to the first iteration of the standard graph (preceeding).

on QED http://www.benpadiah.com/MISC_diagrams/pages/QED.html A photon has zero electro-magnetic charge, but can display properties of either positive (right hand spin, wavelength) or negative (left hand spin, particle) polarity. For the purposes of this brief exposition I will refer to photons as having both positive and negative charge simultaneously, such that they cancel each other out.

Here we see the standard and modified Feynman diagrams for the annihilation of a positron (p+) and an electron (e-) to form a photon (ph+/-). In the original Feynman, the temporary annihilation period results in altered spin vectors for the electron and positron from before to after their duration in combination as a photon. The modified version shows how a pair of photons can be created by the same annihilation of a single positron and electron pair; I have modified this to show that the spin vectors of each can possibly remain unaltered from before to after their duration in combination as the particle-wave photon. We should observe that, in both Feynman models, the electron accounts for the negative characteristics (left hand spin, wavelength) of the photon, while the positron accounts for the positive characteristics (right hand spin, particle) of the photon.

In the diagram labeled "ELECTROMAGNETISM" we see three figures. The third figure is the standard model of the electron's negative charge (e-) orbit around the positive charge magnetic dipole (m+). In figure two, we see this diagram modeled over a time duration, and seen at an angle, so that we can see that the electrical charge's orbit is perpendicular to the precession of the magnetic dipole (the electrical and magnetic forces operate at right angles to one another). In figure 1, I have borrowed a model depicting the orbit of earth around the sun over time to demonstrate why the electical and magnetic wavelengths operate at right angles to one another, and why photons operate at a third right angle to both forces. The model indicates that the unification of these three-dimensional forces lies in the spiral wavelength of a fourthdimensional force.

Now, to return to the Feynman diagram structure, in this diagram I depict the zero-duration event of a photon striking an electron. This diagram displays how the EM field's spin vectors can possibly remain unaltered from before to after the collision, and that the photon's spin vectors can possibly be altered from before to after the collision. The duration of their annihilation is marked by the exchange between the spin vectors of two tachyons operating at right angles to one another. The point of annihilation of the photon and electron is a temporal (charged) singularity, while the point of intersection of the two tachyons is a gravitational (non-charged) singularity. According to this model, whenever a photon and electron strike one another, a tachyon travelling from and a tachyon travelling to the gravitational singularity at the center of our local universe exchange directions.

Here is a model showing how this can be accomplished even for a photon in one galaxy and an electron in another when each passes into the event horizon of a black hole. An electron travelling in the same direction parallel to a photon can actually cross paths and annihilate with that photon through the wormhole of the event horizon such that the photon's and the electron's spin vectors can actually switch places over astronomic distances.

The modified model of the Feynman diagram for particle-antipartical annihilation has become the standard model for string-theory. Here we see that vibrational harmonic resonances surround the quantum field of particle-wave duality of the electron, positron and photon. These vibrational harmonic resonances (called superstrings) are comprised of extra dimensions which have not "expanded," and which surround and conserve the three-spatial dimensions of our local universe.

In my model of string theory we see how the electrical (e-) and magnetic (m+) charges of the EM field combine to form a photon before and after the annihilation of a photon into two tachyons operating at right angles to one another. Once again, the point t=empty-set represents the gravitational singularity, while the entire diagram itself represents a temporal singularity. Because energy is entering and exiting the model simultaneously, we see that a temporal singularity (such as occurs upon the collision of a photon and electron) is the same thing as a wormhole.

further musings on QED

In this image we see two possible ways to rotate an electromagnetic dipole. 44

In the top figure we rotate the dipole centrifugally around an axis intersecting the midpoint of the dipole and operating at right angles to it. This process is sped up until the positive polarity and negative charge reverse so rapidly that the result is a dually charged electromagnetic torus. In the lower figure we rotate the dipole centripetally around one end of the dipole. The resulting torus will be either positive or negative around the outside, and the opposite charge in the center, essentially a monopole.

In this image we see the electrical field lines of a dipole and the magnetic field lines of a torus. It should be noted that, should the dipole and the monopole convector be swapped, the phase portrait of the field lines for each will be the same, even though now the dipole will be magnetic and the monopole electrical.

When the dipole is inserted through the monopole it becomes clearer how the field lines of the electromagnetic force act at right angles to one another. Along the poles of the dipole the forces are oriented parallel such that they overlap perfectly. However, around the middle of the dipole, as well as surrounding the circular monopole of the torus, the field lines operate at right angles to one another. 45

Direct Current:

Alternating Current:

possible QED experiment

musings on the photo-electric effect

advanced QED

a magnification:

Modeling elipses using cones and photo-optics

In figure 1 we see the focus of light shown from the point of a cone through different types of lenses at the base. The convex lens in fig.1A refracts the light from a single concentrated beam into a more diffuse field. The concave lens in fig.1B refracts the light from a diffuse beam into a single concentrated point. In fig.1C we see the combination of these effects rendered by a dualsided convex lens.

In figure 2 we see the same cones from a different angle so as to better view the way the light is refracted through their lenses. Here, the vertical diaganol line perpendicular to the conical bases is shown from a 45° angle above the cones and between the bases and the perpendicular plane. For simplicity's sake, the bases of the cones are depicted as circular, and consequently, the planes reflecting light from them as being a planar elipse, however any shape area can be substituted for the conical bases, and consequently any angles and vertecis of the lens. The elliptical nature of the reflecting surface will be better deomnstrated from the angle depicted in the next diagram. The types of lens A, B, C remain the same as in diagram 1.

In figure 3 we see the same cones from 45°s above and 45°s to the right. Whereas in figures 1 and 2 we saw only one version of each of the three cones, in figure 3 we see two versions of each of the three cones. The different versions depict two perpendicular arcs drawn on the surface of the lens and seen from two different positions in the rotation of the conical bases. For example, fig.3A and fig.3B depict perpendicular arcs on the convex lens seen at 90° and at 45° of rotation, or rather, at perfect vertical/horizontal orientation (3A) and at diaganol orientation (3B). The rest of this figure should be clear.

modeling photo-electric QED with lenses

Here we see the standard model of photon-electron collision. The photon (ph+/-) is depicted at the temporal singularity of it striking the electron (e-). As the two make contact, the event horizon of the orbital shell probability cloud of the electron collapses down into the charged particle of the electron. This can be depicted by using angles of incidence to portray the angular momentum of the electron's orbital vector. The red lines here indicate the angle of incidence of the electron in the "shock" wave probability well created by the impact of the photon. Notice that thered lines, otherwise parallel, are warped into intersection in the presence of the photon. The green lines represent the angle of incidence for the photon relative to the nucleus (m+) at the center of the atom. Notice that it collapses sharply such that the electron's probability cloud is shown to conserve the photic momentum vector. The blue lines depict the relative paths for the photon's projected trajectory following the temporal singularity of its striking the electron. The photon's projected trajectory will combine with or deduct from the prior angular momentum of the electron, and here we see this dual probability expressed as being similar to the double-slit experiment.

Here we see the photon, electron, and nucleus depicted as conic sections. The photon is an ellipse, the nucleus is a circle, the electron's orbital shell is the base of the cone, and the electron itself is a parabola extending between the conic base and the blunt elliptical top. These are all depicted in black lines, while the relative incidence vectors from above are presented according to the same red, green and blue angles. Here, the parallelism of the photic and electric "shock" wave point wells is indicated, and the photon, electron and nucleus depicted at their relative angles to one another. The right angle, for example, between the parabolic electron point and the magnetic orbital conic base represents the orthogonality of the electromagnetic force.

This picture is essentially looking at the first figure from a yet different angle. It has all the same components, (ph+/-), (e-) and (m+) and their incidence vectors. However, here the idea expressed in looking at the first figure from the side of seeing the constituent components as flat (rather than, as tradionally, spherical) is enhanced one step further by their portrayal as a set of lenses. (Ph+/-) is a convex lens, (m+) is a concave lens, and (e-) is a convex/concave lens. Here we see the initial incidence vectors transformed into refracted beams of light. 52

What is significant about this model is that, by depicting the spherical components as relatively flat lenses, and their relationships as refracted beams of light, we are creating an experimentally verifiable model by which to look at tachyons as gravitic. For example, the red lines of the electron incidence vector here become a beam of light (gravitic tachyons) projected from the photon lens, through the electron lens, and onto the nucleus lens. From there, the same beam of light is reflected back (the green lines) through the electron to the photon. Finally, the same beam of "light" is bounced back down to cast the relative "shadow" of the electron's orbit from the lens of the photon. Note that all of these "beams" remain consistent to the angles of incidence in the first diagram, but that now, rather than spherical or conic section constiutent components, we are modeling the same notions with lenses.

So, returning from this again briefly to the spherical (Bohr) atomic model, we see the photon above connecting with the electron (right) in its orbit (outer circle) around the nucleus at the centre. Compare this with the side view model using lenses and note how the incidence vectors can be interpolated logically.

Here is a somewhat more complex model using all the same constituent components as we just saw in figure 2. Here the convex photon, concave nucleus and orbital trajectory are all represented as intersecting at a right angle. Although this situation is nt the one that occurs naturally when a photon strikes an electron (the photon itself never coming into direct contact with the nucleus), this diagram is nonetheless helpful in assisting us to see the peculiar warping of the photoelectric incidence vector (the blue lines) under different hypothetical conditions. In other words, if the angle of orientation between the photon and the nucleus were to prove to be orthogonal (based on, for example, the forces for which each is a particle carrier being oriented perpenidcularly to one another, as is the case with the electromagnetic force), then the temporal singularity of the photon-electron collision could be expressed as the sum over the difference of the two opposite probabilistic projected trajectories, one taken by the photon, the other by the electron (again, the lines in blue).

This diagram represents the same orientation of the constituent components as the last, and holds all the same referential labels and relative relationships as the preceding. This diagram is possibly, however, the clearest in its ability to illustrate the incident vectors as refracted "beams" of gravitic tachyons. 54

the double-slit experiment

Chaos from Nothingness http://www.benpadiah.com/MISC_diagrams/pages/chaosZPE.html

From the simple enough binary pattern established by the Feigenbaum splitting method, we can extrapolate an infinite series of halves. This generates a computer programming language similar to the Chinese I Ching (left) as well as a basic circuitry system based on this progression series similar to the three mother letters of QBLH (right).

Here we depict the Feigenbaum doubling method along the (more familiar) x-axis, but we limit the angle of expansion through the y-axis along the z-axis to a ninety degree slice. We label the origin point of the slice as the null or empty set, and the great arc of the forty-five degree angle we label infinity. We can still, even in this model, observe the three distinctive layers, of "bits," of the standard Feigenbaum set. 56

When we increase the limit of the angle from forty five degrees to three-hundred sixty, we form the image above, which is essentially the shadow of a Bucky-ball, as well as the Radiolarin named for him. It is also a relatively decent depiction of a spherical Menger sponge, which has infinite volume and zero mass. Here again, we are still able to see the three distinct Feigenbaum "bits." Remember, the same terms apply here as above, the origin of the circle is null or zero, and the circumference of the circle branches off into infinity.

There are basically three ways of measuring the Feigenbaum statistical "bits" of information. The first is by looking at the central ring of the first set or "bit" of binary splitting branches. The second measure we extrapolate is the circumference. The third is the radius, however since we are measuring between zero and infinity, we have to substitute the DeSitter metric measure for the event horizon between the binary branching patterns as their wavelngths arc across the radius measure.

shadow time

Consider what Peter Carroll, in Liber Chaos on page 29, calls "pseudo" time and Shadow Time. "Pseudo" time, here, is the perpetual present-tense, where there is no future and no past in physical, material reality. Shadow Time, then, is the "etheric" probabilistic projection, orthogonal to pseudo-time, of the concepts of futrue and past. The interaction of "pseudo" time and Shadow Time is such that, in each present-tense event, which is reckoned as one Planck time in length, there is a collapsing Schroedinger wave-function from probability zero (impossibility) to probability one (certainty). The operant observer principle is not mentioned. "Pseudo" time interjects itself as upon a numberline which can be reckoned as the expanding lightcone of entropy, or the standard, forward-flowing arrow of time. Shadow Time and "pseudo" time must be divided by something akin to a strange attractor, in order for the collapsed waveform of probability (img559) to alternate, or spiral, around the Shadow Temporal axis.

By replacing the simple wave function collapse along the present-tense slice, or Poincare section, with a strange attractor (img551), then we can see that the system acquires an increasing degree of freedom (Chaos), and in this way creates the opportunity for probabilities to assemble in random harmonic patterns from one present to the next.

The results of this kind of model, wherein inversion is included, mainly deal with the constituent components of matter-energy quantum pair-bonds from one Planck time to the next. For example, the direct relationship between probability one certainty and probability zero uncertainty.

However, as Wilson points out (Quantum Psychology, 159), "since quantum 'laws' do not have the absolute nature of Newtonian (or Aristotelian) laws, all quantum theory must use probabilities." He continues, "the Aristotelian 'yes' or 0% and 'no' or 100% represent the certitude Occidentals have traditionally sought. Quantum experiments refuse to yield such certitude, and we find ourselves always with some probability between 0% and 100%." Hence this model theoretically represents a centurial gradient of probabilistic outcomes rather than plain binary. 59

Since the present-tense ("psuedo" time) strange attractors can connect to the immediately past or immediately future strange attractors (Shadow Time) it also represents the combination of six strange attractors (plus and minus for each present, negative and positive for past and for future) that, when graphed in three dimensions over time, as coordinate triplicites plus spin, can be decomposed to form a 4-Matrix measuring the four possible probability states: "yes," "no," "yes AND no," and "NEITHER yes NOR no."

Modeling zero-point with gyroscopic motion in a fluid dynamic medium

Here we see the product sum over histories for the fluid dynamic medium perturbated by the motion of the three-arced gyroscope. This model fits the motions of the scalar polar model on the left (above), where the greater vertical arc remains static and the other great arcs move in opposite directions relative to one another. I have taken the liberty of labeling these spirals by the short-hand notations for PI (the arithmetic spiral) and PHI (the exponential spiral).

here is some more info on the Lorenz strange attractor: To iterate a polynomial (such as x^2 + y^2 = z^2, for example), resolve the answer for the variable z, and then substitute it for the initial variable x. Resolve the equation, and repeat. the essential formula governing the iteration of polynomials dictates that the variable z approaches the value of z^2 + c, where c is some constant. When the value of z and the constant value (c) are expressed as Complex numbers, then this graph becomes more complicated. A complex number is one whose value is determined by the equation, z = x + (?-1)y. In this equation, y is called the "imaginary" component of z, because the square of any Real number is always greater than zero. Complex numbers function identically to real numbers when the negative root is cancelled by (?-1)^2 and the multiple substituted simply by -1. If we graph the polynomial iteration for z > z^2 + c, where c = 0, then the result will be a circle, where any value for z < 1 will approach zero exponentially, because when c = 0 then z will assume the set of squared powers, such that (z, z^2, z^4, z^8, etc). Now, such a circle is one form of attractor known as a regular periodic attractor. However, for the same equation (z > z^2 + c) governing the polynomial iteration where z > 1 and where c != 0, then the regular periodic attractor will assume a new shape, known by its graphic appearance as a fractal. A fractal pattern is a form of another type of attractor, called a "strange" attractor because it is aperiodic. Now, when we say that a normal attractor is "periodic" we mean simply that it repeats, or has a regular periodic orbit. This is expressed such that, for any point value x, if its position in a closed system repeats after T steps, then it is said to have period T and to consist of T distinct points. 61

So, if we have an attractor which cycles over the time dimension such that it forms a regular pattern, then we say that it is a closed system in "phase space," or the area occupied by all possible points in the system. For example, the "phase space" of the combination of two strictly periodic attractors, say, two circles, combines to form the quasi-periodic toroidal attractor. A torus is simply a circle in four dimensions. A "strange" attractor can be created by joining two (or more) periodic attractors along a quasiperiodic axis. For example, imagine two forms of initial condition for the motion of a point in phase space. One "regime" is a circular (or "equiangular") spiral, and the other is a Fibonacci or Phi spiral. Now, when these two spirals are conjoined, then imagine that a point travelling along one spiral can, at certain points of intersection, "cross-over" or "leap" from one spiral to the other. An interesting feature of strange attractors is that they are fractals, which means, simply, that they possess self-same similarity of structure on all scales. This applies not only to the graphed diagrams of strange attractors in phase space, but also to their rate of bifurcation between statechanges over time. For example, the diaganol angle on a two-dimensional Cartesian graphed coordinate system where x = y is a one point steady-state attractor. Any other form of iterated polynomial graphed in this same system creates a series of "steps" or right-angle relationships to the initial attractor. The iterated graph of a parabola, such that y = rx(1 - x), where 0 < r < 4 determines the parabola's grade or degree of slope, with periodicity >= 2 produces fractal bifurcation of these "steps." The simple parabola itself decomposes into a fractal bifurcation when used to chart "perioddoubling" wherein periodicity (usually represented by lambda, the variable in physics used to denote wavelength) is plotted on a graph as function of x, or (x), and x is <= a single maximum value. According to this model, the bifurcation between "period-doubling," when magnified at specific intervals (known as Feigenbaum numbers), "renormalises," that is, its fine structure is indistinguishable between various scales.

Here are some diagrams depicting its construction:

on cosmology http://www.benpadiah.com/MISC_diagrams/pages/cosmo.html

4D CONE

To measure the relativity of three points at one time, construct one three-dimensional axis for all three, then trace out the point of intersection for one random angle from each point, and construct an axis system for this point. To measure a duration, follow the same procedure for a single point at three different times. The circle of the base of this cone will always be a measure of the fourth dimension. It acts as an angle at which the three other directions meet, a plane connecting them all as a tetrahedron. For example, in a 3-sphere, this angle is confined to 45 degrees of the circumeference of the sphere for 3 right angles.

The Einstein-Hubble cosmology of the universe is a simple enough one for most people to understand. The "big bang" began with a singularity, and the universe expanded out from there. Therefore, we can look at the history of the universe as a "light cone" where we can look backward from our present universe (indicated by the eye in the diagram). Einstein proposed in general relativity that the surface of space-time was curved by mass. Edwin Hubble furthered this idea when he discovered the spectral red-shift of galaxies. The point of intersection between Einstein's theory and Hubble's observation is that the present surface of the expanding universe is spherical. This would place the "big bang" at its core (indicated by the observer's eyeline).

Stephen Hawking expanded upon these ideas in modelling his cosmology of the universe. He expressed the Einstein-Hubble light cone as bowed in shape, between the early inflation, expansion, and later inflation periods.This would indicate alterations in the rate at which the universe was expanding. He calculated that an observer looking down into a black hole would be able to see to the point of singularity, at which spacetime became infinitely distoted by gravity. His theories contributed to the search for the point of unification of the four elemental forces and the discovery of blackholes with gas jets at the center of spiral galaxies. 67

Roger Penrose worked with Stephen Hawking on developing a general cosmology based on these observations. They came to the conclusion that the best model of the lightcone might be tetrahedral, based on great arcs drawn around the circumference at 90 degree angles to one another. This compactified frame has already been proposed to account for the collapse of only certain dimensions due to the contraction of superstrings. I call this the conservation of dimension.

This is my own personal model. You may recognise it (though only very small) from the center of tau-sub-tau. Here it is enlarged and labeled. I believe it should more or less speak for itself.

Some features that all of these cosmological models share in common are depicted here. The local universe has conservation of curvature. It expands due to entropy. By moving faster than entropy, one will enter a wormhole into the future, and by moving slower than entropy, a wormhole into the past. Every blackhole leads to the same singularity at the origin point of our universe. Through the singularity, quanta are fed into the flow of entropy and collect cosmic dust in their electromagnetic fields. This cosmic dust forms a galaxy, and when two or more galaxies align, then the view from the center of each will be as though seeing forward and backward in time simultaneously.

ANGLE OF TIME

Here we see that the gas jets of a black hole extend out at different angles from the poles due to polar precession. When the gas jet's angle of arc is attracted toward the black hole at the center of another galaxy (thus forming a "baby universe" between the two galaxies) then the angle of the arc is actually much smaller than the angle of arc connecting the gas jet around toward the pole of a star in the galaxy of which the black hole is its center.

here we see that a "baby universe" (or networked connection between two or more galaxies within one overall parent universe) is formed when any of the listed conditions is met. 70

GRAVITY=TIME

This diagram represents the four stations of earth in orbit around the sun at the four stations of the sun's orbit around galactic core (labeled BH at center).The substations in between the sun stations represent the precession of the poles of the black hole's axis and it's reversal between electromagnetic south and north, being other names for positive (plus) and negative (minus). Consequently all of the polar pressions depicted for the sun and for the earth in their respective stations are determined roughly according to the arrangement in the following diagram.

Here we see that, when North equals "positive" then the northern pole of earth will be attracted toward the sun, that the northern pole of the sun will be attracted to galactic core, and the northern pole of galactic core will be facing in the same direction as toward the sun.

And here is a diagram that depicts this relationship from a sideways vantage. Here, we see the blue and green wavelengths are equivalent to gravity waves, or tachyons, which are oriented perpendicularly to one another. The green waves are a rough approximation because of the rotation of the arms of the galaxy, and the blue waves are a rough approximation because of the varying degrees of thickness of the galaxy's spiral arms. Between these two, and perpendicular to both of them, is the average field that exists between them. Depicted in red, it displays one quadrant of a full corscrew spiral down the length of wave A and B. This one quadrant represents one of the four "positive," "negative," "north" or "south" orientations of the bodies' poles in their respective stations. When the field (red average of blue and green waves) = "positive," then the sun and earth's same hemisphere will be oriented toward one another, and the sun and galactic core's poles will also be oriented toward one another in the same hemisphere. 72

In this diagram we see the precession of earth's polar orientation relative to the ecliptic (blue) zodiac. We can also extrapolate from this diagram the orientation of this (or any) combination of earth's polar precession [relative to orbital perihelion (equinox) and aphelion (solctise)] and the duration of rotation of our solar system around galactic core (green zodiac). In other words, the blue zodiac can simultaneously stand for earth's orbit around the sun and the ecliptic zodiac of our solar system, and the green zodiac represents the plane of the Milky Way galaxy seen from our solar system as we orbit within it around galactic core. So, the Z axis is perpendicular to the plane of the galaxy (G), which is perpendicular to the plane of the solar system (S), which is perpendicular to the direction of the rotation of the solar system around galactic core (t), which is perpendicular to the orientation of the poles of earth relative to those of galactic core (Z). Of course, the equator of earth is also perpendicular to all of these, although, this "prime" measure of perpendicularity obviously only occurs from time to time, and the rest of the time these measures are not in such an exact alignment. In any event, this model describes in its motion the way to symbolically predict all net astrological alignments possible involving gross galactic astronomy.

Solar Precession Relative to the Zodiac In the top diagram we see the earth (shadowed by night) rotating around the sun, clearly marked, at the center. The orbital path of the earth around the sun is depicted as the zodiac, although the ecliptic plane of the zodiac, technically, does differ somewhat from our own current equator. The poles of the earth are shown on the outer four stations, and labeled positive and negative. It should be noted that the positive and negative stations reverse depending on whether the earth is ascendant or descendant to the equator of the sun. Because the planet's orbit is reckoned to be more elliptical than oblate, the red and green shaded ratios on the diagram represent the porportions of Kepler's Third Law of motion by which to guage the duration of time taken, and thereby the speed, of the planet in one of said positions. The time it takes for the orbital plane of earth around the sun to precess from the plus to the minus electromagnetic polarity is approximately 26,000 years, or about twice the time it takes for the geographic pole to precess around in one full rotation. The reason for this will be explained in the following images. At the bottom of the page is another diagram, however what it depicts is clearly labeled.

Topology http://www.benpadiah.com/MISC_diagrams/pages/topology.html

the 3-star The 3-star is basically the intersection of 3 perpendicular planes.

Begin by taking one plane, adding another at right angles, and then a third at right angles to both. The 3-star has many practical purposes for measuring not only 3-space shapes, but also as a basis for plotting the coordinate systems of 4-manifolds.

Here, we see the 3-star used to mount 3 perpendicular toroids, forming a gyroscope. The points of intersection of each toroid with the other three are marked A1, A2, B1, B2, C1 and C2. It should be briefly noted that the toroid CA intersects the toroid AB at A1 and A2, the toroid AB intersects the toroid BC at B1 and B2, and the toroid BC intersects the toroid CA at C1 and C2. It may be deduced then, that the covering set for each of the intersecting plane spaces can be folded around so that A1 connects to A2, B1 to B2, and C1 to C2. Doing so renders the three perpendicular toroids as shown, such that they form a gyroscope whose operation describes a hypersphere. 76

Here is a more detailed examination of the three toroids.

Modeling the fourth dimension with light and motion We can depict and thereby easily understand the three standard accepted dimensions comprising the spatial component of our continuum. We model this as the intersection at right angles of three planes, and depict it using diagrams such as the 3-star described above. However representing the fourth dimension, whether we understand it spatially or temporally, or spatially and temporally, is not quite as easy. For example, it is possible to understand four spatial shapes by depicting the possible shadows that they cast into the third dimension, and then modeling these, usually as flat diagrams drawn in the second spatial dimension. However, these 3-space shadows of 4-space objects, no matter how thoroughly we can model them in either stasis or motion, are limited as 2-spatial diagrams, as well as being limited to representing the 4-space shapes only at certain angles. For example, a cube seen from above a corner casts a hexagonal shadow on a sheet of paper, and so, too, is the 3space tesseract the shadow of only one possible position of the, otherwise much more difficult to visualise, 4-cube. Another way of modeling the fourth dimension is by examining 3-spatial shapes over time. Let us understand the temporal dimension as beiing the measure, perpendicular to the three intersecting planes of 3-space, of a 3-spatial object from itself. For example, imagine a cube sitting on my desk. The difference between this cube and itself can be rendered as the passage of time. Of course, in this model, we cannot see any immediate difference between one state of the cube and another from one moment to the next. So, if we want to model dynamic change over duration we must work with moving parts. Consider, instead of the cube, a gyroscope sitting on my desk. Now, as the gyroscope is set in motion and spins, there is an immediately visualisable difference between itself in one state and itself in another from one moment to the next. Hence, we can use motion as a measurement of the temporal direction, which, again, operates at a fourth right angle to the standard intersection of planes in 3-space. Now, ideally, to consider the full scope of the fourth dimension, we should combine the shadow effect of one dimension being visualisably diagramable on the next dimension beneath it, AND the measurement of motion as being perpendicular to the 3-spatial matter upon which it acts. So, for example, let us consider the following two diagrams. 77

The first measures several different directions of motion on the model of the gyroscope. Here we see that the three planes intersecting at right angles may be depicted as the axes of three great arcs. Each great arc has its own orientation of motion, and the momentum caused by the motions of the combined three great arcs causes further dynamic rotation to occur. In the case of this diagram, that motion is depicted by the insertion of a 2-space axis at 45Â° between the perpendicularly oriented axes of the three great arcs. As the great arcs revolve around the origin point of these conjoined axes, their motion creates the momentum that drives the rotation of the 2space axis. The 3-arcs revolve and the 2-axis rotates. To add even one more step of motion to the model, I have pinioned the entire gyroscopic contraption upon a stable pivot, such that the 3-arcs revolve, causing the 2-axis to rotate, causing the whole gyroscope to revolve around the pivot point. (for reference, the revolution of the great arcs is depicted by the three colour arrows, the rotation of the 2-axis by the solid black semi-circular arrows, and the revolution of the contraption around the pivot by the black outlined white arrows.) This diagram maps the confluence of nine directions of motion, and if the pivot point had been rendered such that, rather than being stable, it followed a precessional concourse (imagine the pivot-beam tracing out a circle around the base of its stable stand), then the diagram would model motion in at least ten different directions. Of course, it is equally easy to continue adding motion in as many different directions as possible, however for the purpose of this exposition we needn't worry about the implications of doing so. 78

The second diagram depicts the same essential model, that being the gyroscope, as well as the shadows that it casts in five of the six cardinal directions. It should become evident upon consideration of this diagram that only three of these shadows are actually necessary for a thorough examination of light's effects on the model, the other three shadows merely being equal and opposite mirror images of the first three. For the sake of brevity the motions of these shadows are not depicted, however it is hoped that the viewer of this diagram should be able to piece these together for themselves.

There are four basic stations of motion for the tripartite gyrsoscope. These represent different spin states for the 3 great arcs of the gyroscope. The right hand (Rh) and left hand (Lh) spin states are simple spinors with only one direction for all three great arcs. These create simple vortecis in a fluid medium, where the rotation of the spin of the gyroscope drives the liquid substance into a simple vortex in the center of the gyroscope. The other stabe positions of the gyroscope involve either one or two of the great arcs moving in an opposite direction from the other(s). These, depicted on the left and right of the diagram, orient themselves with positive and negative positions where the fluid medium disrupted by the motion of the gyroscope will be drawn toward the positive positions and away from the negative positions. In the model on the left, the positive poles are positioned relative to the static horizontal great arc, while in the model on the right, the positive poles are positioned relative to the static position of the greater vertical arc. Because there are dual polar positions for the other models, we can call them scalar rather than spinor vector motions.

some topology

In this figure we see 4 coordinate system plane spaces tangent to a toridal 4-manifold. Intersecting the origin points of the four coordinate system plane spaces are four vertices. These four vertices represent the magnitude (inward arrows) and the direction (outward arrows) of four resultant vectors (labeled A, B, C, and D). While the depicted coordinate system plane spaces are all parallel, three of the vectors (A, C and D) occur at right angles to them, while one of the vectors (B) occurs parellel to its vertex's plane space coordinate system. These four angles define the 4-dimensionality of the toroid manifold. The 4-dimensionality of the toroid manifold is further depicted by its motion: the interior circle operates clockwise, while the exterior circle operates counterclockwise. This form of motion is callled "counter-rotation." The motion on the surface of the torus "involutes" from around the outside toward the hole at the center. The involution of the toroid's surface causes the magnitude of the inward vectors to reflect off the toroid surface along an outward direction. The resultants of the direction/magnitude of the vectors are all perpendicular to the toroid surface, and since they all operate at right angles to one another as well as to the surface of the torus, the four resultant vectors measure the 4dimensionality of the torus manifold.

In this figure we see the four coordinate system plane spaces conjoined as an interior and exterior face of two cubes conjoined in a 4-manifold tesseract. An axis has been added that penetrates the torus between points P and Q and points S and R.

In this figure we see the phase portrait of the torus mapped onto a single plane space. The points P and Q and points S and R are the same as in the last diagram.

Here is the standard measure of how a torus is mapped onto a flat, two-dimensional plane. The complete plane is called the "plane space." There are two ways to express vibrations of a plane space. These are topologically and using matrix mathematics. The two are really the same, since the "neighborhood set" "spans the space" only when its "vector matrix" is "orthonormal." 82

Formal Propositions http://www.benpadiah.com/MISC_diagrams/pages/equations.html

Equations

1 Joule of Energy = 1 Kilogram of Mass traveling at a velocity of light-speed squared or 89,875,517,873,681,800 square meters per second squared (where the square of 1 second is 1 second). (E = Mc^2) Light-speed, or 299,792,458 meters per second, = the square-root of 1 Joule of Energy per each square-root of 1 Kilogram of Mass. (c = !E/M) 1 Kilogram of Mass = 1 Joule of Energy exhibiting stationary density at the square-root of lightspeed or 17,314.51581766005 square meters per second. (M = E/!c)

Metaform

IMG113

This diagram represents several basic ideas of cosmology. the red lines represent the axis between 2-time and 3-space, or, in other words, how we perceive our own universe, beneath the photic speed of light. the green lines represent the axis between 1-time and 4-space, or, in other words, the possible contents of the portion of our universe (or multiverse) that exists above the speed of photons in a vacuum. The blue line represents, essentially, the Main Sequence for the sum over histories (space/time) for our own universe, and thus, after a fashion, also represents the point of Critical Mass of its expansion. the lines labeled A are asymptotes, while those labeled B are quarter circles. the lines 1 & 2 are parellel, and are perpendicular to lines 3 & 4, which are also parallel. This may also be understood by the formula that all four lines are perpendicular to one another, and the diagrammatic shape is not a square, but a quarter-turned tetrahedron.

Time Diagrams

Super-Reimann metric tensor & static DeSitter tensor combined.

I. here we see the event horizon of space defined by the Maxwell equations for Einstein photic light to the Reiman metric tensor (the flat circle on the left). Permeating this at right angles is the sum over histories of all photic light particles (the toroidal 4-manifold). II. This picture represents the mutual attraction/repulsion of the strong and weak nuclear forces. The weak force (the flat circle on the left) is both attracted to and repelled by the strong force (the flat circle on the right). This diagram viewed from above one of the flat circles would represent the Lorenz strange attractor. This models both the gravitic force on galactic scale and on atomic scale. III. this diagram represents the same as the lowest or fourth diagram on the time_diagrams.jpg linked to above. It depicts the same force as guides the sum over histories toroid for the subphotic local universe, guides the gravitic strange attractor of galaxies and atoms and, as seen here, the curling of the hyper-spatial trans-temporal (hyper-dimensional) superstrings around the vibrational frequencies of matter-energy into Calabai-Yau shapes, of which this trefoil is a Poincare section. 88

IV. This diagram shows the one sided trefoil surface unfurled, or the extra dimensions which wrap around quantum particles opened up.This represents the intersection of three toroids (as seen in position I) forming the three right angles of our spatial dimension surrounded by a holographic extra dimension of time. These are the conditions inside the singularity of a black hole.

This is a hypersphere, the product of the overlap of four origin points emitting waves (sides emitting red, top and bottom emitting yellow), depicted as the motion or entropy of a dual-orbital, or two-dimensional, sphere, formed of two elliptical circumferences (the green and blue ellipses). In its centre is the DeSitter metric tensor where the red and yellow waves intersect within the blue and green circumferences. The black outline of the square around the circular hypersphere field is the basic model for how an electromagnetic field warps around a stationary gravity well. The idealised black square outline depicts the boundaries, or limits, of the wavelengths emitted from their midpoint perpendicularly. 89

Squaring the Circle using the Pythagorean Triangle The third step of twinning involves the "law of three." According to the "law of three," there are three physical "dimensions" (or six directions: up, down, right, left, front, back). Under the law of three there are, therefore, patterns of three in nature, such as past, present, future creating spirals over time. Therefore, the perceptions we have of these patterns will also obey the "law of three" (or six fundamental questions of reasoning: who, what, when, where, why and how). I describe more about the six fundamental questions of reasoning in my book the FSOM, so I will not go into them in greater detail here. I will talk, instead, more about the "law of three." Here are ten examples of "triplicities" (objects obeying the "law" of three), from some natural sciences, as listed in my TOD. Thelemic: light love life Hindu: sat (being) chit (mind) ananda (bliss) Hellenic Hebrew: tetragrammaton primamaton anaphaxaton Egyptian: kha ka/ba akh Hebrew: nefesh ruach neschema Buddhist: boar cock snake Hindu Gunas: tamas (stability) rajas (restlessness) satvas (orderliness) alchemical: salt sulphur mercury magical: chain scourge dagger masonic: jubela jubelo jubelum Here is how the "law" of three relates to the rule of fours, such as the four elements, or the three spatial directions over time.

According to the Zohar, for example, there are four kabbalistic "worlds." According to the reckoning of the elders in the Zohar, there must therefore be a "law" of three in each of the four "worlds." And so they established judgment, balance, a pendulum and three supernals, three for each of the four, and thus, there were ten below three in one, and twelve altogether. This was pleasing to the elders because they saw that the "three in one" of the supernal tenth was the Right Knowledge of the Pythagorean Triangle, whose ratio of lengths between its three points is 3:4:5, and whose calculation is a^2+b^2=c^2. This is a somewhat more scientific depiction of the essential topologies associated with the relative force-carrying particles for the four elemental forces of our local universe: http://www.benpadiah.com/MISC_diagrams/img534_copy.jpg More about this can be found on my site here: http://www.benpadiah.com/MISC_diagrams/pages/equations.html#RdS This was all according to the reasoning of the elders, that is the Safeds of Spain, butit was based on observations made by their ancestors, that is, to us, the ancients. Just as we now study their Zohar, the Sefardic Septamanians studied the apocryphal scriptures of the Hermetic Pythagoreans. So, Moses DeLeon came to the conclusion that the "law of three" applied to the "four worlds" because the Pythagorean theorem could be applied to all of nature. However, the four worlds themselves extend beyond only our local universe, which is simply the fourth comprised of nine subtended to the tenth comprised of three. This lowest, or material world, (Atzilut in the above diagram) was called Malkuth and was comprised of the four-elements as a shadow of the three dimensions of space and the fourth of linear time. This was hinted at in the knowledge of the ancients, who tell us that the law of seven derives from the law of three applied to the four worlds no less than does the formation of the twelve. The four worlds of Hebrew Kabbalah and the seven worlds of Hellenic Greek Hermeticism were combined as the seven planets and the twelve houses of the zodiac even long before the time of the ancients, the predecessors of our own elders in keeping with this tradtion. This is because 3+4=7 just as does 3x4=12. Therefore, we cannot necessarily say that the "law of three" the nine to which we can subtend the tenth of three, and thus the four. We cannot, therefore, even determine the difference the sixth sense of the psyche if we cannot distinguish the questions and the six physical directions.

does indeed apply to itself to form by which form the three plus one of between the five physical sense and difference between the six psychic

So, what does this all have to do with "twinning"? Has it not been said that, in order to "selfdeuplicate" we must go "three deep"? What, then is the third step of the process of selfduplication? It is simply realising for yourself, as if in an epiphany from a point of view above, that you are "twinned" with another, or with the idealogy of a group. To realise this, you must remember to "count yourself." When you "look through the eyes of another" you see through two pairs of lenses. Do not forget to count that which is doing the "looking" and "seeing" itself. Now, the law of three, in itself, is quite simple. I have demonstrated how we cannot necessarily COMPOUND the law of three. That would be radioactive. That would be a pyramid scheme. No, we cannot multiply or double the "law" of three in itself. We cannot COMPOUND it, we cannot SQUARE it, we cannot DUPLICATE the law itself. The law of three is the law of three in itself alone. There is no law of four, no law of ten, no law of twelve, thirty-six, sixty-four or seventytwo for the law of three. There is only the law of the three for the law of three. From this third point of view, there is only that which is beneath and within itself. There is only the law of three in itself. 91

There is something interesting about the law of three: all the attributes so far discussed can be applied to triangles. The triangle itself is inherently six: three angles and three lengths. This holds true even on a curved surface, such as a sphere, where the angles can each be 90 degrees (forming 1/4 of the hemisphere). Here we see the 3 and the 4 of the ancients again, resembling the (3+2)x(4+2)=3x(3x4) of the five by six zodiacal dekans. Here are the four basic fundamental types of triangles:

The equilateral trinagle's base angle is 60 degrees. The right triangle's non-right angles are both 45 degrees. The Pythagorean triangle's base angle is 30 degrees. The base angle of the isoceles triangle is twenty. Here are some special cases of these types of triangles where certain angles and certain lengths combine to form certain theorems. 92

Now, let's consider the special cases whose equations are really restatements of one another, or rather, are the same. Even though the ratios of lengths and angles are drastically different, we see equation H also applies to diagram E, that equation E also applies to diagram B, that equation I also applies to diagram H and vice versa, that the equation E holds true for the special case (3,3,9) of diagram G, etc. Let's consider a couple "special cases" in particular: diagrams A and F. We can see that equations A and F are interchangeable, and that the equation for A is actually the same as a^2+b^2=c^2, the Pythagorean theorem describing the base thirty Pythagorean triangle. Also notice the equations H and I are interchangeable between the diagrams H and I, and are a means of "factoring out" the squares of the Pythagorean theorem. All of these being triangles, and all applying to the "law of three" all these different ratios represent different patterns of harmonic vibrational frequencies that occur not only within and through the realm of the mind (brain-waves) but also permeate and fill in the realm of matter. In other words, the "third step" of twinning involves mind over matter. The first step is telepathy. The second step mind-control. The third step is telekinesis. I'll come back to that in a moment. Because all of these triangles have three angles and three lengths, any pair of three attributes can be assigned to any of these diagrams at once. For example: consider the special cases H and I. Now apply Kthr to the great angle, Chkmh to the mdeian angle, and Bnh to the least angle. Then define the seven subtended sefirot as length of the hypotenuse seven. Between Bnh and Kthr there are four (YHVH), but between Chkmh and Kthr there are only three (YHV). Again, three plus four is seven and three times four is twelve. Let's consider two special cases more. Many of you are familiar with the saying "to square the circle" or "square circles." This saying dates back to antiquity, and the measurements of Pythagoras and his contemporary geometres. Although little of this material is available to us nowadays, more of it was available to the elders before its confiscation or destruction in the interim since then. Consider this "alchemical" depiction of the "philosopher's stone" as a geomtric depiction of the "squared circle." 93

The saying "to square the circle" actually dates back to the ancient reckoning of pi as a ratio between a circles circumference and its area. Pi, estimated by the ancients as approximately 22/7, is determined by comparing the radius of a circle and the square of the transcendental number, Pi. To determine the area of the circle relative to the area of the square, you have to "square the circle." This is done, basically, like the following:

You can see many of the "special cases" of triangles involved in this diagram. This diagram basically depicts three areas: A) the greater square, B) the lesser square, C) the area is the same for both the middle square and the circle. Further examination of the difference in degrees yields the origin for the difference between the 360 degree circle and the 365.25 day year. The primary components of the "squared circle" are the 3,4,5 Pythagorean triangle and the 45/45/90 degree right triangle. Although neither of these are "special" in this case, nonetheless we can see many, now familiar, triangles hidden within. Of course, it is also common knowledge that if you graph the squares formed by these two types of triangle: the 3,4,5 Pythagorean and the 45/45/90 degree equilateral against one another, then you will form the phi/pi spiral. However more on this matter need not be said now. So, after all this, what does it mean to "self-duplicate" or to "twin"? It means to go "three deep." This means that, through learning to see through another (telepathy), to speak through another (mind-control), and finally by applying (teleknises) the law of threes (triangles) to physical objects (vibrational ratios) we can learn mind over matter. More about squaring the circle: 95

Here is the crux of the Pythagorean Triangle, conjoined at the measurement of the "Golden Mean" with a right equilateral triangle. The non-right angle measures are 45/45 and 30/60, such that between the "upper" and "lower" triangles there is a measure of fifteen degrees difference. In the above diagram all the units numbered "one" are the same measure. There are two ways to apply the "Golden Mean" of the Pythagorean triangle to determining the area of a circle. One is to measure the radius of the circle as the base, longer, horizontal leg of the triangle. The other is to measure the radius of the circle as the hypotenuse of the Pythagorean triangle. Although these are both essentially the same model, you can see that, applying the former method (radius=base leg) to the basic theorem, you will be measuring a smaller amount of space than if you measure by applying the latter method (radius=hypotenuse).

Therefore notice that the use of the "Golden Ratio" harmonic number sequence of 3:4:5 yields base 6,7,8 square circle areas. The second type of "Golden Ratio" application of the Pythagorean triangle has already been shown (at the end of my last post, above). It is the same as the one I now give here (below) in that the radius is five in both.

Here we can see that, by measuring the circle's radius as the hypotenuse of the "Golden Ratio" Pythagorean triangle, we can yield the larger areas of the base 8,10,12 squared circles. Therefore, these are the two ways to yield a "square circle" from applying the Pythagorean triangle with its "Golden Ratio" of 3:4:5 to split the fifteen degree angle difference between 45/45 fractal expansion (base twelve hypotenuse) and the 30/60 gnomonic expansion (base ten hypotenuse). These lengths for the hypotenuse/radius of the square/circle are derived by continuing to follow the gnomonic and fractal expansion rates up to base 12,13,16 areas. Here are all the fundamental measures for the Pythagroean square/circle formula for any figure up to circle radius eight.

All of this is quite a bit more useful than you might at first think of it. Consider that, using this method, you can find the area of the earth. Consider the following chart, incorporating actual distances and degrees relative to our little blue jewel.

The red line indicates the tilt of earth's rotational axis from perpendicularity to the ecliptic plane of the rest of the objects in our solar system. The green line represents the precession of the gravitational, geographical pole. The blue lines represent the current variable offset, 11.5 degrees, between the geographic and the electromagnetic poles. Here is a depiction of some of the smaller applicable scales of squared circle, particularly those from the two square (with area four) through the eight cube (area sixty-four). Note the special cases applied at certain points of intersection between the squared circles.

As you can see, by attaching an arc from each "special case" in point in the above diagram (depicting the very small square circle areas), one forms two spirals, one spiral arm whose angle of inclination (36 degrees) is determined by the angle of the Pythagorean triangle (let this spiral be called PHI), and one spiral arm whose angle of inclination (45 degrees) is determined by the other angle, of the "equiangular" triangle (let this second spiral be called PI). Here is a diagram depicting the complete two spirals for up to area 256 (square circle sixteen):

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the Pythagorean Spiral

It should be noted that arithmetic expansion occurs unilinearly, as a diaganol vector on a cartesian coordinate graph. Now, exponential expansion occurs when the exponent of the integer is increased. For example, if one takes the exponential expansion of the numberline of arithmetic expansion, such that, 1^2, 2^2, 3^2, etc... where it then forms the summed numberline, such that, 1, 4, 9, etc.... A third kind of expansion rate exists, however it has been considered more or less esoteric, because it forms a very specific inherent pattern when graphed. This kind of expansion rate is associated with the Fibonnaci sequence of numbers, such that, 1, 1, 2, 3, 5, 8, 13, etc... where the rate of expansion is determined by the addition of only the preceding two numbers to form the sum digit third in the sequence. When this sequence is graphed as points around an origin in a cartesian grid, it forms a spiral, and this spiral has been found everywhere throughout nature, from the branching patterns on plants to the proportions of the human body. This is the pattern I call, for shorthand, phi. Phi over Pi can be easily demonstrated using simple mathematics. Apply the Pythagorean theorem, thus: (for any right triangle with legs A and B and hypotenuse C) a^2+b^2=c^2.

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This can be depicted with a triangle surrounded by three squares. The base (number per side) of each square will be a sequential progression, such as 1,2,3. Thus, for example, the square of 3 plus the square of 4 equals the square of 5. As I have described, there are three forms of expansion rate in three dimensions. One is arithmetic, one is exponential (Pi) and one is the Fibonnaci sequence (phi). First let's look at the arithmetic rate as it relates to the Pythagorean theorem, and then we will see how to combine the exponential and Fibonnaci sequences using the Pythagorean theorem and thus to create a three dimensional representation of phi/pi.

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This graph shows us the arithmetic expansion rate in two dimensions. From the square of 3 units (base three) in the lower right to the base 8 square in the upper left, the basic unit of measure (the square of base one, or the one by one square, also called one squared, written 1^2) is here the same throughout. Now, let us begin to look at what happens to this arithmetic expansion rate graph when we apply the Pythagorean theorem. To do this we must first construct the following graph, complementary to the preceding graph, and then we can combine the two using the Pythagorean theorem.

Although it differs from the shape of the last one, this graph also Just as the last graph had shown us the squares from three to squares from base five to base nine. While the graph of three expansion along a forty-five degree angle, this graph, from base 5 105

expresses arithmetic expansion. eight, this graph shows us the to eight showed us arithmetic to 9, shows the same expansion

rate when it occurs along a right angle, that is, a straight vertical line. Thus, here, the arithmetic expansion rate is expressed by the growth from one square to the next by one base unit at ninety degrees from the vertical line. Now let us combine these two graphs, the one from base three to base eight along a forty-five degree angle and the other from base five to base nine along a ninety degree angle, using the Pythagorean theorem, and see what we find out for the arithmetic rate of expansion.

Here we can see the result of combining these two arithmetic rates of expansion to form a Pythagorean triangle, from which we can then derive the Pythagorean mathematical theorem of trigonometry. In the lower right corner we find the standard Pythaogrean triangle formed by the base three and four squares when we apply the base five square to them at a 54° angle. The Pythagorean triangle is thus formed from the 90, 36 and 54° angles between the base 3, 4 and 5 squares. However, beyond this we observe the 54° angle at which we have tilted the previously vertical graph and the 45° angle of the other graph diverge, and that there appears to be no further congruency between the measurements of the two sets of squares. Therefore, the results of applying the Pythagorean theorem to the arithmetic expansion rate is that it yields a discrepancy between the alignments of any of the squares formed of the original base unit along an increasing degree defined by the difference between the 54 and 45° angles. All 106

of this can be used in two-dimensions to model how artihmetic expansion occurs relative to the Pythagorean theorem. In arithmetic expansion, the size of the base unit is constant, and each interval increases the measurement by one of the same size basic units, the one by one square, or, simply, 1^2. So, next let us look at the exponential rate of expansion, and apply the same, step-by-step process as before to compare it to the arithmetic rate of expansion using the Pythagorean theorem triangle that we used to build the arithmetic expansion model, and finally we can compare the exponential model with one based on the Fibonnaci sequence, also by using the Pythagorean theorem triangle.

First, let's begin with the basic 3,4,5 square Pythagorean triangle. Here we see that each of the three sides of the triangle is comprised of a single base unit of measurement, the one square. All this means is that, if we draw a five by five square, and turn it at a 54Â° angle, it will complement the squares of 3 and 4 that use the same unit measurement. This assertion is self-evident to the arithmetic expansion, and taken for granted in most Euclidian geometry. However, for this next part we will need to consider the base unit (1^2) being alterable and relative. 107

Having looked at the standard Pythagorean theorem triangle of sides 3,4,5, and seen that, using a constant base unit, we cannot combine the 45 and 90° graphs of the arithmetic expansion rate, let us now consider how to plot a similar graph for the arithmetic (pi) expansion rate using the Pythagorean theorem triangle.

This is a series of expanding squares constructed at the angle of inclination of the Pythagorean theorem triangle (54°), instead of at the 45° angle of inclination of the arithmetic rate of expansion graph. Here we can see that, beginning with the smallest increment of Pythagorean theorem triangle larger than the 3^2+4^=5^2, or simply the 3,4,5 triangle, that is, beginning with the Pythagorean triangle whose shortest side length is 4, in the lower right corner, and then continuing up at 54° to the square in the upper left, we see there is a definite expansion from one increment to the next, and that this rate of expansion yields a larger sized square than the same number of increments on the 45° angle arithmetic expansion graph. In point of fact, the expansion rate only appears to have increased between the arithmetic and the exponential rates of expansion. In truth, it is not the size of the squares that is expanding. It is the base unit. The 1 square.

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This fact becomes more apparent when we observe the exponential rate of expansion at ninety degrees (right-angled) rather than at a 54° angle. Again, this graph is the same graph as that for artihmetic expansion wherein there was the addition to each increment of one base unit at right angles to the vertical sequence. The only difference between the artihmetic and exponential expansion rates is that, in the arithmetic expansion rate, the base unit (1^2) has a constant measurement, however, in exponential expansion, the base unit increases at a rate determined not by 45 or 90° angles, but instead by 36 and 54° angles. Therefore, just as the last graph represented exponential expansion at 54°, this graph depicts exponential expansion at 36°. 109

However, I can't stress enough that this 36째 angle exponential expansion rate graph is identical to the 90째 angle arithmetic expansion rate graph, and that the only real difference is the angle of perspective from which we, the subjective observer, are viewing the same specific shaped objects. In short, the base unit appears to increase only because of foreshortening of perspective. In other words, the one square is always the one square, but if we rotate it, we can observe various different relationships based upon it occuring at various different angles relative to it. So, let us take the 36 and the 54째 angles of exponential expansion rates and combine them using the Pythagorean theorem.

This is the result of combining the 36 and 54째 angle exponential expansion rate graphs. Here we see that, because the expansion rate occurs at the same angle as that of the Pythagorean theorem triangle, the squares formed by the exponential expansion rate perfectly align to form a series of incrementally increasing Pythagorean theorem triangles. We can say that these triangles and squares are all congruent to their smaller and larger counterparts because the only difference between them is the size of the base unit (the one square) that comprises their fundamental measurement. In this graph we see that, had we stayed with the same base unit (the one square) that we used to construct the smallest increment Pythagorean theorem triangle (the 3,4,5 in the lower right corner), then none of the areas of the rest of the squares of equivalent triangles would be exactly equal to a round integer of 1X1 base units. Again, in point of fact, the only other known Pythagorean triangle than the 3,4,5 Pythagorean triangle (aside from the multiples of the 3,4,5 triangles such as 6,8,10) that has squares of exactly even integers in the same unit base (the one square) as the 3,4,5. This other Pythagorean theorem triangle does not occur until the smallest square is equal to 693 1^2 base units, the larger square equals 2045 base units per side, and the middle sized square between them is equal to 1924 square base units.

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So, having now completed the process of contrasting the arithmetic and exponential expansion rate Pythagorean theorem triangles, let us compare the sequence of Pythagorean triangles that expands exponentially (pi) with the Fibonnaci sequence type of triangle (phi), to determine how they relate and compare. First, we take the "straight" sequence of exponentially expanding equivalent Pythagorean triangles and label it as above. The blue lines represent where to cut to separate between the increments. The red lines represent where the increments fold when they are correlated using the phi spiral. The green lines represent where the folded incremenets align. So, to complete this step, print out the above diagram, cut it into five separate pieces along the blue lines, then tape the back of one each larger increment to the front of each smaller increment following the series of numbers indicated in red (thus: tape the front, coloured side of "1f" to the back, blank side, behind the label "1b" of the next smaller increment, and tape the back, blank side behind the label "4b" to the front, coloured side of "4f" on the next larger increment), and arrange the cut and taped pieces such that they appear thus:

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Represented in red, you can see in this cut and taped together version the beginning formation of the equiangular spiral that I will compare to the accute angled spiral in the next section. The equiangular spiral is equivalent to the exponential expansion rate of Pythagorean triangles. Thus, the equiangular spiral is equivalent to pi, since pi is equivalent to the exponential expansion rate of Pythagorean triangles. However, as you can see, the letters along the alignments of green arrows do not all match each other in the cardinal quadrants. There is still one more step of topological transformation to complete before we see the final phase of this model comparing the pi exponential expansion rate with the Fibonnaci rate of expansion.

To complete the final step in creating the phi/pi model in three dimensions, and thus combining the equiangular spiral of the Pythagorean exponential expansion rate with the accute angled expansion rate associated with Fibonnaci's sequence, we only have to fold and paste the model four times, connecting the letters of each increments three green arrows with the same lettered arrows of the increments smaller and larger than it. The result of this is that now the red outlined Pythagorean (or pi) spiral is complemented by the three necessary directions (the green arrows) to form the accute, or triangular, Fibonnaci (or phi) spiral.

Infinite division of congruent similarities There are essentially two terms that pertain to the Greek concept of "wholeness" denoted by the prefix "holo." Holography refers to two-dimensional coordinate mapping, such as onto a piece of paper or film. Holography allows us to view an image from multiple perspectives, as well as being able to retain the entire image in any piece of the film paper that is cut off into a smaller piece. The other currently known of form is the hologram, or three-dimensional image capture rendering in real space and time, and not using any filmic or flat medium. 112

Essentially, my theory on these, apparently only two, forms of "wholeness" rendering is that holography is like the flat shape of a two-dimensional fractal image. There are many different types of two-dimensional or flat fractal sets, ranging from the Julia to the Mandelbrot to the Cantor and Menger. However what all of these have in common is that their central shape, their strange "attractor" (to borrow a term from chaos theory) is interiorly non-reducable. The centre on the inside of any fractal pattern is a place of infinite replication known as the Feigenbaum scaling function, whereby any finite number of smaller or larger segments can cover the "attractor" of the logistic map. In order to depict these "strange attractors" in three space, they have to be rotated and stretched such that measure a 3D slice of a 4D Quaternion fractal set as it is rotated (that is what renders an attractor "strange"). Here is a depiction of such a quarter twisted "strange attractor" extracted from the "dark space" of a two-dimensional fractal:

-source: http://commons.wikimedia.org/wiki/Image:QuaternionJuliaWP.png So, essentially, you can think of a holograph as a flat, two-dimensional image, such as the standard Mandelbrot or Julia sets of fractals, and you can associate the term hologram with a three-dimensional rendering of a quaternion rotation of the fractal set's strange attractor. Here are a brief example of each to demonstrate the essential difference:

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the moving image on the left is a fractal. The "magic eye" image on the right is a 3-d rendering of a "strange attractor." Now, I postulate their to be a third type of "implicit wholeness" theorem that can be geometrically depicted. In the same way that the holograph is the fractal edge and the hologram the strange attractor interior, so too are there a special set of fractals based on transcendental and transfinite iterations. These occur only when very certain, specific numbers are slected to form the angle and and iteration ratios that determine the edging pattern of the fractal. For example, when the number of repititions of angle replication is set to phi (1.618), then the clear pattern of a fibonnaci sequence or "Golden" spiral will appear. Another form of regularly cycling fractal pattern is based on the square root of two (the diaganol of a square with legs of one and area of one squared); we see this pattern occurs in the branching patterns of plants.

On the left we see the branching pattern of the square root of two mimicked on a computergenerated image of a fractal construction of a tree. On the right we see the phi or Fibonacci spiral branching off from a fixed iteration of the Mandelbrot attractor fractal set. 114

Just as the flat edged of the fractal and the volume of the quaternion slice represented two different forms of "holo" or "wholeness" replication, th fractal's jagged edge being the holograph and the depth of the strange attractor being the hologram, so, too, does the set of all regular patterns produced by feeding back transcendental numbers onto a fractal. The name for this third type of holography and holograms is "holognomic." Here is a depiction of how such a reguarly repeating pattern can be graphed from any fractional number sequence. Here, in this design of my own, we see the iteration of "infinitely repeating halves" is used. Such shapes as are generated by graphing transcendental numbers are called "gnomons."

The "gnomonic" or harmonic "holo" -patterns can be thought of as the result of "auto-correlation" or of making one part of the "holo" refernetial to another, such that tab A fits slot B, etc. The finest example I've seen of applying one form of "holo" pattern to another, in order to form a more averaged ratio of difference between them, ie. a gnomon, is an effect known as the "buddhabrot." The Buddhabrot design takes the mandelbrot fractal set and converts it into a refraction (or 3-d appearing) holograph. By taking one form of holograph or self-referential pattern and applying it to another it elevates the combination of both to a higher level than their combined sum alone. The difference is the measure of the holognomon. Here is the holograph of the Mandelbrot fractal, known as the "Buddhabrot."

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-original image source: http://www.superliminal.com/fractals/bbrot/bbrot_rgb_small.jpg -site context: http://www.superliminal.com/fractals/bbrot/bbrot.htm

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Number Theory Consider the following chart:

On the left we have the column of sequential integers. Let this column represent, say, time, or, say, initial members of an organisation. The column on the right represents the rate of expansion over time, or per each different original group member. The left column reads up and down, and the right column reads side to side. Here is how to read them: begin with square one, expansion rate two. Next in sequence would be to start with two, and measure the expansion rate four, etc. The chart goes up to integer five, rate of expansion thirty two, but of course you can see how applying this method works. The three middle columns show some relationships we can draw across the board between sequential integers and expansion rates. 1+1=2, 2+2=4 and 3+3=6 represent the initial integers and expansion rates. The difference between each initial integer and the next across is the corresponding row on the left column. The difference between each integer sum and the next across corresponds to the expansion rate column on the right. Basically to read across a row, you count the difference between the sums by the factor to the right. Thus: (2,4,6)=2; (32,64,96)=32. The expansion rate in the right column is the same as that for the 1+1=2 column because this entire table represents only multiples of two. 1+1=2 and 2+2=4 establish the base two system as a root function. The rest of the chart is based on multiplying each sum by two. So, in short, this chart represents the corner-stone of the Base-Two system. There are similar charts that can be constructed for all the multiples, and it will show that, for each multiplicative step by its base number system (3,4,5,6... etc.) it will increase in the same fashion: As the sequence of integers expands "arithmetically," the column on the right, the expanson rate, will increase "exponentially."

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on the tables of the eighth and the ninth The tables of the eighth and the ninth tell us many things about the physical construction of our universe on a purely mathematical level, the level which serves as the bridge between the external formation of the material world and our relatively internal domain of consciousness. My arrangement of these tables therefore allows for the application of numerology, or the realization of relationships between the digits en-soi, or in themselves. This existentialist approach to computation is as ancient as the human practice of collecting objects in countable sets, and constitutes an esoteric equivalency to the exoteric fact of pure traditional mathematics. Regardless of the excuse of its origins, this practice?s credentials are frowned upon in the light of pure mathematics as an escape from the understanding provided by methodological calculation. They are, quite to the contrary, no more than a reapplication of methodological computation on an entirely other level, self-contained and non-threatening to the approach of exoteric mathematica. Thus, while their examination may be seen as untraditional, it is at least not unacceptable. One of the fundamental insights of the tables is provided by comparison between the two. Their primary similarity is the repetition of pattern; their primary difference being the nature of these patterns. As the eighth demonstrates quantifiable decline in sequence, so the ninth yields first exact self-replication, and then increasing self-referential sequentialism. These patterns are apparent and undeniable. If the calculations are repeated in any other setting the conclusions will be exactly the same, and therefore the patterns displayed in their relationships will be identical. Numbers do not lie. They lack that motivation for symmetry. It is possible to see these sequences of both as related to one another dimensionally. The perpetual numerological decline of the eighth and the perpetual numerological duplication and factorial increase of the ninth may be seen as ascent through the first three dimensions respectively. The diminishing table of eights constitutes collapse into a singular point in space. Naturally one can question how there can be any differentiation at all in the measurement of a point, but the answer is a simple one indeed, for without it there would be no scale-correspondent maps. The repetitive aspect in the nines resembles the extension of a line in space. At all points along the line two of its dimensions are canceled, leaving only the third behind to mark its position. This is quite obvious in the graphing of a straight vertical or horizontal line in a two dimensional Cartesian coordinate system, where either the x or the y coordinate pairing remains undefined. It may be less obvious in a diagonal, where every point on the line has a defined x and y coordinate pair that differs from every other coordinate pairing of a point along the line. However the distance formula shows that any two points on a graphed straight diagonal will cancel one another out leaving only one integer behind, that being the value of the line itself. This may seem trivial now, but it is essential for understanding the next comparison of the ninth table to dimensionality. One need only to consider that a line in a two dimensional coordinate system, straight or diagonal, is equivalent to a plane in a three dimensional coordinate system to begin to apprehend why. Although it is a confirmable fact that the integer sums of the multiplicative quantities produce a doubling of results along factorially related lines throughout the entire ninth table, this only becomes really evident after the process has entered its third repetition, when the quantities involved are of such an amount that their sums render divergent factors. What this process is in fact describing is the event of entrance into the third dimension from the second. At first, as the plane is defined as two lines of value nine, the factorial sums elevate rapidly through the established sequence. As the shape becomes clearer while the coordinate system is rotated the 119

number of quantities between the factorial transitions becomes greater, allowing, as it were, more time to pass between phase shifts. The key to understanding the shape that is described is contained within the non-numerological pattern of table nine. The factorials diverge first after product 180. That is, their sums if taken by pure integer alone or by combined integer render different results, although both results that occur elsewhere as quantities within the initial multiplicative table. At quantity 108 the products break from the ascending sequence and linger at sum 18, or, if they are taken as a pair of paired integers, begin to decrease along the same factorial lines, beginning at 63, but skipping every other factor, such that the next result yielded is 45, rather than 54, etc. This is the case until 180 appears again as the multiplicative function. The result for 198, that immediately following 180, is peculiar, as it constitutes a different rate of change than has been previously established. Although it describes the results for all the quantities of multiplication between 181 and 240 (interrupted only by 200 * 9 = 1800), this still describes a much shorter set of numbers than are contained within the 36 factorial grouping from 241 to 390 which follows, 58 as opposed to 148 respectively. This anomaly also helps to point out that the phase shifts in sums don?t occur cleanly at factorially defined breaks, but are governed only by the dictates of the digits themselves. Although this opens up the realization of still another, more subtle pattern â&#x20AC;š the difference between the clean factorial 243 and the true break of 240 for the 27 factorial set being 3, and the difference between the clean factorial 396 and the true break of 390 for the 36 factorial set being 6 â&#x20AC;š it is unnecessary at this point to go into it in detail. It is more instructional, from a dimensional emergence perspective, to examine the factorial breaks that are clean, and those are 180 and 360. These numbers are most immediately recognizable as the definitions of the circumference in degrees of a half circle (the angle measure of a straight line) and a full circle (or, if you like, the angle measure of a straight line that reverses its own direction). But they have more, deeper connotations than this. The most fundamental polygons are the triangle, the square, and the pentagon, composed of 3, 4, and 5 angles respectively. The hexagon, with 6 interior angles, is somewhat more complex, and can actually be tallied to be the sum of 2 triangles. The sum of the angles of a triangle is 180. It is always, exactly, and only 180. The sum of the angles of a square is always 360 (90 + 90 + 90 + 90). This is true for any 4 sided rhombus, according to the formula that (n - 2)180 = sum of the interior angles for any object with n number of sides. Applying this same formula we can conclude that the sum of the angles of a pentagon is 540, or 3 * 180, 1 & 1/2 times around the circumference of the unit circle. These numbers are as old as the act of measurement itself, and they are absolute. So we can see how the clean fractional breaks in the ninth table pertain not only to angles that describe arcs, but to those describing well defined shapes, particularly the fundamental polygons, as well. Moreover these fundamental polygons comprise the sides of the only five regular solid polygons that can exist in three dimensions. All other solid polygons, like the hexagon in two dimensions, are only combinations of these first five. The Platonic solids consist of the tetrahedron, comprised of triangles, the cube, composed of squares, the dodecahedron, comprised also of triangles, the icosahedron, also composed of triangles, and the dodecahedron, comprised of pentagons. So we have not only the description of angle measures on the unit circle, but also the sums of the angles of the faces on each of the five platonic solids, and all in a pure numerical form. It is easy to follow the progression of factorial breaks up through their ascending sequence and see how the sums constitute measures within a deepening three dimensional space, describing the unfolding of the platonic solids according to the sums of the interior angle measures of their faces multiplied by their facial sum.

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It would be more probable, at this point, for a practitioner of calculatory mathematica to caution that the bad habits of a writer may be transferred to their readers, than it would be that they could provide hard evidence that this data is more conjecture than implicit organization. In either event it cannot be doubted that the dimensional bridge between function and form is crossed in the table of nines. The bridge described herein is not altogether complicated, but is increasingly complex as more governing rules are discovered to determine each additional dimension. For example, the point may be seen as of any size under magnification; the line as definitive of angle, and implying the edge of a perfectly flat plane. Three dimensional objects are formed simultaneously of matter and energy â&#x20AC;š comprised not only of charged particles, but of waves with measurable frequency related to this charge. This relativity of fundamental components and distance becomes substantial in the fourth dimension, the final to be considered here, where an object has form in both space and time, according to the measurement of intervals. All of these are descriptive of the same process, differing only in the complexity of dimension. To understand the relationship between the table of the nines and the fourth dimension, it is necessary to lay a minimal foundation. Some formulae and models describing progression should be given, as it is by progression that time is measured. It is this process that has thus far been described, and which constitutes our bridge. Growth in biological organisms is measurable according to an exponential law for equiangular spirals that gradually approach ratio based orientation. An example of this is that governing the formation of a nautilus shell, given as r = ae kq , where r is the radius of curvature, a the area, e the natural number (2.71) found in exponential balances, k the kinetic energy, and q the polar angle of predicted curve continuation. Although it seems completely unfamiliar, this formula has been underlying our progress all along. It describes the function of the ninth table whereby the lower dimensions are evoked rapidly, and the forms of the higher dimensions more slowly. What this formula means is that the spiral of the nautilus shell is curled more tightly around its origin, rapidly forming a circularly bound core. As it continues, the arc of the shell expands to break its circular condition, and rate of growth slows. This is the same exact process as described in the numbers of the ninth table. Now, what does this similarity have to do with the fourth spatial dimension? If the rate of growth of either a nautilus shell or the ninth table?s factorial breaks were plotted as a sine wave, it would have high initial frequency and minimal oscillation, followed by lower gradual frequency and more pronounced oscillation. This chart describes the unfoldment of progression for either equally well. It is, itself, also a three dimensional shape. If it were plotted in a three coordinate system the sine wave would orbit about the x-axis, alternating positive and negative in the y-axis, with an expanding radius in the z-axis. If it were displayed with its compliment as well, numerically negative products of all the multiplicatives or geometrically the cosine wave, it would take the familiar form of the double helix of DNA. Before we go into the implications of the gnomon described in the ninth table, let us first pause briefly and consider the construction of a sinusoidal wave. In trigonometry, the study of triangles, lies the basis of wave measurement. A wavelength, l, may be measured as two right triangles extending up to the endpoints from the base line of the standing wave, sharing either a common point, where the wave form crosses the base line, or the vertical leg, where the wave form reaches its peak or trough. These triangles are equivalent for a sine wave depicting circularly bounded progression, and isosceles â&#x20AC;š possessing two equal length legs apiece. The method of creating the sine wave itself comes from a technique for deriving a right triangle?s non-right angle measurements from the measures of its legs and hypotenuse. The sine function for an angle is the opposite leg over the hypotenuse; the cosine function for the same angle being the adjacent leg divided by the hypotenuse. These two functions, as it was stated before, are complimentary. 121

This is expressed in the relationships between sine and cosine for a triangle ABC whose right angle is C: sin A = cos B and cos A = sin B. When the numerical solutions of these functions are graphed, a sine wave appears, which is a spiral in three dimensions. As a measurement of interval, this is also a fourth dimensional shape. Now we may consider the progress through dimensions of the described spiral as it breaks from circular boundary conditions and what form it takes after it does so. The first point that should be made is the distinction between degrees and radians. Both are measurements of the arc of a circle, or the radial expansion of a spiral bound by circular or exponential conditions, but the former is fixed and the latter is open. Radians, unlike degrees, are dependent upon the transcendental measurement of pi, an irrational integer whose value is roughly 22/7 or 3.1415926. Pi, or p, is an expression for the ratio of the circumference of a circle to its diameter. The relationship between degrees and radians is such that each pradian = 180°. It is interesting to note that a spiral column of tetrahedrons, such as Buckminster Fuller modeled for the geometry of a double helix, undergoes 1/3 full rotation while 22 of its faces are exposed. Thus p is the limiting factor for the spiral in its early stages of progression, as it crosses the threshold of the first two dimensions. As it enters into the third, the forms it describes become exponentially complex, and the time interval elapsed between factorial breaks therefore begins to widen. The limiting factor remains a transcendental number, an irrational integer similar to p, but one that describes an open set for growth along dimensional lines. This integer is phi, f, and will be shown to occur in pure mathematics, quantum mechanics, and the helialical pattern of DNA. It is this number that governs the third dimensional forms derived from the ninth table, as well as the fourth dimensional pattern of their progress. In order to decant f from the table of nines we need only convert the products given to degree measurements, a process already implied by the clean factorial breaks occurring at the 180 and 360 multiplicatives. Once this is done a chart for the sine and cosine functions of these angles may be assembled, and, using larger, later occurring sums for these, f will be revealed. The (2 sin)2 and (2 cos)2 sums are preferred, and yield results containing both phi and phi prime, f1, that is the reciprocal of phi, or f/1. Some of the most notable results of this table are, for 9° or p/20 radians: 2 - *(f + 2), 2 + *(f +2), sin and cosine functions respectively; for 18°, p/10: f1 + 1, f + 2; for 45°, p/4: f + f1 for both; for 72°, 2p/5: f + 2, f1 + 1; and for 81°, 9p/20: 2 + *(f + 2), 2 - *(f + 2). The complimentarity of the sin and cosine functions is readily apparent here. These figures represent distance relationships between points on what is called the Golden Triangle of 36° by 72° by 72°, after f, which is itself the Golden Ratio of (1 + *5)/2, with approximated value of 1.61803. Some other properties of note possessed by f are that multiplied by its reciprocal, the value of which is -0.61803, its product is -1, and that squared it is equal to itself plus 1. The spiral described by f is thus the so-called Golden Spiral, and is exponentially bounded rather than circularly, its points being f, f2, f3, f4, f5, and so on. These constitute the bridging of dimensional gaps described by the tables of eights and nines, and the slower, more numerous three dimensional forms that arise by angle sum recurrence during the later stages. It is interesting to note that a line connecting two points on opposite sides of a third point in a pentagon forms a line that, intersected by another such line drawn from the third point, forms the Golden division. Thus, not only is f determinant of the rate of third and fourth dimensional progression, but a fundamental building block in the Platonic forms themselves.

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Basic Metaphysics a lecture by: Jonathan Barlow Gee part 1: the Torus

To understand the shape and properties of 4-space manifolds, we begin with an examination of this familiar diagram, displaying the standard, elliptical orbit of a body in space around a dualfocal point. The labels are as follows, clockwise from left to right: the sphere in orbit is labeled "D," the orbit itself is labeled "M," and the dual focal points are labeled "ST" and "TS." The meanings of these labels is adumbrated on at length in my book, "The Metaphysician's Desk Reference," however their significance to this model is unimportant here, considering only the form of the model itself.

The next form of this model shows us the orbital plane from along the edge, from which vantage we may see a further twist in the shape of the orbit, such that it resembles a sideways figure-8, the symbol of infinity. Here the labels remain the same as before: "D" signifies now two positions of the orb in its orbit around the dual foci, labeled here also "ST" and "TS." The twisting orbit is labeled "M." In this form of the model we may see that, due to the optical illusion formed by looking at the usual elliptical loop from along the edge, two points on opposite sides of the ellipse appear to converge in the center between the twin focal points.

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In the third iteration of this model, we symbolise this super-position of opposite points on the original ellipse by inserting a third position for "D," the orbiting sphere. The label and shape of the orbit, "M" remains a sideways figure-8, however now we see that this super-position point of illusory intersection can be symbolised by a third position of "D," the body in space, that actually signifies two fore-shortened and visually overlapping points in the elliptical orbit, "M," from the original diagram. Thus, we may now also see that this third, super-position point for "D" along "M" has an equator co-circumferential to the circle originally signifying the dual focal points, such that this super-imposed mid-point position for "D" sits within the center between the focal points "ST" and "TS."

In this next form of the original model, we further elaborate on the super-positioned mid-point of the orbiting sphere, "D," by porportioning upon it the usual depiction for the effect on its poles of "precession," labeled "P." The semblance of this addition shows that an upward pointed cone tapers asymptotically in an arced surface from a circular base below the mid-point of "D," the orb, while another, identical cone points downard from above, such that the tips of the two arced cones intersect in the center of the mid-point of "D."

In this form of the model, we may see the co-circumferential equatorial circles signifying the conical angle of the orb, "D's," polar precession are perpendicular on the left and right sides of the twisted figure-8 orbital ellipse, "M," to the position of them relative to the central overlapping position of the sphere, "D," in its orbit. In short, we see the mid-point position of "D" operates at a right-angle to the orientation of "D" on the left and right. Thus, as I attempt to prove in my book, the MPDR, the orbit of the precessiing poles of "D" forms the exterior, and the sideways figure-8 angle of the ellipse form the interior, of a standard 4-space torus. What we are seeing in this final form of the original model is one half of a torus. 124

In this complete model of a torus, we see the manner the 4-space torus evolves from the 1-space singularity. First, the point expands into a line, shown here operating at an angle along a vertical axis in purple. This line is then rotated at this angle around its signularity origin-point to form a plane, which is shown as the red spiral in the middle signifying polar-precession. This red spiral line traces the plane-space surface of the line's rotation around its mid-point as a wavelength, shown in yellow, signifying the equator of the torus. The red spiral line on this plane surface also rotates around itself, and this is shown expanded as the blue spirals on the left and right connected around the toroid equator as a green coil. This gives us all the motions that a point on a torus can move along.

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The essential concept of the torus is that is a 4-space expansion of the 1-space singularity in 4 directions. The resulting shape is essentially as apears here and is called a torus or "hypersphere." The usual nomenclature distinction between these terms is that a "torus" is a "hypersphere" seen from along its equatorial side, while a "hypersphere" is a "torus" seen from above its polar axis. In this diagram, again from my book the MPDR, we may see that a toroid equator surrounds the "nested" hypersphere. The concept of the torus or hypersphere is that it is a sphere within a sphere, where both spheres have the same volume, symbolising a single sphere moving in the invisible direction of time.

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The torus, as we see here from above one pole on the left and from above the opposite pole on the right, obeys the laws of "wave mechanics" and in turn determines the wavelength motion of spherical particles. We see the three directions of possible motion that a point on a torus can travel in as a large blue arrow, around the toroid's exterior circumference, as a series of small green arrows signifying particle rotation inside the toroid, either in the inner or outer hypersphere, and as a red wavelength measuring a spiral line drawn along the plane surface of the toroid equator. To return to the complete toroid form of the original model from the MPDR, we see the combination of the motions within the inner-sphere and upon the exterior-sphere cause the polar precession of a point as it moves along all these possible directions over time. As a point moves along an elliptical orbit, the poles of the point precess such that over time they reverse orientation, first at a right-angle, then to 180Â°'s opposite from their original orientation, and back again. This perpetual cycle forms the overall model of a 4-space torus (from the side) or hypersphere (from above). Because this depiction itself is flat, we may see it as a shadow of this higher dimensional shape cast onto a 2-d plane space. Because a torus is 4-d, it also casts a 3-d shadow. The shadow of the hypersphere is the simple 3d orb, but the shadow of the torus, a hypersphere seen from the side, is shaped like a circularly self-connected tube. This tube has a circular circumference, however if you were to trace the motion of a single point on the exterior surface of the toroid circumference you would follow it as it formed a phi spiral.

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If you were to take the phi spiral formed on the exterior surface of a torus as it revolves inward on itself, and combine it with the pi spiral motion of a point on the interior of the torus as it revolves around in a circular tube, the result would be this diagram, showing the shadows of these two types of spiral (the exponential phi and the arithmetic pi) on a flat plane space with their twin origin points exactly overlapped. The significance of combining these two spirals as flat shadows in this way is to depict the point where the interior of the torus becomes the exterior of the torus as a perameter where both spirals (inner pi and outer phi) coincide. For short-hand I refer to this pair of spirals throughout all my writings on metaphysics as a single "phi over pi spiral."

When we double such an already combined "phi / pi" spiral with an exact duplicate of itself in mirror reflection by overlapping both centroids onto a single origin-point, we arrive at this depiction, which is best thought of as a shadow cast by the motions of a point on a hypersphere seen from above (or below) one of its poles.

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When we double the "phi /pi" spirals at a point along their axis line, but not exactly overlapping one another on a conjoined origin-point, we arrive at this model, best seen as a shadow-depiction of point-motion along the toroid edge, or the hyperpshere seen from its side.

If we flatten the motion of a point on a torus' surface into a plane, the result is this "autocorrelated" mapping, the so-called "7-colour coding pattern" of the surface of a tours. When the space labeled 1 is connected to the space labeled 7 inside a coil formed by the spaces between 1 through 7, we see yet another form of the torus, or hypersphere seen from along its equator. This 7-colour coding maps onto the surface of the round tube-torus such that it forms the phi spiral upon it.

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part 2: the Tesseract To begin our examination of the 4-space manifolds based on geometric polygons besides the simple circle expanded from the singular origin-point down the center of a line, we look first toward the tesseract or "hypercube." The 3-space shadow of the 4-space hypercube is the regular cube with six, square-shaped sides, 12 edge-lines, and 8 corners where 3 edge-lines connect 3 square-faces all at 90Â° angles. The standard terms of measurement for the dimensions implied by these 3 edge-intersecting corners oriented at right angles to one another are "length (or distance), bredth (or width), and depth (or volume)." The hypercube is a symbolic depiction of a single cube changing over time, and is represented by two cubes overlapping.

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The standard form of the "hypercube" is as one cube "nested" within another cube. This depiction is an optical illusion, symbolising the true shape of a "hypercube," however is, itself, merely a flat, 2-d shadow of the true form of such. A real 4-space hypercube is comprised of twin cubes, both of equal volume, while in this standard, symbolic depiction in 2-d of the shape of the 4-d manifold, the inner-cube appears to have smaller volume than the larger cube that it is "nested" within.

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Because of the intersection of the 3 vertices at the origin of "length, width and depth" there are 3 shadows in 2-space that can be cast to show the true form of the 4-space hypercube. The first was the "nested" hypercube seen from above one the mid-point of one of its sides. The second is this combination, where one cube sits on top of a second of exactly equal area, which signifies the shadow cast from a hypercube when viewed above the mid-point of one of its linear edges. This format is called the hypercube's antipode position.

The third form of 2-d shadow cast by the 4-d hypercube is the tesseract. The outline in flat 2space of the tesseract is octogonal, or 8-sided and 8-pointed. This outline contains an arrangement of horizontal, vertical and diaganol lines that forms, at the center of the shape, a smaller star-pattern called an "octogram." The octogon and octagram are, to the 4-d hypercube, the equivalent to, in 2-space, the line and dot. The octagram contained within the octogon itself also contains an octagon, and this pattern can be self-replicated on smaller and larger scales infinitely because it is a gnomon, a form of parallelogram similar to the more organic patterns of fractals. 134

The apparently octogonal shadow cast by the 4-space tesseract is also an optical illusion that symbolises the dual-cube arrangement. In this depiction we see that one cube (shown in red) is oriented at a 45Â° angle to the other cube (shown in blue). As each corner of one cube shifts along this angle to connect to the same corner of the other cube, it forms the lines inside the octogon that define the 2-d depiction of the tesseract. The pattern depicted in this flat, 2-d shape is merely the shadow of a hypercube seen from above one of its corners in 4-space.

Because the 4-d hypercube spans as a measure of distance the change we consider a temporal duration by using the motif of two cubes of the same volume, it changes the shape of the shadow it casts in 3-d by motion, just as in 2-space its shadow is determined by orientation to the hypercube of the pont of view. The hypercube's shadow in 3-space thus appears to transform endlessly just as, in 2-space, it had exactly 3 shadows cast from different points of view. While the "nested, antipode and tesseract" hypercube patterns in 2-d reflect the "faces, edges and corners" of the 3-d shape's "length, width and depth" form in 4-d, the true form of the tesseract expresses the 4th dimensional direction, or change over time, as motion in the orientation between two equal volumed cubes.

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Here is a depiction of the motion of the tesseract through itself, as one cube trades places with the other by both passing through one another. Here is another simulation of a hypercube to demonstrate the change in size between the two cubes as they pass through one another effects motion using a sphere moving forward in a line to signify the "standard arrow" of entropy, ie. time. Because the hypercube is a 4-space manifold object, it is not visually depictable without using motion. It is comomnly known as having countless 2-d visual depictions describing shadows cast by the hypercube when it is seen from various angles. These sorts of 2-d pictures hint at further applications of studying the tesseract's shadows by applying them to a 3-d model as directions of motion. When we use this method we may see that there are 4 axes of the 3-d shadow of the 4space cube. The first 2-space shadow we follow as a direction of motion along an axis in the 3-d shadow of the 4-d hypercube shows the single cube rising vertically. Next we see the hypercube's motion along the antipode angle, where it elongates to the length of the double-cube as its mid-point rising up the vertical axis. The next form of spatial shadow we see cast within the 3-space model is a triangle morphing along the horizontal axis. The 4th iteration of motion inside the 3-d hypercube's shadow is also along the horizontal axis. This "morphing" process is called a "slice" that cuts from corner to corner of the 3-d shadow of a tesseract. At one corner it begins as a tetrahedron, expands in its mid-point into an octahedron, and then collapses again to a tetrahedron at the opposite corner. 136

The presence of the dual tetrahedrons at opposite corners of the tesseract "slice" is significant because it implies the presence of another form of "hypershape," the hyper-tetrahedron, shown here as one tetrahedron of smaller volume "nested" inside another of larger volume, signifying the higher dimensional equation of their volumes as change along the invisible axis of time. The same "nesting" of the hypercube shows it from above one side, and is one of 3 such axial shadows in 2-d cast from the hypercube because the cube connects 3-edged sides at each corner.

A tetrahedron has a total of only 4 sides, with each corner connecting three edged faces like the cube. The "nested" tetrahedrons of different volumes show the 4-d hyper-tetrahedron's 3-d shadow from above one of its sides. A "stelloctahedron" is the co-origin point "nesting" of two equal volume tetrahedrons. It is the equivalent for the hyper-tetrahedron of the equal volume pair of cubes at antipode for the hypercube. It shows the hyper-tetrahedron from above one of its edges. 137

The final form of shadow cast by the hyper-tetrahedron explains why it appears within the "slice" from corner to corner of the hypercube. Because a stelloctahedron can be formed between 8 equi-distant corners, it can also be "nested" within a 3-d cube. When this is done and the shadow of the form is cast onto a 2-d plane space, the shape it assumes is this, a unicrursal hexagram formed of two pentagrams, one upright above and one inverse below, surrounded by a hexagon. The exterior circumference of the hexagon is formed from the shadow cast by the cube. When this is removed the interior lines that remain connecting the 6 points are the shadow in 2-d of the stelloctahedron in 3-d. This signifies the cubical hyper-tetrahedron's shadow when cast from a point of view above one of its corners. This is also why it appears in the diaganol "slice" from corner to corner in the hypercube.

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part 3: the Hypercross

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The tesseract shape in 2-d signifies the motion of a 3-d cube along the 4-d axis of time. The "nested" hypercube shape in 2-d signifies the stationary mid-point of this motion as the shadow of a 3-d model where the two cubes are not of equal volume. At antipode point, seen from above one of the cube's edges, rather than sides as in the "nested" position or corners as in the tesseract pattern, there appear to be twin cubes of equal volume, one above the other below. This final form was known to ancient metaphysicians, who designed a shape to signify the antipode hypercube in flat, 2-d space using two hexagons overlapping such that the upper hexagon's lowest corner overlaps the mid-point of the lower hexagon, and the lower hexagon's upper corner overlaps the mid-point of the upper hexagon. This shape signifies the antipode hypercube seen from above a corner as a 3-d model, then recast as a shadow onto a 2-d plane space also from a point of view directly above one corner of the 3-d model. They called this model "ha QBLH" meaning literally, the "body of God."

The modern form of this ancient lattice pattern shows the same essential shape with one significant discrepancy: the "middle-pillar" of "sefirot" points on the vertical axis have "slipped down" such that now the lowest point is sub-tended below the lower hexagon. This symbolises, for the hypercube, the same "unfolding" idea as may be applied to tesselating the sides of a 3-space cube. 140

Here we see how the 6 sides of the standard regular cube "unfold" into a single plane-space pattern as a tesselation of the 6 square-shaped sides. The resulting shape is commonly called the "calvary cross" pattern, such that 4 of the square sides surround a 5th, while below the lowest of these is subtended the 6th square side. When a hypercube is "unfolded" into a 3-d tesselated pattern the result is called a "hypercross." Because the shadow cast by the "nested" hypercube is the same in plane-space as that of a 3-space cube seen from directly above one side, the same effect should be expected to occur for the shape of the "hypercross," such that it would cast a 3-space shadow of multiple cubes, but in 2-space assume the same form as the "unfolded" cube, that being the common "calvary cross."

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It is also believed that when a hypercube is "unfolded" into a 3-space shape that the form it would take would be comprised of 8 cubes, one for each corner of the standard 3-space regular cube. This reasoning yields a 3-d version of a "calvary" pattern hypercross where each of the 6 sides of a central cube expand into 6 cubes added to the original 7th, wherein the 8th cube is subtended to the lowest cube. Just as the 2-d "calvary cross" pattern is not a single source shadow, but shows the tesselation of all 6 square shaped sides of the 3-d cube, the 2-space shadow of the morphing tesseract appears the same as that of a rotating cube, and likewise the shadow in 2-space of the 3-d "calvary" pattern model of a hypercross should be expected to show the regular tesselation of 6 squares of the flat 2-space "calvary" cross motif. However the shadow in 2-space of the 3-d model of a "calvary" version of a 4-d hypercross does not show the same shape as the regular 2-d "calvary" cross of 6 squares. Instead the shadow in 2-space shows up as a simple 5-square cross, where 4 squares of equal area surround a central 5th. Thus, the actual shape of the hypercross is not a "calvary" pattern with a subtended 8th cube below the lowest cube of 6 surrounding a central 7th on all its square sides. It is actually, as shown here as a 2-d depiction of a 3-d model, only the 6 cubes surrounding the central 7th. However there does exist an extra, 8th cube. Only in this form, seen from above the hypercross' corner as opposed to along its edge as in the "calvary" arrangement, the 8th cube is hidden in plain site in the spaces between the 6 cubes surrounding the central 7th.

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The 8th cube appears as a frame-work shadow in 2-space between the shadows of the 6 cubes and the shadow of the central 7th in the form of a Roger Penrose, or "impossible," cube, so-called for being an optical illusion in which the edges of a regular 3-d cube are solid, but appear such that the background legs overlap those usually in the front. When such an "impossible" optical illusion shape as the Penrose cube casts a shadow in 2-space, it is the same shadow as that which would usually be cast by the shape of a regular 3-d cube. Just as we saw the antipode cubes as paired hexagons in the diagram of "ha QBLH," so in the 2-d shadow of the 3-d model of the 4-d hypercross we see a hexagon shape cast from above the corner of an "impossible" cube.

part 4: Hypershapes

Aside from the singularity or sphere, whose hypershape is the "nested" hypersphere or torus, there are 5 regular polygonal solid forms in 3-space that may have conterpart hypershapes in higher dimensional "hyper-space." These 5 regular solids are commonly called the "Platonic" solids, and they include the cube (of 12 edges, 8 corners and 6 square shaped sides), the tetrahedron (of 6 edges, 4 corners and 4 triangular sides), the octahedron (of 8 edges, 7 corners and 8 triangular sides), the isocahedron (of 15 edges, 12 corners and 20 triangular sides), and finally the dodecahedron (of 35 edges, 20 corners and 12 pentagonal sides).

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The secret of understanding the hypershape forms of these 5 regular 3-d solids lies with the additional truncated or "snub" versions of these original 5, which are commonly called the 16 "Archimedian" solids. These 16 mixed polygonal solids comprise the space occuring within and between these original 5 solids when they are "nested" within their own shapes or within one another.

We begin by re-examining the cube. Its 6 sides intersecting in 3-vertices per corner are the only use of the regular square shape in any of the 5 "Platonic" 3-d solids. From above one of its 8 corners, the 2-d shadow cast by the 3-cube in flat plane-space is a hexagon. Likewise, because a cube can be divided apart into a pair of intersecting tetrahedrons, a stelloctahedron (the hypershape of the tetrahedron) can be extrapolated from inside a 3-cube just as a hexagram hides inside the hexagon shadow of the cube in 2-space.

When we examine the hypershape of the tetrahedron furhter, we find that, similarly to the "impossible" 8th cube within and between the other 7 in a regular hypercross, if we were to "unfold" the hyper-tetrahedron as a flat, 2-d shadow, it would tesselate onto flat plane-space below the 3-d shape of the stelloctahedron in the shape of an "impossible" triangle hidden within and between the twin tetrahedrons. This "impossible" triangle is a 2-d optical illusion formed by the unfolding and tesselating onto a flat surface of the edges of the stelloctahedron.

The next larger form of regular 3-d solid with all equilateral triangular sides is the isocahedron. Its hypershape casts as many various forms of 3-d shadow as it has edges, corners and sides. Here we see one such depicting this as a yellow cube connecting the interior-corners of a "hyperisocahedron" formed by "nesting" one isocahedron of smaller area upside-down within another of larger area. Thus, just as the tetrahedron can be doubly nested within the regular cube, so too can a cube be singularly nested within a doubled isocahedron.

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The isocahedron can not only "nest" within itself a perfectly regular 3-space cube, formed inside a "hyper-isocahedron" that is "nested" within itself. It can also house the next higher-order shape, the dodecahedron. Here we begin to see the role played by the 16 "Archimedian" solids as occuring between these dual "nested" forms, such that between the outer isocahedron (in blue) and the inner dodecahedron (in green) we find the now familiar stelloctahedron shape (in purple). Thus, we begin to see that one level of "hyper-shape" occurs between two "nested" forms of regular solid. Likewise, the "Archemidan" solids occur as "slices" from one corner to another of these hypershapes.

The dodecahedron of 12 pentagonal shaped sides likewise can serve as a hypershape which has a smaller, backwards "nested" version of the form within a larger version of itself. However, even in this form, the "hyper-dodecahedron" can also "nest" within itself another of the 5 regular "Platonic" solid forms on the next lower level of complexity, the isocahedron. We see here the outer dodecahedron in green, the inner isocahedron in blue, and the "hyper-dodecahedron" portrayed as the stellation of an "Archiemdian" solid between them in purple. The "hyper-dodecahedron" however has 35 edges and 20 corners in addition to its 12 sides of 5 edges each. Thus, to depict these additional sums as 3-d modeled shadows of the hyperdodecahedron, we see here "nested" inside the regular 3-d dodecahedron (in blue), a cube (in green), a tetrahedron (in yellow) and an octahedron (in red). Again, the spaces between these "nested" regular solids form the "truncated" and "snub" 16 "Archemidian" solids. All "hypershapes" in 4-d can be modeled by combining these 5 basic polygonal solids in 3-space by "nesting" them within one another, and these 5 solids are in turn comprised of the only 3 flat regular polygons, the penatagon, square and triangle. In my book, the MPDR, I refer to such "hypershapes" throughout as "metaforms."

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Each original IDEA is a shard from the transfinite ideal meta-form. Every THOUGHT is attacked from all sides by countless alternative perspectives. All EMOTIONS occur simultaneously as scalar wavelengths in a planar field. POSTULATE: if the functions of "Psi" (ESP potential) are graphed onto geometric forms as diagrammatic lattices, then their sum yields multi-dimensional results. FORMULA: if "idea" is mapped onto the corner-points of a "meta-form" object, and "thought" is mapped onto the edges of same, and "emotion" onto the faces of same, then patterns form; thus: a tetrahedron = 4 ideas, 4 thoughts and 4 emotions. an octahedron = 6 ideas, 12 thoughts and 8 emotions. a cube = 8 ideas, 12 thoughts and 6 emotions. an icosahedron = 12 ideas, 30 thoughts and 20 emotions. a dodecahedron = 20 ideas, 30 thoughts and 12 emotions. these patterns also occur in non-3rd dimensional forms as well. EXPLICATION: The combination of all "actual" motions for all "meta-forms" made of "psi" energy moving amidst and often through one another amounts to the experience of this limitless energy field as our own mental egos. Our own neural networks are only sieves filtering out static and rendering more exact psi-energy, temporal ellipses in their wake. The combination of all "possible" meta-forms of psi and all their "possible" trajectories provides the "inductive" cosmological set, and "deduction" reduces this from infinitude to an apprehensible scale. The result is the perception of "psi" energy in phases within the singularity-well of "ego" as "ideas," "thoughts" and "emotions." There are various ways to define what constitutes an "ideal form" in any dimension. For example: if one begins with the "zero dimension" of one corner-point, and proceeds next to the two types of 1-dimensional extension of such a point: the straight line and the semi-circular arc. However, these lower dimensions are not usually included in the list of "ideal forms," which begins, most commonly with 2-dimensional, planar faces, in the form of the three regular polygons (triangle, square, pentagon). However, the circle, comprised of a single completed arc, can also be counted as a shape at this stage, as the sphere can be in three-dimensions, although it is usually excluded. In 3-dimensions, the "ideal forms" are the regular polyhedra: the tetrahedron (or simplex) of 4 triangles, the octahedron (or orthoplex) of 8 triangles, the cube (or hexahedron) of 6 squares, the icosahedron of 20 triangles and the dodecahedron of 12 pentagons. In 4-space these have corresponding geometrical forms as well; there are 6 "4-space" regular polytopes: the 5-cell (hyper-tetrahedron), the 8-cell (tesseract), the 16-cell (hyper-octahedron), the 24-cell (selfdual), the 120-cell (4-d icosahedron) and the 600-cell (4-dodecahedron). The circle and sphere also have a correspondent 4th spatial dimensional form, the torus or hypersphere. In all dimensions greater than 4, only 3 types of "meta-form" ideal shapes exist; these are extensions of patterns formed in the first three 3-d solids: the simplex (hyper-tetrahedron), the orthoplex (hyper-icosahedron) and the tesseract (hypercube). Therefore, in the 5 extra-spatial dimensions from the 5th through the 10th dimension, there are only 15 "ideal forms." To assemble the 5 "ideal forms" in 3-dimensions, and do so each one at a time, one would only need as many components (corners, edges and faces) as the largest / most complex of the solid forms - in 3-dimensions this being the dodecahedron of 20 corners, 30 edges and 12 pentagonal faces; in 4-dimensions the 600-cell of 120 corners, 720 edges, 1200 faces and 600 solid shapes. However, if you wanted to assemble all of the solids in all of the dimensions simultaneously, one 148

would require 190 components (50 corners, 90 edges and 50 faces) in 3-dimensions, and 49990 elements (773 corners, 1362 edges, 2082 faces and 773 solid "cells") in 4-dimensions. Thus, for all the "ideal forms" in both 3 and 4 spatial dimensions to be all assembled one next to the other in a line would require 5180 parts (823 corners, 1452 edges, 2132 faces and 773 solid "cell" shapes). The total of all these parts assembled into shapes along this line, however, is only 11 "ideal forms" in the 3rd and 4th dimensions, excluding the sphere and torus. To assemble the 15 "ideal forms" in the 5th through the 10th spatial dimensions, and do so each one at a time, one would only need as many components (corners, edges and faces) as the largest / most complex of the "meta-forms" - in 10 dimensions this being the hypercube of 59,048 components). However, if you wanted to assemble all of these meta-solids in all of the dimensions from 5 thru 10 simultaneously, one would require a total of 4,020 components for the "simplex" model (the "hyper-tetrahedron"), 88,446 components for the "tesseract" model (the "hypercube") and 88,446 components alike for the "orthoplex" or "cross polytope" (the "hyperoctahedron"). This means, to assemble these 15 shapes altogether would require a total of 180,912 components to complete. To assemble 1 each of the 26 "ideal forms" in the first 10 dimensions, one would therefore need 186,092 components.

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Advanced Metaphysics a lecture by: Jonathan Barlow Gee Electrons

In the Bohm model of the atom, we see here the magnetically-attractive atomic nucleus surrounded by the magnetically-repelled electron-cloud of varying orbital shell energy levels.

Of course, we know this model is greatly over-simplified from the real-world orbital paths taken by the electron in its orbital shell. We can only predict these roughly using interactions from the photo-electric effect.

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Thus, as we see in the standard Bohm model, there is a sub-strata allocated in this diagram of mine between the positive nucleus and the negative electron, and this inner-shell reflects the ability of an electron to temporarily store a photon.

First, let us examine only the electro-magnetic effect that causes attraction between atoms to form covalently-bonded molecules. For the purpose of demonstration, we assume the magnetically positive poles to form an axis around the middle of which develops the magnetically negative electron's orbital shell's rough equator. We see the magnetically attractive poles here in green and the equator of the electron shell at a right-angle to them we see as a red circle surrounding the green line.

Thus, there are two forms of electro-magnetic conductivity that can occur as a result of aligning electrons, based on whether the electrons are aligned along the axis of their poles or along their equatorial circumference.

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When the wavelength is of magnetically alinged electrons that are oriented along their magnetically attractive polar axis, the result is called Direct Current, which offers unrestricted capacitance, limited in an inverse exponential amount by distance determined by the medium through which the electricity is channeled.

The other form of electro-magnetism occurs when the electrons are being magnetically aligned along their equatorial circumferences, and is called Alternating Current, because magneticallypositive capacitance will "alternate" with magnetically-negative resistance along this form of a wavelength.

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Because the positively magnetic polar axis is perpendicular to the equatorial circumference of the negatively magnetic electron's orbital-shell, and because these can both be aligned into patterns along wavelengths by orienting them using magnetism, the electro-magnetic force can be graphed as two wavelengths, as we see here in green (for the magnetically positive polar axis) and red (for the magnetically negative electron's equator) along a single central axis for both, however that operate at a right-angle to one another. Because these combined forms of electron alignment according to magnetism combine to form the single force of the electro-magnetic spectrum, and because of the discovery of the electromagnetic force's interaction with electro-magnetically neutral photons, the so-called photoelectric effect, we cannot discuss the combination of these three components to form the electromagnetic force's full spectrum without discussing the photo-electric effect as well. To do this we will next discuss photons, which combine with AC and DC forms of aligned electron wavelengths to comprise the full electromagentic force's spectrum.

Photons

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For photons, first we examine them according to a form of electronic schematic designed by Richard Feynman. According to Feynman's initial premise, a magnetically neutral photon occurs when a magnetically attractive positron combines for some period of time with a magnetically repulsive electron. We see here how, according to Feynman's model, a positron and electron can co-operate for a duration as a photon wavelength, and then once again break apart to emit a single positron and a single electron.

This positron-electron pairing may be what results in the effects we observe from the classical "double-slit" experiment. Often touted as proving "light acts holographically" it appears that when interupted by interference with a solid material object, a photon wavelength will break into multiple parts, and each part will travel simultaneously through all of the multiple "slits" or permeations through the materal interference. The light expresses this doubling effect in the form of losing half its luminosity, and it seems to be this that accounts for the "inverse square law" of light dimminishing with distance from the source of its emission.

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Here we see a potential extended form of the "double-slit experiment" using 6 material barriers, each with either 2 or 3 permeations, or "slits." According to my predictions, we can extrapolate various frequencies of wavelengths by controlling their refraction in this way. The resultant "wave pattern" formed on an emissions receiver at the opposite side of the interfering obstructions would have as many points of origin as obstructions, and we can see using the 6 walled form of the "double slit experiment" that the emission spectrum on the reciever would resemble the hexagonal arrangement of the quantum chromo-dynamics applying to stable forms of quarks. 158

To further break this model down, we can see how a wavelength divided by 4 layers of obstructive interference would refract into 4 points of origin surrounded by ripples outward of less and less often placed randomly scattered positron-electron repairings.

Here we see that, by applying this same method of light refraction using 4 walls of interference to graph a pattern on a flat reciever, we can measure a Lorentz transform of the surface motion on the topology of a torus aligned along a single axis penetrating it perpendicularly to its interior-axes and to its exterior equator.

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The topological patterns of the torus, such as we see here, can seem complex. There is a horizontal wavelength penetrating the torus along its lattitudinal equator in yellow. The outermost circumference of the torus is comprised of a coil that in the diagram is coloured green from the outside and blue inside. The latitudinal spiral in red traces from the equator to the precessing polar axis, the off-set vertical angled purple line.

the Photo-Electric Effect

Having now studied how magnetic electrons behave in AC and DC voltage currents, and how photons combine a positron and electron that can be infinitely divided in halves, diminishing its luminosity according to the "inverse square law," let us now study the combination of the electromagnetic force spectrum and the photo-electric effect. 160

The photo-electric effect is known to us simply by colours. The wavelengths of photons that transmit light reflected to our eyes from all the objects we can see are caused to assume certain colours of the rainbow-spectrum by the composition of the elements they are reflecting off of. This reaction occurs such that all the atomic elements, and thus all the larger molecular forms of matter, reflect one hue of light and refract it to bounce off at another angle and another frequency, carrying a frequency of wavelength our eyes would interpret as a different colour. The spectral chromatic effect can only occur because photon-wavelengths of one frequency / colour will reflect - off electron energy-shells of various different levels and sums of electrons on each, per atomic element - as a different frequency / colour of light.

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The photo-electric effect can be modeled as I have here, using a green circle to signify the electron's orbit in its energy shell-level, red to signify an approaching photon prior to impact with and absorption into the electron's orbital energy shell-level, and blue to signify a retreating photon following ejection and emission from the electron's orbital energy shell-level.

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This aspect of the photo-electric effect, that the massless photon is absorbed into the electron's energy shell for any duration at all, regardless of how breifly, before being immediately reflected off its surface, has long puzzled quantum mechanics. The frequency of the photon wavelength prior to impact appears as one chromatic hue on the visible spectrum and another after reflection off its surface because the trajectory of the photon wavelength is, for a fraction of an instant, combined with that of the electron inside its orbital energy shell. The result is the electron assumes mass enough to be measured during this duration while the photon and electron are combined. Such is the essense of quantum-mechanics.

This occurs along the electron's orbital energy shell level at an arc-radian angle determined by a ratio of before and after collision trajectories of the light-wave (and thus its chromatic tone), that quantum mechanics call "theta." As theta dimminishes asymptotically toward zero the closer to the nucleus the photon wavelength penetrates, the angle of refraction, denoted usually by "phi," expands.

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The duration of time the photo-electric effect can last can thus also approach a zero sum. In this Feynman-type diagram I have attempted to model the photo-electric effect where "t" (its duration) asymptotically approaches a zero point at a right-angle of origin. The electron is modeled as the green and red perependicular wavelengths, while the photon is modeled as a single wavelength in blue.

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Quantum mechanics refer to the apparent zero-time duration of the photo-electric effect wherein the photon is absorbed into the electron, illuminating it, prior to its being reflected off its surface in an altered trajectory - "quantum tunneling." Here is a model of quantum-tunneling that may be applied on an inter-galactic as well as sub-quantum level. We see a negative electron and a neutral photon can combine to form a positive "quantum tunnel" which acts, in a large enough mass aggregate, like a worm-hole that cuts vast distances in space down to approaching zero duration travel-time.

To return to the photo-electric effect, we find that the frequency of a photon wavelength translates into the momentum of the photon particle during the zero-time event while any photon collides with any stable electron energy shell.

The momentum and trajectory of the photon particle combine with the prior trajectory of the magnetically negative electron to cancel its charge just long enough for it to be observed. 165

In the same exact moment, the photon wavelength bounces off the electron energy shell with a frequency comprised of a combination of its previous particle trajectory and that taken by the electron orbiting in its energy shell.

Following the photo-electric effect, the frequency (and colour) of the wavelength is altered, and the electron ceases being a measureable particle, and resumes its apparent occupation of all points on its orbital energy shell simultaneously.

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a Light Cone

First let us consider the three possible curvatures of the fabric of the space-time continuum. All of these occur in different places in space, around various sorts of orbs or clouds, however the one that adds up to occur the most determines the overall shape of the majority of the space-time continuum itself. The "saddle" form signifies a cyclcial "big bang" and "big crunch" alternatng explosion and implosion of the entire universe. The "sphere" geometry signifies a single, fixed aspect-ratio for the expansion of space over time, such that, at a certain pre-determined "critical mass" point, the universe would simply stop changeing over time, and everything would hold still for eternity. The final potential geometry defining space-time shapes it into flat planes, usually in spirals, such as in the accretion of stars into spiral galactic discs and the double-helix formation of molecules into biological DNA. This morpology means a time-line of pereptual expansion at a fixed rate.

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Secondly we consider the method we use to slice backwards in time through all these potential geomtric patterns for the fabric of our spacetime continuum, that is, creating a 4-d cone with a circulating base and a singularity at its apex. The axis of rotation of the base is assumed to be a factor determined by the base' mid-point's relation to the conical apex. We call this axis the "observer's line of sight." The circulating base revolves around the mid-point according to a trajectory determined by the angle of an axis connecting the base' mid-point to the conical apex.

When we apply the 4-d "light-cone" method to measuring the expanding history of the universe, we trace the origin point of our continuum's surface to a central singularity, the "big bang," and for short-hand depiction assume the traditional expansion model of a sphere. In this model, the curvature of present spacetime causes a lens effect to occur as we look back through time toward the first spark of the "big bang," similar to a "double-slit experiment" with no interfering obstacles.

Modern cosmology has improved on this method only by modifying the curvature of spacetime's effect on the model of the light-cone to depict various levels at which the rate of expansion has slowed or sped up. 168

It should also be recalled that, in different locations in space, various different types of curvature can occur as well. For example, in addition to the direct line of sight from an observer looking backward through time toward the "big bang," we see the formation of wormholes, babyuniverses, and inter-galactic alignments.

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These same effects can all be incorporated as occuring inside this model, of my own design, based on a strong gravitational curvature to the continuum's historical light-cone caused between the "big bang" and the point of "critical mass" by the singularity of the "big bang" event itself. The resultant "curved light cone" shows the division of the 4 elemental forces between the "big bang" and the point of "critical mass." The observer's line of sight is shown as the dotted arc.

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In this model, also of my own design, we see the light-cone history of the universe from the "big bang" until "critical mass" (in black) occurs inside a larger temporal pattern, depicted as "aleph sub n" in blue at a 45Â° angle, and as "aleph sub sigma" in red at a right angle. This pattern forms the interior "hole" in the "aleph sub omega" hypersphere, depicted as a green circle. Surrounding this entire cosmological process is the "tau sub tau" tesseract symbolises the method of measuring time.

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Cosmological Model

We begin from the furthest measurement outside of the pattern that comprises the history of our universal time-line, at the level of the "tau sub tau" tesseract symbolising the measurement of time over space. As we begin to zoom inwards, we pass the exterior ring of the "aleph sub omega" torus, and approach the interior pattern of perpetual recreation in the inner-most engine of our universal pattern. As we approach closer still, we see our present universe in the circle on the left of the diagram, the "big bang" singularity event expanded in the central circle, and in the circle on the right is expressed a geometric representation of the "nulliverse" on the opposite side of this cyclical pattern from our own present universe.

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In this exerption of only the engine of creation pattern from the innermost core of the "tau-subtau" tesseract, we see the "big bang" and "expanding singularity" is given as the "aleph sub sigma" torus in red; the present universe and its warped light-cone are depicted as the "aleph sub zero" and "aleph sub-infinity" time-lines on the half of the diagram in green; the "nulliverse" of pure anti-matter zero-point energy is shown as the opposite half of the torus in the diagram as the "aleph sub n" sub-torus in blue.

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In this expanded depiction of the geometry of the "aleph sub sigma" warped-torus form, showing the overall shape of the space-time continuum as an expanded pattern over its entire cyclical pattern, we see the warped light-cone of our own universe's history on the lower left, the nulliverse of ZPE on the right, and between them the "engine of creation" expansion of the "big bang" event that began the expansion of our present universal singularity.

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Here we see only the heart of the engine of creation pattern, the expansion of the event of the "big bang" beginning the expansion of our universal continuum from the pre-universal singularity. We see electric force lines shown as a torus in blue; conical magnetic field lines in red, and between them the warped "saddle shape" of a pereptually re-cycling continuum's geometry.

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Following from this, to the left side of the "aleph sub sigma" torus diagrams seen previous, we return to the original, warped form of a "light cone" model I proposed, depicting the division of the 4 elementla forces between the "big bang" and the point when the universe reached "critical mass."

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Finally, expanding on the central circle showing the universe's complexifcation within a temporally stationary contraction of space, we find this torus, seen from above the mid-point of its central "hole," showing "baby universes" occuring perpendicular to black holes, interconnected by parallel dimensional worm-holes in a multiverse of n potential alternate time-lines. Multiversal time-space defines the outer circle in this diagram, and the inner circle, depicting the torus shape's central "hole," represents the space-time speed limit of our own universal continuum's photon fabric.

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the 4 Forces

To examine the division of the 4 elemental forces of energy in the universe - following the "big bang," but prior to the point when the universe reached "critical mass" and began to devour itself from within like a hungry stomach - we can use graphs such as this one, with only modifications by myself from the original model proposed by Michio Kaku in his book "Hyperspace." The evolution of the elements during the first "Planck time" following the "big bang" is plotted as proceeding from the upper left to the lower right along the diaganol of the square lattice. The first to form is Einsteinian gravity, the so-called "gravitational constant" of "general relativity;" the second is Maxwell radiation, classed as photic and EM radiation; the third are Yang-Mills type particles, comprising the weak nuclear force of fission; the fourth and final to form prior to critical mass were the quark and lepton real particles of solid matter, held together through, presumeably, nuclear fusion. The question-marks along the diaganol axis, where the vertical columns and horizontal rows signifying the 4 elements intersect, signify the energy-level at which the elementary energy forces re-combine and approach total re-unification.

In this diagram, an extension of the previous diagram to signify 5 elemental forces prior to the sixth state of "critical mass," we find the traditional order of formation of the 4 forces following the "big bang," constituting the prime or fifth element. First following the "big bang," gravity formed, then photic light, then the nuclear force carrying particles, then quarks and leptons, and finally "critical mass" occured. Replacing the question marks from the previous diagram are miniature versions of a transform given by Stephen Hawking and Roger Penrose. 178

In this picture we see my own hand-copied depiction of this Hawking-Penrose transform. This graph shows the way spacetime drops off rapidly into a deep warping surrounding the eventhorizon of a black-hole. Here, again, we see the form of a Poincare slice of a torus, the same results produced from a 4-walled "double-slit experiment."

By substituting the Hawking-Penrose transform graph showing the event horizon's warping of spacetime surrounding a black-hole for the previous, question marked diagram, we arrive at this model where the one Hawking-Penrose transform applies to the re-unification energy for all 4 elemental forces. The results are the different warpings to the fabric of space-time shown on the right. 179

In this final form of the preceding diagrams, we see the division of the 4 elemental forces during the first expansion immediately following the "big bang." We see the Hawking-Penrose transform as the Poincare section of a torus, symbolising the temperature of energy-excitation at which the 4 elemental forces re-unify. Gravity we see in black, EM in blue, fusion in green and fission in red. On the left side of this diagram we see also applied the gnomonic pattern of expansion rate at which the 4 elemental forces divided from one another in their intitial temperature conditions just after the "big bang."

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Black Holes and Worm Holes

Black-holes occur when a star grows old and large enough it implodes and creates a deep gravity well in spacetime. Black holes swallow matter and energy and invert them into equal quantities of anti-matter and so-called "dark energy." This inflates a relativistically smaller singularity to form a "baby universe" inside each black hole. Wormholes occur on the surface of black-holes' event-horizons, and result in deep-space gamma ray eruptions occuring apparently at random. These gamma-ray eruptions occur in between them when spiral galaxies formed around the equators of black-holes align with one another. 181

In this diagram we see the poles of a black hole emitting gas jets that circle around in the deepest voids between galactic filaments to form baby universes. The angle of the gas jets' bending around at which a "baby universe" begins to form is given as "theta," the same as the interaction of the "photo-electric effect" in quantum-mechanics. As theta approaches zero, the gas-jets arc further and further out into deeper and deeper space, encompassing larger and larger arcs interacting with and connecting them gravitationally to other distant neighboring galaxies.

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Here we can see how this process relates to our own Milky Way galaxy, wherein the same gas-jets that form the deep-space "baby universes" (such as that on the left) connect in shorter arcs (such as those on the right) to our own star's poles, and from there to those of our planet, etc.

In this picture, we show the gravitic wave-lengths connecting our star to the black-hole at centre of our Milky Way galaxy. In green and blue, at right angles to one another, we "positive" gravity waves A and B, combining to form "positive" gravity field A, in red. As gravity waves emitted from galactic core by the central black hole there pass through our star, sun, they result in ocassional reversals of the solar electromagentic poles.

the see the the

Just as wormholes form on the surface of black holes at the cores of spiral galaxies, so too does the sunspot cycle reflect the effect of the north and south oriented magnetic poles of the black hole and all its galaxy's stars. As the north and south poles of the galactic core's black-hole precess over time, they alter the effects on the sunspot cycle of their galaxy's stars that eventually result in electromagnetic polar reversal in stars and their accompanying planets. As the poles precess their orientation over time, each electromagentically reverses with respect to the others. 183

For a planet in a roughly circular solar orbit, and for a star in a spiral galaxy, there will be a total of 8 planetary pole reversals per every 4 solar pole reversals, per each single pole reversal of the black-hole at galactic-core.

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EM-pole reversals of the gas jets of a black-hole at the core of a spiral galaxy occur when one galaxy aligns with another at a right angle. When the gas-jets of two distant black-holes align at right angles, a worm-hole forms between our own universe and a "baby universe" between them resulting in a gamma ray explosion in deep space between these galaxies.

When an EM pole reversal occurs for a black-hole in the centre of a spirla galaxy, it prommotes the precession of the polar axis, and also results in a rippling effect as all the rest of the stars outward from the core of the spiral galaxy undergo an EM pole reversal at their own intervals. When two galaxies align at right angles from one another's core black holes, the galactic core's black hole's pole reverse, and from there the pole reversal emanates outward toward the other stars in the galaxy's spiral accretion disc, effecting first each star, and then the planets of each star, and finally the moons of any such planets that have any.

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