[ hyper ] by Michael Buckley

emergent formation : non-linear methodologies Arch 704: Research Studio | Spring 2014 | University of Pennsylvania School of Design Department of Architecture critic: Cecil Balmond Ezio Blasetti

In this model, a building looks back to its philosophy of making, revealed as the poetic essence of an appropriate technology. Both in formulation and material, from art object the architectural project moves towards improvisation and process. Being an experiment, structure frames in episodes, and architecture stitches together as layers of evolving stabilities. With computer form making and advances in technology the building becomes more a laboratory of ideas. Science is not denied in the is lab, nor humanism. What blinds both is the obligation to negotiate with pattern, small scale for the human, deep scale for the abstract. Cecil Balmond, Crossover project objectives This studio will investigate non-linear systems and self-organization at both a methodological and tectonic level. This exploration will take the form of design research into algorithmic methodologies and will be tested through a concrete architectural proposal. Design research is not defined here as a linear scientific process with objective outcomes, but rather as the iterative, non-linear and speculative process with the ability to reassess and shift our disciplinary discourse. This semester, the studio will collaborate with the artist Perry Hall to strengthen the interdisciplinary dialogue between architecture and art.

conceptual framework Our world is increasingly being understood as an emergent outcome of complex systems. Similarly, both analytical and generative tools for the definition of spatial and architectural systems have been established within our discipline. Although this design approach is extremely sensitive to existing models of self-organization in material, biological and physical systems, our intention could not be further than the mere replication of â&#x20AC;&#x2DC;natureâ&#x20AC;&#x2122;. On the contrary, with the deployment of non-linear computational design methodologies we seek new singularities in the extended territory of contemporary architectural production. At the same time, this research allows us to transcendent traditional disciplinary boundaries since research in complexity itself is an emergent language shared between multiple fields of scientific and artistic interest. The inter-disciplinary nature of the studio brings together expertise from architecture, art, engineering, complex systems theory and new technologies of manufacturing. Algorithmic architectural research has the unique capacity to develop a deeper dialogue with a variety of scales, from the material and microscopic, to the emergent and macroscopic.

emergent formation : non-linear methodologies

project

methodology:

The major project for the semester will be the design of an art conscious building program in Kesington Gardens in Hyde Park, London. The project will be concerned with the relationship of art and architecture, formation and sculpture. The Serpentine Gallery Pavilion Program provides an extensive list of recent precedents in collaborations between artists and architects. The studio will propose a hybrid program between a public plaza, a conference center and an art gallery. Key elements of the programmatic narrative will be internal to the development of each project and are expected to emerge out of the dialogue between the artistic component and the architectural intention.

Our point of departure will be the research which was developed during the seminar on Form and Algorithm in the fall semester. Students will revise their non-linear algorithmic design methodologies based on their previous success as well as their ability to engage the studio project. Further algorithmic workshops will be conducted in the first half of the semester to help in rapidly developing sophisticated scripted techniques. Algorithmic design does not operate through a specific technique or medium - digital or analogue - and the studio will organize a series of workshops that will engage closely with active material explorations in our collaboration with Perry Hall.

the paradox of the ‘arche-fossil’: ‘How can a being manifest being’s anteriority to manifestation? What is it that permits mathematical discourse to bring to light experiments whose material informs us about a world anterior to experience?’ In this short passage, speculative realist philosopher Quentin Meillassoux claims the primacy of mathematics over perception, in the access of qualities of things. Expanding on this notion beyond the ability of description of the world, this research studio will investigate the underlying idea that formal systems and algorithmic processes have the capacity for expansive translatability between mediums. They constitute generative languages whose artifacts can operate on a multitude of scales and readings.

mediums: According to Jeff Kipnis, architectural drawings can function in three different ways: as an innovative design tool, as the articulation of a new direction, or as a creation of consummate artistic merit, leading to the fact that a perfect act of architecture must achieve all three at once. For our purposes, we will attempt a substitution of the architectural drawing by algorithmic processes in this thesis. The pervasive contemporary uniformity of architectural drawings and renderings is inhibiting ideological, aesthetic and expressive difference in our discipline. Challenging notions of architectural representation, the studio will interrogate drawings, models, codes, narratives and other mediums as objects in themselves as well as complementary to each other in a larger dialogue between artistic and architectural production. This conceptual context will attempt to foster unique and multifaceted Projects that capture not only a fully articulated architectural experience but also other aspects of an Architectural Idea which tend to resist direct manifestation into built form.

emergent formation : non-linear methodologies

Rn [ hyper ] 2014 PennDesign Arch 704 Research Studio Spring 2014 University of Pennsylvania Dept of Architecture critic: Cecil Balmond Ezio Blasetti student: Alexander Dâ&#x20AC;&#x2122;Aversa Michael Buckley

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This project attempts to engage and reconcile euclidean coordinate systems in higher dimensions to generate a set of architectonic strategies that capture the properties of higher dimensional spaces and their manifestations in 3D. This exploration is divided into 3 closely related stages which, when taken in sequence, manifest higher-dimensional spaces material in our threedimensional world. These stages are: creation of higher-dimensional geometry, transformation of geometry, and finally projection into lowerdimensional space. Multiple varieties of operations exist within each stage, which explore broader concepts of boundary -- inclusion/exclusion & interiority/exteriority -- notions of superposition, intersection, non-adjacent adjacencies, symmetry and lastly, gains/losses in information while traversing dimensions.

0.0.

introduction

0.1. 0.2. 0.3. 0.4.

creation 0.1.1. hypercube 0.1.2. hyperdimensional lattices 0.1.3. native 4D geometry

0.5.

visualization catalog

transformation 0.2.1. rotation 0.2.2. translation & scaling 0.2.3. boolean projection 0.3.1. orthographic 0.3.2. stereographic 0.3.3. section summary

what are higher dimensions. The most basic definition of a higher-dimensional Euclidean space is one in which an N number of additional basis vectors are needed to describe the extent of the space. X, Y, and Z are canonical basis vectors for 3D space, but any set of 3 mutually perpendicular vectors can define a 3D space. A higher-dimensional space includes an additional basis vector, i.e. vectors perpendicular to all other basis vectors in the space. A higher-dimensional space must then use additional coordinates to define a vector or point in space. In 3D, a point is given by <X,Y,Z> while in 4D it is given by < X, Y, Z , U > and in 6D it is given by < X, Y, Z, U, V, W >. This is difficult to visualize because we inhabit a 3D space, and cannot graphically nor physically construct a higher-dimensional space. The only way we can see a higher-dimensional entity is by transformation and projection of that entity into our 3D space. By doing so, information is either lost or transformed, depending on the methods used. This study does not encompass the 3D Space + 1D Time, called Minkowski Space, which treats the time coordinate differently than in Euclidean geometry.

0.1 | Platoâ&#x20AC;&#x2122;s Cave

http://www.gnostic-centre.com/Plato.jpg

X Y Z 0.2 | DimensionsU

(0)

(0,0)

(0,0,0)

(0,0,0,0)

(0,0,0,0,0)

(0,0,0,0,0,0)

y u z

z y

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0.0. Rn [ hyper ] 2014 [ hyper ]

the content of a figure (polytope) content is defined as the number of unit measure polytopes and fractions thereof that would exactly fill the interior of a polytope or other closed figure. for points, the content is 0 since a point has no interior. the content of a line (1D) is its length. the content of a polygon or other closed shape (2D) is area. the content of a polyhedra or closed figure (3D) is volume. in higher dimensions the definition continues. in 4D it is called bulk, while in spaces higher than 4D, a greek root is prefixed to the name, e.g. pentabulk, hexabulk, heptabulk, octabulk, enneabulk, etc.

0.3 | 2D Projection of 600-cell Vertices

content, if not explicitly stated to be in another sub-dimension, is taken to be the content of the highest dimension of the figure.

definitions of polytopes: 0. vertex 1. edge 2. face 3. volume 4. bulk 5. pentabulk

0.4 | 2D Projection of 600-cell Edges

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a point with number of coordinates needed to define the figure. < X, Y, Z, U... > a 2-tuple defining the start and end points of an edge. < V1, V2 > an n-tuple defining the edges that make up a face. < E1, E2, E3, E4... > an n-tuple defining the faces that make up a volume. < F1, F2, F3, F4... > an n-tuple defining the volumes that make up a bulk. < V1, V2, V3, V4... > an n-tuple defining the bulks that make up a pentabulk. < B1, B2, B3, B4... >

0.1 | Projections of higher-dimensional geometries

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0.1.0 Rn [ hyper ] 2014 terms hypercube.higher-dimensional analog to the cube. 4-cube.4-dimensional analog to the cube. 5-cube.5-dimensional analog to the cube. polytope.a geometric object with flat sides existing in n-number of dimensions.

[ creation ]

this chapter describes higher dimensional geometry that we are working with throughout our investigation. there are many analogous higher dimensional shapes that correspond to a lower dimension shape. for example, take the familiar cube -- a 3-dimensional object -- itâ&#x20AC;&#x2122;s analogous shape in four dimensions is called the tesseract or the 4D hypercube. this is an important geometry for us because it is one of the simplest, primitive polytopes and thus allows for us to understand higher dimensional space through its clarity. there are a few methods to create higher dimensional geometries: extrusion, definition, parametric equation. some geometries that we deal with do not exist in lower dimensions, these are said to be native to higher dimensions. there are many higher dimensional geometries and in our investigation we deal with a range of these to describe higher dimensional space.

lattice.a regular geometrical arrangement of points or objects over an area or in space.

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0.1.0

creation

0.1.1 0.1.2 0.1.3

hypercube

hyperdimensional lattices cubic native 4D geometry 600-cell

hyperdimensional geometry. The image to the right displays a 3D projection of a hypercube. The hypercube is the 4-dimensional version of a cube. It is one a primitive, regular polytope, and provides a good basis for early explorations in higher dimensions. Although it appears to be identical to a 3-dimensional cube the hypercube is much different; at each vertex lies additional vertices that are similarly one unit away in 4-dimensional space, yet collapse to a coincident location in 3-dimensional space. the hypercube is not understood by one simple projection, but by viewing alternate projections in which the hypercube is rotated in 4-dimensional space before it is projected back to 3D space.

3D projection

hypercube.4-cube # of pts: 16

2D projection

xu: 0° yu: 0° zu: 0°

xu: 20° yu: 60° zu: 45°

0.1| views of hypercube a.3D-projection b.3D-projection-rotated c.2D-projection d.2D-projection-rotated

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0.1.1

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hyperdimensional lattices. The easiest lattice to visualize is the cubic lattice, since it is a repetition without transformation (rotation, mirroring, etc) in all coordinate vectors. A 3-dimensional cubic lattice can be “extended” into 4D by extending the definition of each point on the lattice to include an additional coordinate. Thus, a 4D cubic lattice is a multiplicity of 3D cubic lattices separated by one unit each in the 4th dimension. In other words, for each point on a 3D cubic lattice, there exist a number of points in 4D with the same <X,Y,Z> coordinates, the number of which depends on the extents of the lattice.

3D projection

hypercubic.lattice # of pts: 81

2D projection

xu: 0° yu: 0° zu: 0°

xu: 20° yu: 60° zu: 45°

0.1| views of hypercubic lattice a.3D-projection b.3D-projection-rotated c.2D-projection d.2D-projection-rotated

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0.1.2

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native hyperdimensional geometry. There are certain geometries that do not exist within 3-dimensional space. In this way, some shapes can be said to be native to higher dimensions, since they do not otherwise exist. One example of a regular polytope that is native to 4 dimensions is the 600-cell. It is composed of 120 vertices, 720 edges, 1200 triangular faces, and 600 tetrahedral volumes.

3D projection

600.cell # of pts: 120

2D projection

xu: 0° yu: 0° zu: 0°

xu: 20° yu: 60° zu: 45°

0.1| views of 600-cell a.3D-projection b.3D-projection-rotated c.2D-projection d.2D-projection-rotated

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0.1.3

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0.1 | Hypercubic lattice

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0.2.0 Rn [ hyper ] 2014 terms rotation.in short, rotation can be understood as turning around a center. this center and the â&#x20AC;&#x2DC;axesâ&#x20AC;&#x2122; of rotation can be defined in n-dimensions.

[ transformation ]

this chapter describes the various operations that can performed on higher dimensional geometries in order to transform their appearance. for example, one can rotate a polytope in higher dimensional space and then view a projection of that rotated geometry in order to gather a more holistic view the shape than a singular nonrotated projection. in a similar manner there are other transformations that prove to be helpful -- translation, scaling, attraction/repulsion, and boolean operations are transformations that we have investigated.

translation.simply put, translation means moving. every point of the geometry must move the same distance in the same direction. scaling.to increase or reduce proportionately in size. attraction.to be pulled to a source proportionately as a function of distance from the attractor. repulsion.to be pushed away from a source proportionately as a function of distance from the source. boolean.the logic in which values are either true [1] or false [0] that is used to test for intersection of geometry.

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0.2.0

transformation

0.2.1

rotation

0.2.2

translation

0.2.3 boolean

rotation. Rotating geometries before projection is a very effective way to get more views around an object . More views are important to help understand higher dimensional shapes because although they can be projected into 3-dimensional shapes but never fully realized as 4 dimensional objects.

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0 �� 0 � 0 � 0 � ��

ROTATION [pt]

0.2.1

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translation. In a geometric translation every point of the geometry must move the same distance in the same direction. This transformation is only moving geometry; changing position but not shape. This makes translation less useful if used to strictly move geometry rather than to alter it. However, translation can be applied in different ways to manifest effects that do transform geometry.

attraction/repulsion. In the case of attraction or repulsion a translation is made to a source geometry. Essentially, attraction is to be pulled to a source proportionately as a function of distance from the attractor. And conversely, repulsion is the act of moving away from a source proportionately as a function of distance from the locus of repulsion. This can be generalized to n-dimensions and will act the same in each. However, an understanding of 4 dimensional distance in 3D space can lead to perplexing situations; objects that appear coincident in 3D may actually be very far apart 4D space. This seemingly contradictory situation that arises through traversing between dimensions is a paradox that is at the heart of this investigation.

0.1 | 3D Projection of 6-Dimensional Repulsion

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0.2.2

What is n-dimensional attraction? Generate Attractors [pts]

ATTRACT [pt]

n i=1

Attracted Pts

(ai - bi)

Attractor Pt Pt Affected Pt Vector

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boolean. It can be said that the 4 dimensional realm is â&#x20AC;&#x2DC;much roomierâ&#x20AC;&#x2122; than our 3 dimensional space. With this in mind, an intersection in a projection of a hyperdimensional object does not indicate a true intersection in hyperspace. The abundance of apparent interstections in lower dimensions one must take a stance on what intersection can be or mean. This raises questions about what kinds of intersection one can have -- good / bad, pseudo-instersection, et cetera. Diagrammed on these pages is a display of the superimposition of volumes when a hypercube is projected into 3D space.

0.1 | Rotated hypercube displaying volumes

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0.2.3

0.2 | Volumes within Hypercube

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�� 0.1 | Math�� matrices for orthographic projection � �� 0 0

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0 0 � 0 1 Rn [ hyper ]

0.3.0 Rn [ hyper ] 2014 terms orthographic.projection method in which all the projection lines are orthogonal [perpendicular] to the projection plan. Similar to shadows cast by the sun. stereographic.a projection obtained by projecting points on the surface of a sphere from the sphereâ&#x20AC;&#x2122;s north pole to a point in a plane tangent to the south pole. Similar to shadows cast by a light fixture. section.taking a section is the act of cutting an object into slice(s). this slice is then viewed in order to understand a discrete portion of an object.

[ projection ]

this chapter describes the visualization of higher dimensional geometry, which can be done only through projecting the object into the 3rd or lower dimensions. this act is inherently deficient: we cannot ever see the true nature of a higher dimensional object and all of its properties, and the act of projection must distort or change the object in some way to make it visible in our world of perception. each projection has its own method which aims to maintain certain properties at the expense of others. projections applied in series do not solve this problem, but instead amplify it. the properties maintained by one projection technique will be disturbed by another, and, while the second will attempt to preserve its own set of properties, it is operating on the projection, not the original polytope. projections also apply across dimensions. an orthographic projection from 4D to 3D behaves similarly to the same from 3D to 2D. By analyzing the lower dimensional operations, in which both original and projected geometry are visible, we gain an understanding of their workings in higher dimensions. 0.3.0

projection

0.3.1 orthographic 0.3.2 sterographic 0.3.3 section

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orthographic projection. from 3D to 2D the operation of an orthographic projection is very simple. for 3D to 2d, this operation is typically understood as a mapping from <X,Y,Z> to <X,Y,0> thus projecting points from a 3D object onto the XY plane. this operation can be done onto the other two planes, XZ and YZ, just as easily, while projections onto other 2D spaces requires additional mathematical effort. this transformation preserves the connectivity between objects, but does not necessarily preserve distances or areas. most notably, it differs from the stereographic projection in that it does not distort all distances, but rather only distorts distances that are not included in the space which is being projected onto. for example, if two endpoints of a line both share the same z coordinate value, its projection onto the xy plane will not change in length. more distances are preserved the more coordinates are preserved in the transformation.

0.1 | 2D Orthogonal Projection of a 600-cell

from 4D to 3D this projection collapses a set of 4D coordinates or objects into 3D by setting the 4th coordinate to 0, e.g. <X,Y,Z,U> to <X,Y,Z,0>. it again preserves connectivity between coordinates but does not necessarily preserve distance, area, or angle. this type of projection, due to the specific way in which information is lost along only the 4th axis, creates a very dense object. two points <1,1,1,0> and <1,1,1,1> after the transformation become equivalent points in 3D. this superposition of points does not necessarily create superposition of surface, but does transform the original 4D polygon into a 3D geometry with non-manifold edges and many intersections between surfaces and subspaces. 0.2 | 3D Orthographic Projection of a 600-cell

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PROJECTION [pt]

0.3.1

A projection of an object from a higher dimension down to a lower dimension produces a shadow of the object. If the object changes orientation, the project is altered. With this method the projection of n-dimensional points into 3 space is possible, with the rotation of a given object much more can be understood.

Rotations x = 120.0 y = 08.0 z = 40.0

Rotations x = 00.0 y = 00.0 z = 00.0

Original Object Projection Lines 2D Projection

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stereographic projection. from 3D to 2D any closed polygon in 3D (with no non-manifold edges) can be thought of as a specifically related set of 2D surfaces, which can be mapped onto the surface of a sphere. from this, the stereographic function projects each point of the sphere onto a 2D plane that the sphere is resting on, from the projection point at the point on the sphere opposite the point that touches the plane. this type of projection preserves angles locally (though these angles will be distorted through the initial mapping of the object to the sphere), however, it distorts the distances and areas of lines and surfaces through its operation. from 4D to 3D a similar operation is carried out in the 4D to 3D stereographic projection as in the 3D to 2D version. a 4D object is mapped onto the surface of a 3-sphere, the analog of the 3D sphere we are familiar with. this is then mapped onto a hyperplane by projecting through the point on the 3-sphere opposite the point which sits on the hyperplane. this gives a 3D object which preserves local angles (as distorted to be on the sphere) but does not preserve distance, area, or volume.

0.1 | Stereographic projection of a 600-cell

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STEREOGRAPHIC PROJECTION

0.3.2

Stereographic projection maps the sphere to the plane, save the north pole. This is a confromal mapping meaning it preserves angles or intersection. It does not preserve distances nor area. The idea is to point a light source at the north pole, and look at the shadows of points on the sphere as they appear on the plane below.

How a sphere maps to a plane

ie.

Sphere Projection Lines

S Pâ&#x20AC;&#x2122;

2D Plane

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hyper-section projection. from 3D to 2D a plane is intersected with the target geometry and the section is the resulting intersection geometry. in a section, all geometric objects at all dimensions are taken as inputs, and are returned as objects with dimension D-1 from the original. for example, an edge becomes a point, a face becomes an edge, and a volume becomes a face. the process takes each object, for instance a volume, and decomposes it into its constituent D-1 parts, in this case faces. the faces are decomposed into edges and the edges tested for intersection on the plane. this intersection becomes section point. when the multiple edges of a face are intersected with the section plane, they return 2 points, which are then taken to be endpoints of an edge. the faces of a volume, when intersected, return edges which are reconstructed into a face. from 4D to 3D the section operation in higher dimensions is merely an extension to the 3D to 2D operation. each higher-dimensional bulk is split into its constituent D-1 bulks until these decompose into volumes, faces, and finally edges. these are intersected with the hyperplane and the intersected geometry is built back up as point, edge, face, volume, bulk, etc. until the section is comprised of D-1 entities.

0.1 | Serial Sections through a 600-cell

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HYPER-SECTION

0.3.3

The process of taking an n-dimensional section. Is described through the diagram to the right. The object is cut from higher dimensional parts down to its points, then rebuilt according to the way it was cut.

Cutting Plane

Cut

Rebuild

Projection Lines

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0.4.0 Rn [ hyper ] 2014 [ summary ]

we have gained an understanding of what each different projection methods gains/ loses. stereographic projection grants great scalar differentiation, and maintains all local angles at intersections. orthographic projection produces overlap because geometries are collapsed onto each other, while lines that are coplanar to each other maintain their length. the hyper-section transforms our geometry into specific intersected volumes and the projections of these are shells. we have learned how geometric transformations perform in higher dimensional space in comparision to 3 dimensional space or lower, and we have developed an understanding of higher dimensional geometries.

our research has been translated into an attempt at manifesting shadows of the undetectable -- the other.

The remainder of this book is a catalog that displays the multitude of results from our research into hyperdimensions. The images included are generated through the modes of projecting, transforming, and with the hyperdimensional geometries that have been described. These visualizations are all representations of what is imperceivable.

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600-cell.4D # of pts: 120

orthographic.projection2D point visualizations of rotations and reflections

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0.5.0

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600-cell.4D # of pts: 120

orthographic.projection2D point visualizations of rotations and reflections

600-cell.4D # of pts: 120

orthographic.projection2D point visualizations of rotations and reflections

Rn [ hyper ]

600-cell.4D # of pts: 120

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orthographic.projection2D point visualizations of rotations and reflections

600-cell.4D # of pts: 120

orthographic.projection2D point visualizations of rotations and reflections

Rn [ hyper ]

600-cell.4D # of pts: 120

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orthographic.projection2D point visualizations of rotations and reflections

600-cell.4D # of pts: 120

orthographic.projection2D edge visualizations of rotations and reflections

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600-cell.4D # of pts: 120

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orthographic.projection2D edge visualizations of rotations and reflections

600-cell.4D # of pts: 120

orthographic.projection2D edge visualizations of rotations and reflections

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600-cell.4D # of pts: 120

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orthographic.projection2D edge visualizations of rotations and reflections

600-cell.4D # of pts: 120

orthographic.projection2D edge visualizations of rotations and reflections

Rn [ hyper ]

600-cell.4D # of pts: 120

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orthographic.projection2D edge visualizations of rotations and reflections

600-cell.4D # of pts: 120

orthographic.projection2D visualizations of rotations and reflections

Rn [ hyper ]

600-cell.4D # of pts: 120

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orthographic.projection2D visualizations of rotations and reflections

600-cell.4D # of pts: 120

orthographic.projection2D visualizations of rotations and reflections

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

orthographic.projection2D visualizations of rotations and reflections

600-cell.4D # of pts: 120

orthographic.projection2D visualizations of rotations and reflections

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600-cell.4D # of pts: 120

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orthographic.projection2D visualizations of rotations and reflections

hypercube.4D # of pts: 16

orthographic.projection3D rotations in the < XU >

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hypercube.4D # of pts: 16

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orthographic.projection3D rotations in the < YU >

hypercube.4D # of pts: 16

orthographic.projection3D rotations in the < ZU >

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hypercube.4D # of pts: 16

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orthographic.projection3D rotations in the < XU, YU >

hypercube.4D # of pts: 16

orthographic.projection3D rotations in the < XU , ZU >

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hypercube.4D # of pts: 16

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orthographic.projection3D rotations in the < YU, ZU >

hypercube.4D # of pts: 16

orthographic.projection3D rotations in the < XU, YU, ZU >

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hypercube.4D # of pts: 16

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stereographic.projection3D rotations in the < XU >

hypercube.4D # of pts: 16

stereographic.projection3D rotations in the < YU >

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hypercube.4D # of pts: 16

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stereographic.projection3D rotations in the < ZU >

hypercube.4D # of pts: 16

stereographic.projection3D rotations in the < XU, YU >

Rn [ hyper ]

hypercube.4D # of pts: 16

Rn [ hyper ]

stereographic.projection3D rotations in the < XU, ZU >

hypercube.4D # of pts: 16

stereographic.projection3D rotations in the < YU, ZU >

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hypercube.4D # of pts: 16

Rn [ hyper ]

stereographic.projection3D rotations in the < XU, YU, ZU >

hypercube.4D # of pts: 16

section.cut3D translations in the

Rn [ hyper ]

hypercube.4D # of pts: 16

Rn [ hyper ]

section.cut3D rotations in the < XU >, translation in the

hypercube.4D # of pts: 16

section.cut3D rotations in the < YU >, translation in the

Rn [ hyper ]

hypercube.4D # of pts: 16

Rn [ hyper ]

section.cut3D rotations in the < ZU >, translation in the

hypercube.4D # of pts: 16

section.cut3D rotations in the < XU, YU >, translation in the

Rn [ hyper ]

hypercube.4D # of pts: 16

Rn [ hyper ]

section.cut3D rotations in the < XU, ZU >, translation in the

hypercube.4D # of pts: 16

section.cut3D rotations in the < YU, ZU >, translation in the

Rn [ hyper ]

hypercube.4D # of pts: 16

Rn [ hyper ]

section.cut3D rotations in the < XU, YU, ZU >, translation in the

hypercube.4D # of pts: 16

section.cut3D rotations in the < XU >

Rn [ hyper ]

hypercube.4D # of pts: 16

Rn [ hyper ]

section.cut3D rotations in the < YU >

hypercube.4D # of pts: 16

section.cut3D rotations in the < ZU >

Rn [ hyper ]

hypercube.4D # of pts: 16

Rn [ hyper ]

section.cut3D rotations in the < XU, YU >

hypercube.4D # of pts: 16

section.cut3D rotations in the < XU, ZU >

Rn [ hyper ]

hypercube.4D # of pts: 16

Rn [ hyper ]

section.cut3D rotations in the < YU, ZU >

hypercube.4D # of pts: 16

section.cut3D rotations in the < XU, YU, ZU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

orthographic.projection3D rotations in the < XU >

600-cell.4D # of pts: 120

orthographic.projection3D rotations in the < YU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

orthographic.projection3D rotations in the < ZU >

600-cell.4D # of pts: 120

orthographic.projection3D rotations in the < XU, YU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

orthographic.projection3D rotations in the < XU, ZU >

600-cell.4D # of pts: 120

orthographic.projection3D rotations in the < YU, ZU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

orthographic.projection3D rotations in the < XU, YU, ZU >

600-cell.4D # of pts: 120

stereographic.projection3D rotations in the < XU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

stereographic.projection3D rotations in the < YU >

600-cell.4D # of pts: 120

stereographic.projection3D rotations in the < ZU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

stereographic.projection3D rotations in the < XU, YU >

600-cell.4D # of pts: 120

stereographic.projection3D rotations in the < XU, ZU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

stereographic.projection3D rotations in the < YU, ZU >

600-cell.4D # of pts: 120

stereographic.projection3D rotations in the < XU, YU, ZU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

section.cut3D translation in the

600-cell.4D # of pts: 120

section.cut3D rotations in the < XU >, translation in the

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

section.cut3D rotations in the < YU >, translation in the

600-cell.4D # of pts: 120

section.cut3D rotations in the < ZU >, translation in the

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

section.cut3D rotations in the < XU, YU >, translation in the

600-cell.4D # of pts: 120

100

section.cut3D rotations in the < XU, ZU >, translation in the

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

section.cut3D rotations in the < YU, ZU >, translation in the

101

600-cell.4D # of pts: 120

102

section.cut3D rotations in the < XU, YU, ZU >, translation in the

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

section.cut3D rotations in the < XU >

103

600-cell.4D # of pts: 120

104

section.cut3D rotations in the < YU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

section.cut3D rotations in the < ZU >

105

600-cell.4D # of pts: 120

106

section.cut3D rotations in the < XU, YU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

section.cut3D rotations in the < XU, ZU >

107

600-cell.4D # of pts: 120

108

section.cut3D rotations in the < YU, ZU >

Rn [ hyper ]

600-cell.4D # of pts: 120

Rn [ hyper ]

section.cut3D rotations in the < XU, YU, ZU >

109

5-cell.5D # of pts: 32

110

orthographic.projection4D. orthographic.projection3D rotations in the < XV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. orthographic.projection3D rotations in the < YV, UV >

111

5-cell.5D # of pts: 32

112

orthographic.projection4D. orthographic.projection3D rotations in the < ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. orthographic.projection3D rotations in the < XV, YV, UV >

113

5-cell.5D # of pts: 32

114

orthographic.projection4D. orthographic.projection3D rotations in the < XV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. orthographic.projection3D rotations in the < YV, ZV, UV >

115

5-cell.5D # of pts: 32

116

orthographic.projection4D. orthographic.projection3D rotations in the < XV, YV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. stereographic.projection3D rotations in the < XV, UV >

117

5-cell.5D # of pts: 32

118

orthographic.projection4D. stereographic.projection3D rotations in the < YV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. stereographic.projection3D rotations in the < ZV, UV >

119

5-cell.5D # of pts: 32

120

orthographic.projection4D. stereographic.projection3D rotations in the < XV, YV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. stereographic.projection3D rotations in the < XV, ZV, UV >

121

5-cell.5D # of pts: 32

122

orthographic.projection4D. stereographic.projection3D rotations in the < YV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. stereographic.projection3D rotations in the < XV, YV, ZV, UV >

123

5-cell.5D # of pts: 32

124

orthographic.projection4D. section.cut3D rotations in the < XV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. section.cut3D rotations in the < YV, UV >

125

5-cell.5D # of pts: 32

126

orthographic.projection4D. section.cut3D rotations in the < ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. section.cut3D rotations in the < XV, YV, UV >

127

5-cell.5D # of pts: 32

128

orthographic.projection4D. section.cut3D rotations in the < XV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

orthographic.projection4D. section.cut3D rotations in the < YV, ZV, UV >

129

5-cell.5D # of pts: 32

130

orthographic.projection4D. section.cut3D rotations in the < XV, YV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. orthographic.projection3D rotations in the < XV, UV >

131

5-cell.5D # of pts: 32

132

stereographic.projection4D. orthographic.projection3D rotations in the < YV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. orthographic.projection3D rotations in the < ZV, UV >

133

5-cell.5D # of pts: 32

134

stereographic.projection4D. orthographic.projection3D rotations in the < XV, YV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. orthographic.projection3D rotations in the < XV, ZV, UV >

135

5-cell.5D # of pts: 32

136

stereographic.projection4D. orthographic.projection3D rotations in the < YV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. orthographic.projection3D rotations in the < XV, YV, ZV, UV >

137

5-cell.5D # of pts: 32

138

stereographic.projection4D. stereographic.projection3D rotations in the < XV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. stereographic.projection3D rotations in the < YV, UV >

139

5-cell.5D # of pts: 32

140

stereographic.projection4D. stereographic.projection3D rotations in the < ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. stereographic.projection3D rotations in the < XV, YV, UV >

141

5-cell.5D # of pts: 32

142

stereographic.projection4D. stereographic.projection3D rotations in the < XV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. stereographic.projection3D rotations in the < YV, ZV, UV >

143

5-cell.5D # of pts: 32

144

stereographic.projection4D. stereographic.projection3D rotations in the < XV, YV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. section.cut3D rotations in the < XV, UV >

145

5-cell.5D # of pts: 32

146

stereographic.projection4D. section.cut3D rotations in the < YV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. section.cut3D rotations in the < ZV, UV >

147

5-cell.5D # of pts: 32

148

stereographic.projection4D. section.cut3D rotations in the < XV, YV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. section.cut3D rotations in the < XV, ZV, UV >

149

5-cell.5D # of pts: 32

150

stereographic.projection4D. section.cut3D rotations in the < YV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

stereographic.projection4D. section.cut3D rotations in the < XV, YV, ZV, UV >

151

5-cell.5D # of pts: 32

152

section.cut4D. orthographic.projection3D rotations in the < XV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. orthographic.projection3D rotations in the < YV, UV >

153

5-cell.5D # of pts: 32

154

section.cut4D. orthographic.projection3D rotations in the < ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. orthographic.projection3D rotations in the < XV, YV, UV >

155

5-cell.5D # of pts: 32

156

section.cut4D. orthographic.projection3D rotations in the < XV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. orthographic.projection3D rotations in the < YV, ZV, UV >

157

5-cell.5D # of pts: 32

158

section.cut4D. orthographic.projection3D rotations in the < XV, YV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. stereographic.projection3D rotations in the < XV, UV >

159

5-cell.5D # of pts: 32

160

section.cut4D. stereographic.projection3D rotations in the < YV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. stereographic.projection3D rotations in the < ZV, UV >

161

5-cell.5D # of pts: 32

162

section.cut4D. stereographic.projection3D rotations in the < XV, YV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. stereographic.projection3D rotations in the < XV, ZV, UV >

163

5-cell.5D # of pts: 32

164

section.cut4D. stereographic.projection3D rotations in the < YV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. stereographic.projection3D rotations in the < XV, YV, ZV, UV >

165

5-cell.5D # of pts: 32

166

section.cut4D. section.cut3D rotations in the < XV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. section.cut3D rotations in the < YV, UV >

167

5-cell.5D # of pts: 32

168

section.cut4D. section.cut3D rotations in the < ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. section.cut3D rotations in the < XV, YV, UV >

169

5-cell.5D # of pts: 32

170

section.cut4D. section.cut3D rotations in the < XV, ZV, UV >

Rn [ hyper ]

5-cell.5D # of pts: 32

Rn [ hyper ]

section.cut4D. section.cut3D rotations in the < YV, ZV, UV >

171

5-cell.5D # of pts: 32

172

section.cut4D. section.cut3D rotations in the < XV, YV, ZV, UV >

Rn [ hyper ]

173

174

Rn [ hyper ]

175

176

Rn [ hyper ]

6-lattice.6D 64-tree

Rn [ hyper ]

orthographic.projection6/3D increase in # of sample points

177

6-lattice.6D 64-tree

178

orthographic.projection6/3D increase in # of sample points

Rn [ hyper ]

6-lattice.6D 64-tree

Rn [ hyper ]

orthographic.projection6/3D increase in # of sample points

179

6-lattice.6D 64-tree

180

orthographic.projection6/3D increase in # of sample points

Rn [ hyper ]

5-lattice.5D 32-tree

Rn [ hyper ]

orthographic.projection4D. stereographic.projection3D increase in # of sample points

181

5-lattice.5D 32-tree

182

stereographic.projection4D. orthographic.projection3D increase in # of sample points

Rn [ hyper ]

6-lattice.6D 64-tree

Rn [ hyper ]

orthographic.projection6/3D. section.cut3D increase in # of sample points

183

6-lattice.6D 64-tree

184

orthographic.projection6/3D. section.cut3D increase in # of sample points

Rn [ hyper ]

6-lattice.6D 64-tree

Rn [ hyper ]

orthographic.projection6/3D. section.cut3D increase in # of sample points

185

6-lattice.6D 64-tree

186

orthographic.projection6/3D. section.cut3D increase in # of sample points

Rn [ hyper ]

6-lattice.6D arbitrary initial rotation

orthographic.projection6/3D tracing rotation in around specified axes

< XU, YU, ZU, UV, UW >

< XV, YV, ZV, UV, VW >

< XW, YW, ZW, UW, VW >

< XY, XZ, XU, XV, XW >

Rn [ hyper ]

187

6-lattice.6D arbitrary initial rotation

orthographic.projection6/3D tracing rotation in around specified axes

< XY, YZ, YU, YV, YW>

< XZ, YZ, ZU, ZV, ZW >

< ALL >

< UV, UW, VW >

188

Rn [ hyper ]

6-lattice.6D arbitrary initial rotation

< VW >

< XU >

< XV, YV >

< XV >

Rn [ hyper ]

orthographic.projection6/3D tracing rotation in around specified axes

189

6-lattice.6D arbitrary initial rotation

orthographic.projection6/3D tracing rotation in around specified axes

< XW, VW>

< XW, YW, VW >

< XZ, YU, VW >

< XW, YW, ZW, VW >

190

Rn [ hyper ]

6-lattice.6D arbitrary initial rotation

orthographic.projection6/3D tracing rotation in around specified axes

< XY, XU >

< XV, XZ, YZ, UV, UW, VW >

< XY, XZ, YZ, UV, UW >

< XY, XZ, YZ, UV >

Rn [ hyper ]

191

6-lattice.6D arbitrary initial rotation

orthographic.projection6/3D tracing rotation in around specified axes

< XY, XZ, YZ >

< XZ, XZ >

< XY, YZ >

< XY >

192

Rn [ hyper ]

6-lattice.6D arbitrary initial rotation

< XZ, ZU >

< XZ >

< YU, YV, UV >

< YU >

Rn [ hyper ]

orthographic.projection6/3D tracing rotation in around specified axes

193

6-lattice.6D arbitrary initial rotation

orthographic.projection6/3D tracing rotation in around specified axes

< YV, UV>

< YV, ZV, UV >

< YV, ZV >

< YV >

194

Rn [ hyper ]

6-lattice.6D arbitrary initial rotation

< YZ, YU, YV, UV >

< YZ >

< ZU >

< XY, ZU, VW >

Rn [ hyper ]

orthographic.projection6/3D tracing rotation in around specified axes

195

600-cell.4D # of pts: 120

196