Platonic Ratio
A presentation by Det Springende Punkt
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Welcome to the temple of arithmetics! We have travelled 500/ 1,500/ 2,500 years (Gregor Reisch / Boethius / Pythagoras) back in time. The woodcut is from Margarita Philosophica (The Pearl of Philosophy ) from 1503 by Gregor Reisch. Here we meet Boethius to the left and Pythagoras to the right. Plenty of attention has been given to the interpretation of the image content as a showdown between abacus (right) and Arabic numerals (left). Frankly: In an illustration of two men and one woman - in this case even a muse or a goddess - whom is the most worthy of our focus?
... yes, it's almost too easy, after all she is the central figure! But what should we pay attention to? It is not necessarily just a platitude to point out that one should not dwell on what runs through the woman's head (it says 'Typys Arithmeticae'), and the content of her hands also just reflects these two fundamental ways of calculating. No, what is interesting about women and goddesses, is what springs forth from their lap; the fertile, the life-giving! These aspects even a Carthusian prior in times of great upheaval may be sentient of! Gregor Reisch was born about 1467, died 1525 and was active in southern Germany and Switzerland.
We need to go below the belt to find the arithmetic gem, the two sequences 1-2-4-8 and 1-3-9-27, gushing forth from the muse. We note that 1 + 2 + 3 + 4 + 8 + 9 = 27 From a perspective of music the two sequences illustrate the three octaves (ratio 1:2), from 1 to 8 and a process of perfect fifths (perfect twelfths, ratio 1:3) respectively. This creation matrix has its roots at least back to the time of Plato, and it is worthwhile to consider that throughout his works he refers to musical symbolism. When in The Republic he appoints 729 as the 'number of the tyrant' it relates to the same progression, as it is an extension of the 3-series (729 = 36), forming the highly dissonant tritone interval. The language of music WAS numbers. A real music notation came later.
... and here's Plato's Genesis animated and set to music! It should be mentioned that the ancient Greeks would have interpreted the two sets of figures as strings lengths, and thus would have come up with a tone row whose ranges are inverse of those where we have allowed to interpret as frequency values. However, this is equalled out as the ancient Greeks conceived music as flowing from the top down - from octave to prime - and often interpreted the numbers as fractions of 1.
Regarding the geometric aspect of the two number sequences, one should keep in mind that in Plato's time a strict and unambiguous mathematical language and notation had not yet been developed. It is obvious that they conceived number figures musically and geometrically. The two sequences have thus been perceived as the first odd and the first even number and their respective squares and cubes. In the Pythagorean understanding that Plato was referring to 1 was not a real number, but unity. Kepler's (1571-1630) third law of planetary motion is an example that it may be fertile to reflect on the square and cube numbers when contemplating the relations of universe, time and space: "The square of the distance is proportional to the cube of the orbital period." The cube of two (23) is actually a perfect illustration of the binary language used in computers, where the information unit byte consists of eight bits.
Here's the remains of Plato's Academy in Athens. You will hardly find a single cubic stone left on that square!
... but ...
... here is a central section of Raphael's (1483-1520) fresco The School of Athens from 1511, where Plato - Leonardo da Vinci was used as his model - is pointing towards the immaculate realm of ideas while he carries his cosmological work Timaeus, which was largely influenced by the Pythagorean tradition.
The demiurge combined three elements: - two varieties of Sameness (one indivisible and another divisible), - two varieties of Difference (again, one indivisible and another divisible), - and two types of Being (or Existence, once more, one indivisible and another divisible).
In this dialogue from around 360 BCE. it appears that the world builder, the demiurge, created the world on the basis of three different components, three kinds of being:
From this emerged three compound substances, intermediate (or mixed) Being, intermediate Sameness, and intermediate Difference. From this compound one final substance resulted, the World Soul.
And he proceeded to divide after this manner: -First of all, he took away one part of the whole [1], and then he separated a second part which was double the first [2], and then he took away a third part which was half as much again as the second and three times as much as the first [3], and then he took a fourth part which was twice as much as the second [4], and a fifth part which was three times the third [9], and a sixth part which was eight times the first [8], and a seventh part which was twenty-seven times the first [27]. The illustration is a Platonic lambda chart from Franchino Gaffurio's (1451-1522) work, Theorica Musica, from 1492, perhaps just as groundbreaking as Columbus' discovery of the same year ...
... and it contains mysteries which run deeper than the flourishing ideas about hidden codes in da Vinci's paintings. Here is one of them, supposedly a portrait of Franchino Gaffurio with sheet music, painted around 1490. Franchino and Leonardo met in Milan, where Franchino was Maestro di Capella at the cathedral. Read, watch and listen to more of da Vinci's Music Code.
Here is a woodcut showing Franchino with twelve (tonal) students, and it is probably not too farfetched to associate it with a mural by Leonardo created in a refectory just around the corner. With the organ pipes, numbers and compass there should be no doubt about that music, arithmetics and geometry can meet! More about Franchino and the image at Seven Liberal Arts - Music
Do you know your classical Pythagorean means by heart? Do not despair, the concept has slipped out of daily usage, but the old Pythagorean, Platonists and Aristotelians cultivated the understanding of proportions as important concepts of how the world may be created and bound together. Franchinos numbers, 3:4:6, is an example of 4 as harmonic mean between 3 and 6 But let's take it all from the beginning:
The geometric means - Such as 2:4:8 and 3:9:27 have the property that the smallest part, a, is related to the intermediary, b, in the same manner as the intermediary, b, is related to the large part, c Or more mathematically: a:b::b:c
In other words, one finds the geometric mean, b, between the extremes, a and c, by: b =√ac Example: 4 = √(2x8). 4 is therefore the geometric mean between 2 and 8 The ancient Greeks called this proportion for logos, and in Plato's thinking it was at the basis of the creation of the world! You may read more about what it also may reflect, not at the beginning of this presentation but at the end! ;-) The golden proportion is a special case within the geometric divisions, where a:b = b:(a+b). Here the greater part, c, is the sum of the two smaller. Franchino's woodcut displays the harmonic mean 4 within the frame of the octave 3:6: b = 2 : (1 / a + 1 / c). 4 = 2 (1/3 + 1/6) This also corresponds to the sequence 6:8:12
The three classical Pythagorean means:, 'the harmonic', 'the geometric' and the 'arithmetic' here illustrated as lengths within the octave 6:12
Finally, the sequence 6:9:12 is an example of the arithmetic mean: b = (a + c) : 2, as 9 = (6 +12) : 2
By pairing Plato's Lambda with another Pythagorean structure, the tetractys, the pattern is completed. As illustrated the three classical means are included.
Pythagoras and Arithmetica presenting the arithmetic table which became well known through Nichomachus of Gerasa's work Introduction to Arithmetic The image is of unknown origin, from a work presenting the Seven Liberal Arts.
The understanding of the Pythagorean and Platonic perspectives were revived in Italy during the Renaissance. Cosimo de Medici (1389-1464, right) gave Marsilio Ficino (1433-99, left) his support for the creation of a Platonic Academy in Florence. Now we have traveled back in time to Switzerland, Italy and ancient Greece. So let us now take a journey through the dimensions of space in an almost timeless sequence:
Pre-dimensional, point: One basic geometrical element
The first dimension, line: Two (end-) points and one line between them. Total: 2 + 1 = 3 basic geometric elements.
Two dimensions, square: Four (corner-) points between them, framed by the four edges: one plane. Total: 4 + 4 + 1 = 9 basic geometric elements.
Three dimensions, cube: Eight corner points between them the twelve edges frame the six faces, which together define one space. Total: 8 + 12 + 6 + 1 = 27 basic geometric elements.
Four dimensions hypercube: When the figure - as here - is rendered in two dimensions, one must take into account that it is 'the shadow of a shadow' of the actual figure. Sixteen points, thirty-two edges, twenty-four faces, eight spaces and one hyperspace. A total of 16 + 32 + 24 + 8 + 1 = 81 basic geometric elements.
... Still the figure is displayed in two dimensions, but the animation (time dimension) helps to form an idea of the hidden space dimension. It's probably perfectly normal to react with a mixture of joy, dizziness and nausea!
Plato's lambda matrix has several facets, primarily a starting point for understanding the basic musical structure, but it is not being referred much to any longer. Here the attention is drawn to the last two columns and the demonstrated relation between dimensionality and the number of geometric elements, which, as far as is known, is a discovery of the Det Springende Punkt, with reminiscent of Euler's Polyhedra Formula. As can be seen, one can drive the progression beyond the three dimensions (and yet further), while maintaining the pattern. This progression through basic geometric shapes is of course only one of several possible: Instead of square and cube we could have travelled through the two and three dimensions by trigon and tetrahedron to a hyper-tetrahedron, which would have given a different result. Essential is the evident and fundamental coupling between numbers, geometry and music; processes in time and space .... And may be after all it wasn't so crazy by the ancient Pythagoreans to consider the world's genesis a result of the meeting between the first 'female' and the first 'male' figures 2 and 3.
The Cantor set is a fractal pattern which displays both the 1-2-4-8-... progression and the 1-3-9-27-... progression From bottom up each level consists of 1, 2, 4, 8 etc. white squares. At the same time the width of the full figure is constituted by in all 1, 3, 9, 27, ... units (black and white squares).
... and with remarkable complementarity the Sierpinski triangle: Its axial black triangles cover 1/2, 1/4, 1/8,... etc. of the height. And in the middle is 1 large triangle surrounded by 3 smaller, 9 in the next cathegory, followed by 27 tiny ones etc.
It is tempting to return to the Temple of Arithmetic, which seen from a compositional view is characterized by the trigonal and the square (3 and 2x2 respectively), but where the perspective obviously gave the artist some trouble. So we turn to Rafael again, who truly mastered it, also because he was an architect. In his plate the maternal womb is also prominent. The thigh of the Immaculate Conception forms the center of the composition. From the spiritual heights - the trigon - she carries the living word of God into the square of matter flanked by the holy Sixtus and Barbara. The base of the trigon rests on the center of the square. The Sistine Madonna refer to the first stanzas of John's gospel: "In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning. Through him all things were made; without him nothing was made that has been made. In him was life, and that life was the light of men. The light shines in the darkness, but the darkness has not understood it. (...) The true light that gives light to every man was coming into the world. He was in the world, and though the world was made through him, the world did not recognize him. He came to that which was his own, but his own did not receive him. (...) The Word became flesh and made his dwelling among us."
... and these verses refer to God's creator words from the Old Testament:
"Having pervaded the whole universe with a fragment of Myself, I remain."
FIAT LUX! ('let there be light!')
(Krishna in the Bhagavad Gita, X, 42)
The illustration is by Robert Fludd (1574-1637).
The 'Word' of in John's gospel is the Greek LOGOS, which was borrowed from the Platonic tradition. It carries with it the meaning of ratio, proportion, intermediary. If by all measures it is viable to assign these kind of attributes to a carnal being is probably a crucial factor, but here and now the errand is not to challenge or run a mission for other peoples' religion, but to throw sidelights and finish the circle! Moreover, one can find passages with related meaning in other religious traditions.