Plato Part Two: The Timaeus

Plato’s Timaeus represents his most difficult work. The passages that are most pertinent to our interest here, are given below, and illustrated accordingly. While it is certainly possible that there may numerous approaches to an illustration of what is intended from these complex passages, what is offered here, reveals that it is possible to make a semblance of sense out these paragraphs, with the isometeric matrix in mind. The most difficult portion of the text reads as follows. this is followed by an elaboration of the elements:

From 34-36

Out of the indivisible and unchangeable (signatures of light and motion?) and out of that which is divisible and has to do with material bodies (concrete elements of the natural world including artifacts), he compounded a third and intermediate kind of essence (frame of reference?), partaking in the  nature of the same (planetary domain?) and of the other, (sun?) and this compounded he placed accordingly in a mean between the indivisible and the divisible and material. He took the three elements of the same (triune unity of the source of light, motion and mind as gnomic projection?), the other (light?) and the essence (motion?) and mingled them into one form, compressing by force, the reluctant and unsociable nature of the other to the same (topological translation?). When he had mingled them with the essence and out of three made one (Eye of God Geometry?) he again divided this whole into as many portions as was fitting (all manner of spoked wheels, shapes, forms, and structural designs), each portion being a compound of the same, the other, and the essence. 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 of 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].”

After this he filled up the double intervals [that is, between 1,2,4,8] and the triple [that is between 1,3,9,27], cutting off yet other portions from the mixture and placing them in the intervals so that in each interval there were two kinds of means, the one exceeding and exceeded by equal parts of its extremes [as for example 1,4/3,2, in which the mean 4/3 is one-third of 1 more than I, and one third of 2 less than 2], the other being that kind of mean which exceeds and is exceeded by an equal number. Where the were intervals of 3/2 and of 4/3 and of 9/8, made by connecting them in the former intervals, he filled up all the intervals of 4/3 with the interval of 9/8. leaving a fraction over; and the interval which is this fraction expressed was in the ratio of 256 to 243.”

“This entire compound he divided lengthways into two parts, which he joined into one another at the center like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting point; and comprehending them into a uniform revolution upon the same axis, he made the one the outer and the other  the inner circle. Now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the other or diverse.”

“The motion of the same he carried round by the side (note given: i.e of the rectangular figure supposed to be inscribed in the circle of the same.) to the right, and the motion of the diverse (note given: i.e. across the rectangular figure from corner to corner) diagonally to the left. And he gave dominion to the motion of the same and like, for that he left single and undivided; but the inner motion he divided in six places and made seven unequal circles having their ratios of two and three, three of each and bade the orbits proceed in a direction opposite to one another, and three [Sun, Mercury, Venus] he made to move with equal swiftness, and the remaining four [Moon, Saturn, Mars, Jupiter] to move with unequal swiftness to the three and to one another, but in due proportion.

From 54 -55

“Now is the time to explain what was before obscurely said: there was a  error in imagining that all the four elements might be generated by and into one another; that is to say, was an erroneous supposition, for there are generated from the triangles which we have selected four kinds- three from the one which has the sides unequal; the fourth alone is framed out of the isosceles triangle. Hence they cannot all be resolved into one another, a great number of small bodies being combined into a few large ones, or the converse. But three of them can be thus resolved and compounded, for they all spring from one, and when the greater bodies are broken up, many small bodies will spring up out of them and take their own proper figures; or again, when many small bodies are dissolved into their triangles, if they become one, they will form one large mass of another kind. So much for their passage into one another. I have now to speak of their several kinds, and show out of what combinations of numbers each of of them was formed. 

“1. the first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side. 2. When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest on their diagonals and shorter sides on the same point as a center, a single equilateral triangle is formed out of six triangles: 3. and four equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles; [55] 4. and out of the combination of these four angles arises the first solid form which distributes into equal and similar parts the whole circle in which it is inscribed. 5. The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. 6.  And the third body is made up of 120 triangular elements, forming 12 solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle. The one element [that is the triangle which has its its hypotenuse twice the lesser side] having generated these figures, generated no more; 7. but the isosceles triangle produced the  fourth elementary figure, which is compounded of four such triangles, joining their right angles to the center, and forming one equilateral quadrangle. 8. Six of these united from eight solid angles, each of which is made by the combination of three plane right angles; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases.”

There are numerous conclusions which may be drawn from this passage. These illustrations provide a translation which is coherent with regard to the historical progression of the knowledge system. The certainty of this process is constantly informed by the historical  progress made in theoretical science, and comparison with other models, products and culturally determined cosmologies, which constitutes an “archaeology” of ancient cosmological models, and their formative influence on the social sphere.

This analysis shows numerous things. That Kepler’s model of the embedded Platonic solids is entirely speculative and flawed. That Plato’s process of apprehending the structural attributes of the pentadic structure, appears to be a “cut and paste” process, that fails to realise the location of the five radials that inform the five pointed star, their formative influence, and the importance of the logo used by Pythagoreans as an indication of the fact that the geometer knew how to construct the isometric matrix to the point of density which supports the 72, 60, 30, 15 and 5 spoked wheels. His project is one that exhibits a process of “reverse engineering”, established on the knowledge given to him, by the High Priest of Heliopolis. The use of the name Timaeus, is a direct reference to the last ruler who witnessed the incursion of the Hyksos, an historical notation that returns after Plato’s death, in the public domain, through the writings of Josephus who relies heavily on the history of Egypt written by Manetho.

We began our treatment of Plato with a return to the so called “Prophecy of Neferti”, which constitutes an historical document referencing an address by a Noble, to the Fourth Dynasty Ruler named “Seneferu”. We also considered reasons for the social discontect of his time, which remained a recurrent theme in extant Egyptian documents. In order to reach into the “swampland” of our distant past, as Karl Popper put it, for the purpose of addressing the same issues which arise from the provision of a geometrical system which fails to make explicit its formative influence based on elementary propositions, which constitute the authority of signatures of light and motion, which remain outside the sphere of human influence, and inside the sphere of human experience, that are then translated into the superstructure which informs the building of the isometric matrix, we are compelled to recognise the elements which inform the ancient “Hypostyle”  fashion, embodied in Fourth Dynasty Architecture, and employed as the foundational style in Ancient Greek architecture, for it is the embodiment of the geometrical model and its products, which informs the progression of the applied sciences, and its application to the social sciences, which remain the structural support for both civil and social engineering, and the progression of technologies, whose formative influence underpins the failure to explicate foundations with reason and logic, and the redirection towards the imaginary domain of first and final causes, which remains inside the sphere of human influence, and outside the sphere of common human experience due to the fact that without disclosure, and indeed the induction of the child to integral foundations, a society functions as a closed system, directed by a hidden elite, who represent a hidden god, under the auspices that there is nothing which remains outside the sphere of human influence, that is significant. As such, signals communicated through various forms of media, poetry, drama, comedy, and all tropologically considered communications, remain subject to the overarching belief that either recognizes the authority of the signatures themselves, as remarkable, and hence outside the sphere of human influence, or indeed not, reducing the formative influences of the knowledge system which encompasses both the theoretical and the applied, to merely man made constructions, indeed as if Plato was the One to whom, all other work regarding geometry, remains as Alfred North Whitehead proclaimed, merely a footnote to Plato’s Timaeus.