The Timaeus is one of Plato's dialogues in which he gives an account of the construction of the universe by a rational craftsman, or Demiurge. The four “elements” (earth, air, fire, water) are associated with four of the five Platonic solids.
The dialogue discusses the five Platonic solids and argues that they are building blocks of nature, in some way. See for instance Platonic Solids and Plato’s Theory of Everything
In modern mathematics the Platonic solids are still a source of rich structure, via the McKay correspondence.
ADE classification and McKay correspondence
Dynkin diagram/ Dynkin quiver | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
---|---|---|---|---|
$A_{n \geq 1}$ | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group $SU(n+1)$ | |
D4 | Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ | quaternion group $2 D_4 \simeq$ Q8 | SO(8) | |
$D_{n \geq 4}$ | dihedron, hosohedron | dihedral group $D_{2(n-2)}$ | binary dihedral group $2 D_{2(n-2)}$ | special orthogonal group $SO(2n)$ |
$E_6$ | tetrahedron | tetrahedral group $T$ | binary tetrahedral group $2T$ | E6 |
$E_7$ | cube, octahedron | octahedral group $O$ | binary octahedral group $2O$ | E7 |
$E_8$ | dodecahedron, icosahedron | icosahedral group $I$ | binary icosahedral group $2I$ | E8 |
By way of this correspondence, in the speculative physics theory called M-theory on G2-manifolds, the Platonic solids do again serve to exhibit fundamental aspects of nature, even if in a more intricate way than Timaeus had imagined.
[35a] and in the fashion which I shall now describe. Midway between the Being which is indivisible and remains always the same and the Being which is transient and divisible in bodies, He blended a third form of Being compounded out of the twain, that is to say, out of the Same and the Other; and in like manner He compounded it midway between that one of them which is indivisible and that one which is divisible in bodies. And He took the three of them, and blent them all together into one form, by forcing the Other into union with the Same, in spite of its being naturally difficult to mix.
[35b] And when with the aid of Being He had mixed them, and had made of them one out of three, straightway He began to distribute the whole thereof into so many portions as was meet; and each portion was a mixture of the Same, of the Other, and of Being. And He began making the division thus: First He took one portion from the whole; then He took a portion double of this; then a third portion, half as much again as the second portion, that is, three times as much as the first; he fourth portion He took was twice as much as the second; the fifth three times as much as the third;
In the Science of Logic Hegel picks up this concept of Something and an Other. His students paraphrased him as attributing this to Timaeus:
EL§92Zusatz Gott hat die Welt aus der Natur des Einen und des Anderen (του ετεου) gemacht; diese hat er zusammengebracht und daraus ein Drittes gebildet, welches von der Natur des Einen und des Anderen ist. (vgl. Timaios, Steph. 34 f.)
Stanford Encyclopedia of Philosophy, Plato’s Timaeus
Wikipedia, Timaeus (dialogue)
Last revised on September 25, 2016 at 16:08:54. See the history of this page for a list of all contributions to it.