PLATONIC SOLIDS AND UNIVERSAL SYMMETRIES

GOLDEN RATIO: THE STORY OF PHI, THE WORLD’S MOST ASTONISHING NUMBER, by Mario Livio (Broadway Books, NY:2002), PP.67-68, 70 “In Timaeus, Plato takes on the immense task of discussing the origin and workings of the cosmos. In particular, he attempts to explain the structure of matter using the five regular solids (or polyhedra), which had been investigated already to some extent by the Pythagoreans and very thoroughly by Theaetatus. The five Platonic solids are distinguished by the following properties: They are the only existing solids in which all the faces (of a given solid) are identical and equilateral, and each of the solids can be circumscribed by a sphere (with all its vertices lying on the sphere). The Platonic solids are the tetrahedron (with four triangular faces); the cube (with six square faces); the octahedron (with eight triangular faces); the dodecahedron (with twelve pentagonal faces); and the icosahedron (with twenty triangular faces). … “To Plato, the complex phenomena that we observe in the universe are not what really matter; the truly fundamental things are the underlying symmetries, and those are never changing. This view is very much in line with modern thinking about the laws of nature. For example, these laws do not change from place to place in the universe. For this reason, we can use the same laws that we determine from laboratory experiments whether we study the hydrogen atom here on earth or in a galaxy that is a billion light-years away. This symmetry of the laws of nature manifests itself in the fact that the quantity which we call linear momentum (equaling the product of the mass of an object and the speed, and having the direction of the motion) is conserved, namely, has the same value whether we measure it today or a year from now. Similarly, because the laws of nature do not change with the passing of time, the quantity we call energy is conserved. We cannot get energy out of nothing. Modern theories, which are based on symmetries and conservation laws, are thus truly Platonic.” Platonic Solids www.mathsisfun.com