Platonic Solids

A Platonic solid is a convex polyhedron. Platonic solids are made up of equal faces and are made up of congruent regular polygons. There are 5 Platonic solids. They are named for the number of faces: tetrahedron - 4 faces, hexahedron - 6 faces, octahedron - 8 faces, dodecahedron - 12 faces, and icosahedron - 20 faces. The ancient Greeks believed that these 5 Platonic solids symbolized the elements, with the dodecahedron symbolizing the heavens.

The Platonic Solids belong to the group of geometric figures called polyhedra.

A polyhedron is a solid bounded by plane polygons. The polygons are called faces; they intersect in edges, the points where three or more edges intersect are called vertices.

A regular polyhedron is one whose faces are identical regular polygons. Only five regular solids are possible: cube, tetrahedron, octahedron, icosahedron, dodecahedron. These have come to be known as the Platonic Solids

The Elements Linked to the Platonic Solids

Plato associates four of the Platonic Solid with the four elements. He writes,

We must proceed to distribute the figures the solids we have just described between fire, earth, water, and air. . .

Let us assign the cube to earth, for it is the most immobile of the four bodies and most retentive of shape

• the least mobile of the remaining figures (icosahedron) to water
• the most mobile (tetrahedron) to fire
• the intermediate (octahedron) to air

Note that earth is associated with the cube, with its six square faces. This lent support to the notion of the four-squaredness of the earth.

[[Archimedean Solids]]

Below is the Greek text and an English translation from the fifth book of the “Synagoge” or “Collection” of the Greek mathematician Pappus of Alexandria, who lived in the beginning of the fourth century AD. This book gives the first known mention of the thirteen “Archimedean solids”, which Pappus lists and attributes to Archimedes. However, Archimedes makes no mention of these solids in any of his extant works.

The earliest surviving manuscript of Pappus’s “Collection” is located in the Vatican Library and dates from the tenth century (Codex Vaticanus Graecus 218). A photograph of a pair of pages from this manuscript can be downloaded from a Web site of the Library of Congress Vatican Exhibit.

Johannes Kepler (1571-1630) was the next to write about the Archimedean solids collectively in his book Harmonices Mundi, although some of the solids were separately rediscovered and discussed by others. Kepler sharpened Pappus’s somewhat loose definition of the solids and gave a proof that there are precisely thirteen of them (Book II, “De Congruentia Figurarum Harmonicarum”, Proposition XXVIII, pages 61-65. He also provided the first known illustration of them as a set (see pages 62 & 64) and gave them their modern names, which are reproduced below. Other representations and properties of these solids can be found at Wikipedia and a site maintained by Tom Gettys.

Pappus’s narration begins . . .

![[img/sg_img-164.png]]	Truncated Tetrahedron	The first is a figure of eight bases, being contained by four triangles and four hexagons.
![[img/sg_img-165.png]]	Cuboctahedron	After this come three figures of fourteen bases, the first contained by eight triangles and six squares,
![[img/sg_img-166.png]]	Truncated Octahedron	the second by six squares and eight hexagons,
![[img/sg_img-167.png]]	Truncated Cube	and the third by eight triangles and six octagons.
![[img/sg_img-168.png]]	Rhombicuboctahedron	After these come two figures of twenty-six bases, the first contained by eight triangles and eighteen squares,
![[img/sg_img-169.png]]	Truncated Cuboctahedron	the second by twelve squares, eight hexagons and six octagons.
![[img/sg_img-170.png]]	Icosidodecahedron	After these come three figures of thirty-two bases, the first contained by twenty triangles and twelve pentagons,
![[img/sg_img-171.png]]	Truncated Icosahedron	the second by twelve pentagons and twenty hexagons,
![[img/sg_img-173.png]]	Snub Cube	After these comes one figure of thirty-eight bases, being contained by thirty-two triangles and six squares
![[img/sg_img-174.png]]	Rhombicosidodecahedron	After this come two figures of sixty-two bases, the first contained by twenty triangles, thirty squares and twelve pentagons,
![[img/sg_img-175.png]]	Truncated Icosidodecahedron	the second by thirty squares, twenty hexagons and twelve decagons.
![[img/sg_img-176.png]]	Snub Dodecahedron	After these there comes lastly a figure of ninety-two bases, which is contained by eighty triangles and twelve pentagons.