Platonic and
Archimedean Polyhedra
The Platonic Solids, discovered by the Pythagoreans but described by Plato (in the Timaeus) and used by him for his theory of the 4 elements, consist of surfaces of a single kind of regular polygon, with identical vertices. The Archimedean Solids, consist of surfaces of more than a single kind of regular polygon, with identical vertices and identical arrangements of polygons around each polygon. In the following table, the Platonic Solids are indicated in red and the Archimedean Solids in green, blue, and purple. Green is for solids that can be produced by truncating the vertices of either Platonic or the blue Archimedean solids. Blue Archimedean Solids are produced from green ones by continuing the trucation until edges disappear and half the vertices merge. Pairs of Archimedean Solids become identical in that procedure. Purple Archimedean Solids result when, in the five triangles per vertex of the Platonic Icosahedron, one triangle is replaced by either a square (the Snub Cube) or a pentagon (the Snub Dodecahedron). The purple Archimedean solids have the interesting property of having right-handed and left-handed forms.
For information about all this, George W. Hart's "Virual Polyhedra" Site is wonderful; and I first learned about Archimedean solids from The Penguin Dictionary of Curious and Interesting Geometry, by David Wells (Penguin Books, 1991). Now there is a nice little book, Platonic & Archimedean Solids by Daud Sutton (Wooden Books, Walker & Company, New York, 2002), that covers the solids with many related facets of their geometry.
| Platonic and Archimedean Polyhedra |
|---|
| Solid | vertices | faces | faces/
vertex | edges | polygons |
|---|
| P1 |  | 4 vertices | 4 faces | 3 faces/
vertex | 6 edges | 4 triangles
(3 triangles /vertex) |
| A1 |  | 12 vertices | 8 faces | 3 faces/
vertex | 18 edges | 4 hexagons,
4 triangles
(2 hexagons & 1 triangle /vertex) |
| P2 |  | 6 vertices | 8 faces | 4 faces/
vertex | 12 edges | 8 triangles
(4 triangles /vertex) |
| P3 |  | 8 vertices | 6 faces | 3 faces/
vertex | 12 edges | 6 squares
(3 squares /vertex) |
| A2 |  | 24 vertices | 14 faces | 3 faces/
vertex | 36 edges | 8 hexagons,
6 squares
(2 hexagons & 1 square /vertex) |
| A3 |  | 24 vertices | 14 faces | 3 faces/
vertex | 36 edges | 8 triangles,
6 octagons
(2 octagons & 1 triangle /vertex) |
| A4 |  | 12 vertices | 14 faces | 4 faces/
vertex | 24 edges | 8 triangles,
6 squares
(2 triangles & 2 squares /vertex) |
| A5 |  | 48 vertices | 26 faces | 3 faces/
vertex | 72 edges | 6 octagons,
8 hexagons,
12 squares
(1 octagon, 1 hexagon, & 1 square /vertex) |
| A6 |  | 24 vertices | 26 faces | 4 faces/
vertex | 48 edges | 18 squares,
8 triangles
(3 squares & 1 triangle /vertex) |
| A7d |  | 24 vertices | 38 faces | 5 faces/
vertex | 60 edges | 6 squares,
32 triangles
(1 square & 4 triangles /vertex) |
| A7s |  | 24 vertices | 38 faces | 5 faces/
vertex | 60 edges | 6 squares,
32 triangles
(1 square & 4 triangles /vertex) |
| P4 |  | 12 vertices | 20 faces | 5 faces/
vertex | 30 edges | 20 triangles
(5 triangles /vertex) |
| P5 |  | 20 vertices | 12 faces | 3 faces/
vertex | 30 edges | 12 pentagons
(3 pentagons /vertex) |
| A8 |  | 60 vertices | 32 faces | 3 faces/
vertex | 90 edges | 20 hexagons,
12 pentagons
(2 hexagons & 1 pentagon /vertex) |
| A9 |  | 60 vertices | 32 faces | 3 faces/
vertex | 90 edges | 12 decagons,
20 triangles
(2 decagons & 1 triangle /vertex) |
| A10 |  | 30 vertices | 32 faces | 4 faces/
vertex | 60 edges | 12 pentagons,
20 trangles
(2 pentagons & 2 triangles /vertex) |
| A11 |  | 120 vertices | 62 faces | 3 faces/
vertex | 180 edges | 12 decagons,
20 hexagons,
30 squares
(1 decagon, 1 hexagon, & 1 square /vertex) |
| A12 |  | 60 vertices | 62 faces | 4 faces/
vertex | 120 edges | 12 pentagons,
30 squares,
20 triangles
(1 pentagon, 2 squares, & 1 triangle /vertex) |
| A13d |  | 60 vertices | 92 faces | 5 faces/
vertex | 150 edges | 12 pentagons,
80 triangles
(1 pentagon & 4 triangles /vertex) |
| A13s |  | 60 vertices | 92 faces | 5 faces/
vertex | 150 edges | 12 pentagons,
80 triangles
(1 pentagon & 4 triangles /vertex) |
Johannes Kepler was the first person since antiquity to systematically describe all the Archimedean solids. However, he made one mistake. While the Great Rhombicuboctahedron certainly looks like a Truncated Cuboctahedron, and the Great Rhombicosidodecadhedron a Truncated Icosidodecahedron, which is what Kepler called them, mere truncation does not produce perfectly regular polygons on the surfaces. A little stretching is necessary. I have organized the table above as though Kepler was right, but this ends up being a little deceptive.
Several Archimedean solids can be broken down into parts that can be rotated against each other to produce new polyhedra with less symmetry. All of these rotations will also produce some vertices with different arrangements of the constituent polygons except one, the "pseudo-rhombicuboctohedron," derived from the rhombicubotohedron, where the arrangement of all the vertices is retained (but there are differing arrangements of the polygons around each square).
| 4 Dimensional "Platonic" Polytopes |
|---|
| Polytope | cells | vertices | edges | faces | duals |
|---|
| 1. 5-cell, Pentatope or Simplex | tetrahedra | 5 | 10 | 10 | self-dual |
| 2. 8-cell, Tesseract or Hypercube | cubes | 16 | 32 | 24 | 16-cell |
| 3. 16-cell | tetrahedra | 8 | 24 | 32 | 8-cell |
| 4. 24-cell | octahedra | 24 | 96 | 96 | self-dual |
| 5. 120-cell | dodecahedra | 600 | 1200 | 720 | 600-cell |
| 6. 600-cell | tetrahedra | 120 | 720 | 1200 | l20-cell |
In four dimensions, the five Platonic Solids have six analogues. Polyhedra become "polytopes." The Pentatope and the Tesseract are relatively easy to understand, and illustrate with projections, as analogues of the Tetrahedron and the Cube. Robert Heinlein's science fiction story, "There Was a Crooked House," is based on the Tesseract. Curiously, there is no 3-d analogue to the 24-cell.
The famliar Pentagram is, strangely enough, a two dimensional (2-D) projection of the Pentatope. Since a Pentatope contains five Tetrahedra, it should be possible to find five distinct two dimensional projections of a Tetrahedron in the projection of the Pentatope. In the diagram at right this can be seen. Highlighted in red are each of the five Tetrahedra, with an independent red Tetrahedron for comparison. While it seems like this should be excellent fuel for fantasy or science-fiction connections between higher dimensional reality and occult practices, I have not noticed any such use of it that way. Even better, if the red lines are taken to be the projection of a Square with two diagonals, then the black lines can make each drawing the projection of a Pyramid.
| n-Dimensional "Platonic" Polytopes, n > 4 |
|---|
| Polytope | number of (n-1) D cells | vertices | duals | 3-d analogue |
|---|
| 1. (n + 1) cell | n + 1 n-cells | n + 1 | self-dual | Tetrahedron |
| 2. 2n-cell | 2n (2n-2)-cells | 2n | 2n-cell | Cube |
| 3. 2n-cell | 2n n-cells | 2n | 2n-cell | Octahedron |
In all dimensions greater than four, there are exactly three analogues to the Platonic Solids. This is, curiously, exactly half the forms we find in 4 dimensions.
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