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 [note].

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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The full pentagram provides a diagram for the suggested "Lizard-Spock Expansion" of the traditional "rock, paper, scissors" game. This was publicized in The Big Bang Theory in 2008 [episode 2:8]. Later the show acknowledged that this was developed by "internet pioneer" Sam Kass, who himself acknowledges the collaboration of Karen Bryla.

Ordinarily, (1) scissors cut paper, (2) paper covers rock, and (3) rock breaks scissors. With the addition of the lizard and (Mr.) Spock (from Star Trek), (4) rock crushes lizard, (5) lizard poisons Spock, and (6) Spock smashes scissors. However, we now have the occurrence of additional matches. Rock still breaks scissors, but now (7) scissors also decapitate lizard, (8) Spock vaporizes rock, (9) lizard eats paper, and (10) paper disproves Spock. The three actions of "rock, paper, scissors" expand to ten actions with the addition of two new moves.

One might ask, "Why expand the system by two instead of just one?" One reason would be the asymmetry of the result. If we only add "lizard," which gives us the square diagram at left, we see that two of the items defeat two others but that the other two only defeat one. This means that in playing the game it would be a very poor strategy ever to choose paper or lizard, which will be defeated twice as often as they can ever defeat another themselves. This asymmetry is inevitable given that each vertex of the square is the intersection of three lines. So each vertex must either defeat two or be defeated by two. Adding both lizard and Spock means that each vertex is the intersection of four lines, which can be divided evenly. Kass and Bryla devoted some thought to the Lizard-Spock Expansion.

The premise on The Big Bang Theory was that "rock, paper, scissors" did not provide enough alternative choices, resulting in too many ties; but then, when used, the "Expansion" resulted in everyone always choosing Spock. Given the preference of players for Spock, for symbolic and personal reasons, the best strategy would be to choose lizard or paper, both of which defeat Spock. However, since another player with similar understanding might choose lizard or paper, lizard is the best choice, since it defeats both Spock and paper.

Now we get further explansions of the game, for instance with the addition of the Zombie and the LHC (i.e. Large Hadron Collider -- the particle accelerator at Cern in Switzerland/France). This allows for three victories and three defeats for each gesture. The Zombie hand is a "limp-wristed gesture, with all fingers opened," while the LHC is a "fist with the thumb and index finger extended," like the hand shape for a gun, since the collider uses a "particle gun" to inject particles into the machine. Another large body of rules must be added, which I will not detail here. This is intriguing but all begins to seem like too much.

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The Chinese Elements and Associations, Note 1