Icosahedron

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Data on various polyhedra are available at

http://cm.bell-labs.com/netlib/polyhedra/index.html

http://netlib.bell-labs.com/netlib/polyhedra/index.html

http://www.netlib.org/polyhedra/index.html

Twelve vertices for a regular icosahedron with edge length 2/Φ are:

(±1/Φ, 0, ±1) (0, ±1, ±1/Φ) (±1, ±1/Φ, 0)

where Φ = (1+√5)/2 is the golden ratio.

Twenty vertices for a regular dodecahedron can be found as the face centers of an icosahedron, or for edge length 2/Φ as

(0, ±1/Φ, ±Φ) (±Φ, 0, ±1/Φ) (±1/Φ, ±Φ, 0) (±1, ±1, ±1)

Five distinct cubes of edge length 2 can be found taking eight dodecahedron vertices at a time, the obvious example being

(±1, ±1, ±1)

Six vertices for a regular octahedron can be found as the face centers of a cube; for edge length √2,

(±1, 0, 0) (0, ±1, 0) (0, 0, ±1)

Four vertices for a regular tetrahedron of edge length √2 can be found by taking alternate vertices of a cube, in two ways, one being

(1, 1, 1) (-1, -1, 1) (-1, 1, -1) (1, -1, -1)

The regular icosahedron, dodecahedron, cube, octahedron, and tetrahedron form the complete list of Platonic solids, the regular convex solids in 3D.

The list of semi-regular polyhedra, essentially the Archimedean solids, is somewhat longer, and properly includes two infinite families (prisms and antiprisms). The Archimedean vertices, and those of the Platonic solids, all can be generated from a single vertex by the action of a finite rotation group and (for some) an inversion (coordinate negation). Two of the Archimedean solids, generated by rotations alone, have distinct mirror images, not counted in the usual tally of 13. The most complex member, with 120 vertices and an edge length of 2/Φ, can be generated using inversion, the icosahedral rotation group given below, and the initial vertex (Φ, 2Φ, 3).

When finite 3D rotations groups are represented as quaternions, some of them form regular convex polychora (hyper-polyhedra), the 4D equivalent of Platonic solids. For example, the rotational symmetry group of the icosahedron is itself a hyper-icosahedron called the 600-cell, with 120 vertices, each a unit quaternion.

even permutations of

(0, 0, 0, ±1) (±1, ±1, ±1, ±1)/2 (0, ±1, ±1/Φ, ±Φ)/2

For tetrahedron rotations, omit the vertices involving Φ to get a 24-cell.

all permutations of

(0, 0, 0, ±1) (±1, ±1, ±1, ±1)/2

For octahedron rotations, augment the tetrahedral group.

all permutations of

(0, 0, 0, ±1) (±1, ±1, ±1, ±1)/2 (0, 0, ±1, ±1)/√2

Larger polyhedral families exist, many detailed by George Hart.

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