If the numbers p, q, and r in the Wythoff symbol are integers, the corresponding polyhedron is convex. All convex uniform polyhedra have been known since antiquity.
The dodecahedron has the formula 3|2 5, therefore, the vertex configuration is {2, 5, 2, 5, 2, 5}. After removing trivial faces, simply {5, 5, 5} remains, that is, 3 pentagons. | |
The cuboctahedron has the formula 2|3 4, therefore, the vertex configuration is {3, 4, 3, 4}. | |
The great dodecahedron has the formula 5/2|2 5, therefore, the vertex configuration is {5, 5, 5, 5, 5}/2, that is, five pentagons, but each vertex is traversed twice (in the form of a pentagram). It is regular. |
The truncated tetrahedron has the formula 2 3|3, therefore, the vertex configuration is {2, 6, 3, 6}. After removing trivial faces, simply {6, 3, 6} remains, that is, two hexagons and one triangle. | |
The rhombicuboctahedron has the formula 3 4|2, therefore, the vertex configuration is {3, 4, 4, 4}. | |
The prisms are also in this class; they have the formula 2 n|2. They have vertex configuration {2, 4, n, 4}, or {4, 4, n} after removing trivial faces. Here, n=5. |
The truncated cuboctahedron has the formula 2 3 4|, therefore, the vertex configuration is {4, 6, 8}. | |
The great truncated cuboctahedron has formula 4/3 2 3|, therefore, the vertex configuration {8/3, 4, 6}, that is, a square, hexagon, and an eight-pointed star. |
The snub cube has the formula |2 3 4, therefore, the vertex configuration is {2, 3, 3, 3, 4, 3}. After removing trivial faces, {3, 3, 3, 3, 4} remains, that is, four triangles and one square. | |
The great snub dodecicosidodecahedron has the formula |5/3 5/2 3, therefore, the vertex configuration is {5/3, 3, 5/2, 3, 3, 3}, that is, two pentagrams (one of them retrograde) and four triangles. |
There is one uniform polyhedron that cannot be constructed
with Wythoff's method. It is the only one with more
than six faces around a vertex (it has 8 of them!).
It has vertex configuration {4, 5/3, 4, 3, 4, 5/2, 4, 3/2}.
It is denoted by the pseudo Wythoff symbol
(|3/2 5/3 3 5/2)
and named the Great Dirhombicosidodecahedron.
It can be found by a modification of
(|5/3 5/2 3).
The Great Dirhombicosidodecahedron consists of 40 triangles (20 of them retrograde), 60 hemispherical snub squares, and 24 pentagrams (12 of them retrograde). It also has the largest number of faces (124) and edges (240) of all uniform polyhedra. The squares come in 20 coplanar pairs, that is, two squares each lie in the same plane, rotated against each other by 38.17 degrees. |
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