Omnitruncated polyhedron

Omnitruncated polyhedron

In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra.

All omnitruncated polyhedra are zonohedra. They have Wythoff symbol p q r | and vertex figures as 2p.2q.2r.

Contents

List of convex omnitruncated polyhedra

There are 3 convex forms. They can been seen as red faces of one regular polyhedron, yellow faces of the dual polyhedron, and blue faces at the truncated vertices of the quasiregular polyhedron.

Wythoff
symbol

p q r |
Omnitruncated polyhedron Regular/quasiregular polyhedra
3 3 2 | Uniform polyhedron-33-t012.png
Truncated octahedron
Uniform polyhedron-33-t0.png Uniform polyhedron-33-t1.png Uniform polyhedron-33-t2.png
Tetrahedron/Octahedron/Tetrahedron
4 3 2 | Uniform polyhedron-43-t012.png
Truncated cuboctahedron
Uniform polyhedron-43-t0.pngUniform polyhedron-43-t1.pngUniform polyhedron-43-t2.png
Cube/Cuboctahedron/Octahedron
5 3 2 | Uniform polyhedron-53-t012.png
Truncated icosidodecahedron
Uniform polyhedron-53-t0.pngUniform polyhedron-53-t1.pngUniform polyhedron-53-t2.png
Dodecahedron/Icosidodecahedron/Icosahedron

List of nonconvex omnitruncated polyhedra

There are 5 nonconvex uniform omnitruncated polyhedra.

Wythoff
symbol

p q r |
Omnitruncated star polyhedron Wythoff
symbol
p q r |
Omnitruncated star polyhedron
Right triangle domains (r=2) General triangle domains
3 4/3 2 | Great truncated cuboctahedron.png
Great truncated cuboctahedron
4 4/3 3 | Cubitruncated cuboctahedron.png
Cubitruncated cuboctahedron
3 5/3 2 | Great truncated icosidodecahedron.png
Great truncated icosidodecahedron
5 5/3 3 | Icositruncated dodecadodecahedron.png
Icositruncated dodecadodecahedron
5 5/3 2 | Truncated dodecadodecahedron.png
Truncated dodecadodecahedron

Other even-sided nonconvex polyhedra

There are 7 nonconvex forms with mixed Wythoff symbols p q (r s) |, and bow-tie shaped vertex figures, 2p.2q.-2q.-2p. They are not really omnitruncated polyhedra: the true omnitruncates have coinciding 2r-gonal faces that must be removed to form a proper polyhedron. All these polyhedra are one-sided, i.e. non-orientable. The p q r | degenerate Wythoff symbols are listed first, followed by the actual mixed Wythoff symbols.

Omnitruncated polyhedron Image Wythoff symbol
Small rhombihexahedron Small rhombihexahedron.png 3/2 2 4 |
2 4 (3/2 4/2) |
Great rhombihexahedron Great rhombihexahedron.png 4/3 3/2 2 |
2 4/3 (3/2 4/2) |
Small rhombidodecahedron Small rhombidodecahedron.png 2 5/2 5 |
2 5 (3/2 5/2) |
Small dodecicosahedron Small dodecicosahedron.png 3/2 3 5 |
3 5 (3/2 5/4) |
Rhombicosahedron Rhombicosahedron.png 2 5/2 3 |
2 3 (5/4 5/2) |
Great dodecicosahedron Great dodecicosahedron.png 5/2 5/3 3 |
3 5/3 (3/2 5/2) |
Great rhombidodecahedron Great rhombidodecahedron.png 3/2 5/3 2 |
2 5/3 (3/2 5/4) |

See also

References


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