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.

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 |	Truncated octahedron	Tetrahedron/Octahedron/Tetrahedron
4 3 2 |	Truncated cuboctahedron	Cube/Cuboctahedron/Octahedron
5 3 2 |	Truncated icosidodecahedron	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	4 4/3 3 |	Cubitruncated cuboctahedron
3 5/3 2 |	Great truncated icosidodecahedron	5 5/3 3 |	Icositruncated dodecadodecahedron
5 5/3 2 |	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		3/2 2 4 | 2 4 (3/2 4/2) |
Great rhombihexahedron		4/3 3/2 2 | 2 4/3 (3/2 4/2) |
Small rhombidodecahedron		2 5/2 5 | 2 5 (3/2 5/2) |
Small dodecicosahedron		3/2 3 5 | 3 5 (3/2 5/4) |
Rhombicosahedron		2 5/2 3 | 2 3 (5/4 5/2) |
Great dodecicosahedron		5/2 5/3 3 | 3 5/3 (3/2 5/2) |
Great rhombidodecahedron		3/2 5/3 2 | 2 5/3 (3/2 5/4) |

See also

• Uniform polyhedron

References

• Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 246: 401–450, ISSN 0080-4614, JSTOR 91532, MR0062446 
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. 
• Skilling, J. (1975), "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 278: 111–135, ISSN 0080-4614, JSTOR 74475, MR0365333 
• Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, Images, dual images
• Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.