# Omnitruncated polyhedron

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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.

More generally an omnitruncated polyhedron is a bevel operator in Conway polyhedron notation.

## List of convex omnitruncated polyhedra

There are three convex forms. They can be seen as red faces of one regular polyhedron, yellow or green 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 true omnitruncated polyhedra: the true omnitruncates p q r | or p q s | 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
Cubohemioctahedron 3/2 2 3 |
2 3 (3/2 3/2) |
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) |

## General omnitruncations (bevel)

Omnitruncations are also called cantitruncations or truncated rectifications (tr), and Conway's bevel (b) operator. When applied to nonregular polyhedra, new polyhedra can be generated, for example these 2-uniform polyhedra:

Coxeter trrC trrD trtT trtC trtO trtI
Conway baO baD btT btC btO btI
Image

## 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, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446
• 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, doi:10.1098/rsta.1975.0022, ISSN 0080-4614, JSTOR 74475, MR 0365333
• 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.

Polyhedron operators

Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}