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[edit]

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 | Uniform polyhedron-33-t012.png
Truncated octahedron
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform polyhedron-53-t0.pngUniform polyhedron-53-t1.pngUniform polyhedron-53-t2.png
Dodecahedron/Icosidodecahedron/Icosahedron

List of nonconvex omnitruncated polyhedra[edit]

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
CDel node 1.pngCDel 4.pngCDel rat.pngCDel d3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
4 4/3 3 | Cubitruncated cuboctahedron.png
Cubitruncated cuboctahedron
3 5/3 2 | Great truncated icosidodecahedron.png
Great truncated icosidodecahedron
CDel node 1.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5 5/3 3 | Icositruncated dodecadodecahedron.png
Icositruncated dodecadodecahedron
5 5/3 2 | Truncated dodecadodecahedron.png
Truncated dodecadodecahedron
CDel node 1.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node 1.pngCDel 5.pngCDel node 1.png

Other even-sided nonconvex polyhedra[edit]

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 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) |

General omnitruncations (bevel)[edit]

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 Truncated rhombicuboctahedron.png Truncated rhombicosidodecahedron.png Truncated rectified truncated tetrahedron.png Truncated rectified truncated cube.png Truncated rectified truncated octahedron.png Truncated rectified truncated icosahedron.png

See also[edit]

References[edit]

Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
CDel node 1.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Uniform polyhedron-43-t0.png Uniform polyhedron-43-t01.png Uniform polyhedron-43-t1.png Uniform polyhedron-43-t12.png Uniform polyhedron-43-t2.png Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-33-t0.png Uniform polyhedron-43-h01.png Uniform polyhedron-43-s012.png
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}