From Surf Wiki (app.surf) — the open knowledge base

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 considered as 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. These forms 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.

Wythoffsymbol p q r	Omnitruncated polyhedron	Regular/quasiregular polyhedra	3 3 2	4 3 2	5 3 2
[[File:Uniform polyhedron-33-t012.svg	100px]]Truncated octahedron	[[File:Uniform polyhedron-33-t0.svg	100px]] [[File:Uniform polyhedron-33-t1.svg	100px]] [[File:Uniform polyhedron-33-t2.svg	100px]]Tetrahedron/Octahedron/Tetrahedron
[[File:Uniform polyhedron-43-t012.png	100px]]Truncated cuboctahedron	[[File:Uniform polyhedron-43-t0.svg	100px]][[File:Uniform polyhedron-43-t1.svg	100px]][[File:Uniform polyhedron-43-t2.svg	100px]]Cube/Cuboctahedron/Octahedron
[[File:Uniform polyhedron-53-t012.png	100px]]Truncated icosidodecahedron	[[File:Uniform polyhedron-53-t0.svg	100px]][[File:Uniform polyhedron-53-t1.svg	100px]][[File:Uniform polyhedron-53-t2.svg	100px]]Dodecahedron/Icosidodecahedron/Icosahedron

List of nonconvex omnitruncated polyhedra

There are 5 nonconvex uniform omnitruncated polyhedra.

Wythoffsymbol p q r	Omnitruncated star polyhedron	Wythoffsymbol p q r	Omnitruncated star polyhedron
[[File:Great truncated cuboctahedron.png	100px]]Great truncated cuboctahedron	[[File:Cubitruncated cuboctahedron.png	100px]]Cubitruncated cuboctahedron
[[File:Great truncated icosidodecahedron.png	100px]]Great truncated icosidodecahedron	[[File:Icositruncated dodecadodecahedron.png	100px]]Icositruncated dodecadodecahedron
[[File:Truncated dodecadodecahedron.png	100px]]Truncated dodecadodecahedron

Other even-sided nonconvex polyhedra

There are 8 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. Instead, the true omnitruncates p q r | or p q s | have coinciding 2r-gonal or 2s-gonal faces that must be removed respectively 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	[[File:Cubohemioctahedron.png	100px]]	3/2 2 3
2 3 (3/2 3/2)
Small rhombihexahedron	[[File:Small rhombihexahedron.png	100px]]	3/2 2 4
2 4 (3/2 4/2)
Great rhombihexahedron	[[File:Great rhombihexahedron.png	100px]]	4/3 3/2 2
2 4/3 (3/2 4/2)
Small rhombidodecahedron	[[File:Small rhombidodecahedron.png	100px]]	2 5/2 5
2 5 (3/2 5/2)
Small dodecicosahedron	[[File:Small dodecicosahedron.png	100px]]	3/2 3 5
3 5 (3/2 5/4)
Rhombicosahedron	[[File:Rhombicosahedron.png	100px]]	2 5/2 3
2 3 (5/4 5/2)
Great dodecicosahedron	[[File:Great dodecicosahedron.png	100px]]	5/2 5/3 3
3 5/3 (3/2 5/2)
Great rhombidodecahedron	[[File:Great rhombidodecahedron.png	100px]]	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
[[File:Truncated rhombicuboctahedron.png	100px]]	[[File:Truncated rhombicosidodecahedron.png	100px]]	[[File:Truncated rectified truncated tetrahedron.png	100px]]	[[File:Truncated rectified truncated cube.png	100px]]	[[File:Truncated rectified truncated octahedron.png	100px]]	[[File:Truncated rectified truncated icosahedron.png	100px]]

References

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.

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

polyhedra

Want to explore this topic further?

Ask Mako anything about Omnitruncated polyhedron — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report