Bitruncation
Operation in Euclidean geometry
title: "Bitruncation" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polytopes", "bitruncated-tilings"] description: "Operation in Euclidean geometry" topic_path: "general/polytopes" source: "https://en.wikipedia.org/wiki/Bitruncation" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Operation in Euclidean geometry ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/16/Birectified_cube_sequence.png" caption="A ''bitruncated [[cube]]'' is a truncated [[octahedron]]."] ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a9/Bitruncated_cubic_honeycomb.png" caption="A [[bitruncated cubic honeycomb]] - Cubic cells become orange truncated octahedra, and vertices are replaced by blue truncated octahedra."] ::
In geometry, a bitruncation is an operation on regular polytopes. The original edges are lost completely and the original faces remain as smaller copies of themselves.
Bitruncated regular polytopes can be represented by an extended Schläfli symbol notation t{p,q,...} or 2t{p,q,...}.
In regular polyhedra and tilings
For regular polyhedra (i.e. regular 3-polytopes), a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.
In regular 4-polytopes and honeycombs
For a regular 4-polytope, a bitruncated form is a dual-symmetric operator. A bitruncated 4-polytope is the same as the bitruncated dual, and will have double the symmetry if the original 4-polytope is self-dual.
A regular polytope (or honeycomb) {p, q, r} will have its {p, q} cells bitruncated into truncated {q, p} cells, and the vertices are replaced by truncated {q, r} cells.
Self-dual {p,q,p} 4-polytope/honeycombs
An interesting result of this operation is that self-dual 4-polytope {p,q,p} (and honeycombs) remain cell-transitive after bitruncation. There are 5 such forms corresponding to the five truncated regular polyhedra: t{q,p}. Two are honeycombs on the 3-sphere, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space. ::data[format=table]
| Space | 4-polytope or honeycomb | Schläfli symbolCoxeter-Dynkin diagram | Cell type | Cellimage | Vertex figure | \mathbb{S}^3 | \mathbb{E}^3 | \mathbb{H}^3 |
|---|---|---|---|---|---|---|---|---|
| Bitruncated 5-cell (10-cell)(Uniform 4-polytope) | t1,2{3,3,3} | |||||||
| truncated tetrahedron | [[Image:truncated tetrahedron.png | 60px]] | [[File:Bitruncated 5-cell verf.png | 60px]] | ||||
| Bitruncated 24-cell (48-cell)(Uniform 4-polytope) | t1,2{3,4,3} | |||||||
| truncated cube | [[Image:truncated hexahedron.png | 60px]] | [[File:Bitruncated 24-cell verf.svg | 60px]] | ||||
| Bitruncated cubic honeycomb(Uniform Euclidean convex honeycomb) | t1,2{4,3,4} | |||||||
| truncated octahedron | [[Image:truncated octahedron.png | 60px]] | [[File:Bitruncated cubic honeycomb verf.png | 60px]] | ||||
| Bitruncated icosahedral honeycomb(Uniform hyperbolic convex honeycomb) | t1,2{3,5,3} | |||||||
| truncated dodecahedron | [[Image:truncated dodecahedron.png | 60px]] | [[File:Bitruncated icosahedral honeycomb verf.png | 60px]] | ||||
| Bitruncated order-5 dodecahedral honeycomb(Uniform hyperbolic convex honeycomb) | t1,2{5,3,5} | |||||||
| truncated icosahedron | [[Image:truncated icosahedron.png | 60px]] | [[File:Bitruncated order-5 dodecahedral honeycomb verf.png | 60px]] | ||||
| :: |
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (pp. 145–154 Chapter 8: Truncation)
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26)
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