# Bitruncation

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> Source: https://en.wikipedia.org/wiki/Bitruncation
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{{Short description|Operation in Euclidean geometry}}
[[Image:Birectified cube sequence.png|thumb|A ''bitruncated [cube](/source/cube)'' is a truncated [octahedron](/source/octahedron).]]
[[Image:Bitruncated cubic honeycomb.png|thumb|A [bitruncated cubic honeycomb](/source/bitruncated_cubic_honeycomb) - Cubic cells become orange truncated octahedra, and vertices are replaced by blue truncated octahedra.]]

In [geometry](/source/geometry), a '''bitruncation''' is an operation on [regular polytope](/source/regular_polytope)s. The original [edges](/source/Edge_(geometry)) are lost completely and the original [faces](/source/Face_(geometry)) remain as smaller copies of themselves.

Bitruncated regular polytopes can be represented by an extended [Schläfli symbol](/source/Schl%C3%A4fli_symbol) notation {{math|'''t'''{{sub|1,2}}{''p'',''q'',...} }} or {{math|'''2t'''{''p'',''q'',...}.}}

== In regular polyhedra and tilings ==
For regular [polyhedra](/source/polyhedron) (i.e. regular 3-polytopes), a ''bitruncated'' form is the truncated [dual](/source/Dual_polyhedron). For example, a bitruncated [cube](/source/cube) is a [truncated octahedron](/source/truncated_octahedron).

== In regular 4-polytopes and honeycombs ==

For a regular [4-polytope](/source/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](/source/Dual_polyhedron).

A regular polytope (or [honeycomb](/source/Honeycomb_(geometry))) {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](/source/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](/source/3-sphere), one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space.
{| class="wikitable"
!Space
!4-polytope or honeycomb
![Schläfli symbol](/source/Schl%C3%A4fli_symbol)<BR>[Coxeter-Dynkin diagram](/source/Coxeter-Dynkin_diagram)
!Cell type
!Cell<BR>image
![Vertex figure](/source/Vertex_figure)
|-
!rowspan=2|<math>\mathbb{S}^3</math>
|[Bitruncated 5-cell](/source/Bitruncated_5-cell) (10-cell)<BR>([Uniform 4-polytope](/source/Uniform_4-polytope))
|t<sub>1,2</sub>{3,3,3}<br>{{CDD|node|3|node_1|3|node_1|3|node}}
|[truncated tetrahedron](/source/truncated_tetrahedron)
|60px
|60px
|-
|[Bitruncated 24-cell](/source/Bitruncated_24-cell) (48-cell)<BR>([Uniform 4-polytope](/source/Uniform_4-polytope))
|t<sub>1,2</sub>{3,4,3}<br>{{CDD|node|3|node_1|4|node_1|3|node}}
|[truncated cube](/source/truncated_cube)
|60px
|60px
|-
!<math>\mathbb{E}^3</math>
|[Bitruncated cubic honeycomb](/source/Bitruncated_cubic_honeycomb)<BR>([Uniform Euclidean convex honeycomb](/source/Uniform_convex_honeycomb))
|t<sub>1,2</sub>{4,3,4}<br>{{CDD|node|4|node_1|3|node_1|4|node}}
|[truncated octahedron](/source/truncated_octahedron)
|60px
|60px
|-
!rowspan=2|<math>\mathbb{H}^3</math>
|[Bitruncated icosahedral honeycomb](/source/Convex_uniform_honeycombs_in_hyperbolic_space)<BR>(Uniform hyperbolic convex honeycomb)
|t<sub>1,2</sub>{3,5,3}<br>{{CDD|node|3|node_1|5|node_1|3|node}}
|[truncated dodecahedron](/source/truncated_dodecahedron)
|60px
|60px
|-
|[Bitruncated order-5 dodecahedral honeycomb](/source/Convex_uniform_honeycombs_in_hyperbolic_space)<BR>(Uniform hyperbolic convex honeycomb)
|t<sub>1,2</sub>{5,3,5}<br>{{CDD|node|5|node_1|3|node_1|5|node}}
|[truncated icosahedron](/source/truncated_icosahedron)
|60px
|60px
|}

== See also ==
* [Rectification (geometry)](/source/Rectification_(geometry))
* [Truncation (geometry)](/source/Truncation_(geometry))
* [Uniform 4-polytope](/source/Uniform_4-polytope)
* [Uniform polyhedron](/source/Uniform_polyhedron)

== References ==
* [Coxeter, H.S.M.](/source/Coxeter) ''[Regular Polytopes](/source/Regular_Polytopes_(book))'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} (pp.&nbsp;145–154 Chapter 8: Truncation)
* [Norman Johnson](/source/Norman_Johnson_(mathematician)) ''Uniform Polytopes'', Manuscript (1991)
** [N.W. Johnson](/source/Norman_Johnson_(mathematician)): ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
* [John H. Conway](/source/John_Horton_Conway), [Heidi Burgiel](/source/Heidi_Burgiel), [Chaim Goodman-Strauss](/source/Chaim_Goodman-Strauss), ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26)

== External links ==
* {{mathworld | urlname = Truncation | title = Truncation}}

{{Polyhedron_operators}}

Category:Polytopes
Category:Bitruncated tilings

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Adapted from the Wikipedia article [Bitruncation](https://en.wikipedia.org/wiki/Bitruncation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bitruncation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.