{{Short description|Operation in Euclidean geometry}} [[Image:Birectified cube sequence.png|thumb|A ''bitruncated cube'' is a truncated octahedron.]] [[Image:Bitruncated cubic honeycomb.png|thumb|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 {{math|'''t'''{{sub|1,2}}{''p'',''q'',...} }} or {{math|'''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. {| class="wikitable" !Space !4-polytope or honeycomb !Schläfli symbol<BR>Coxeter-Dynkin diagram !Cell type !Cell<BR>image !Vertex figure |- !rowspan=2|<math>\mathbb{S}^3</math> |Bitruncated 5-cell (10-cell)<BR>(Uniform 4-polytope) |t<sub>1,2</sub>{3,3,3}<br>{{CDD|node|3|node_1|3|node_1|3|node}} |truncated tetrahedron |60px |60px |- |Bitruncated 24-cell (48-cell)<BR>(Uniform 4-polytope) |t<sub>1,2</sub>{3,4,3}<br>{{CDD|node|3|node_1|4|node_1|3|node}} |truncated cube |60px |60px |- !<math>\mathbb{E}^3</math> |Bitruncated cubic honeycomb<BR>(Uniform Euclidean convex honeycomb) |t<sub>1,2</sub>{4,3,4}<br>{{CDD|node|4|node_1|3|node_1|4|node}} |truncated octahedron |60px |60px |- !rowspan=2|<math>\mathbb{H}^3</math> |Bitruncated icosahedral honeycomb<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 |60px |60px |- |Bitruncated order-5 dodecahedral honeycomb<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 |60px |60px |}

== See also == * Rectification (geometry) * Truncation (geometry) * Uniform 4-polytope * Uniform polyhedron

== References == * Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} (pp.&nbsp;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, {{ISBN|978-1-56881-220-5}} (Chapter 26)

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

{{Polyhedron_operators}}

Category:Polytopes Category:Bitruncated tilings