File:Birectified cube sequence.png A bitruncated cube is a truncated octahedron . File:Bitruncated cubic honeycomb.png 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 1,2 {p ,q ,...} 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.
Space
4-polytope or honeycomb
Schläfli symbol Coxeter-Dynkin diagram
Cell type
Cell
Vertex figure
𝕊 3 Bitruncated 5-cell (10-cell)Uniform 4-polytope )
t1,2 {3,3,3}File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png
truncated tetrahedron
File:Truncated tetrahedron.png File:Bitruncated 5-cell verf.png
Bitruncated 24-cell (48-cell)Uniform 4-polytope )
t1,2 {3,4,3}File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png
truncated cube
File:Truncated hexahedron.png File:Bitruncated 24-cell verf.png
𝔼 3 Bitruncated cubic honeycomb Uniform Euclidean convex honeycomb )
t1,2 {4,3,4}File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png
truncated octahedron
File:Truncated octahedron.png File:Bitruncated cubic honeycomb verf.png
ℍ 3 Bitruncated icosahedral honeycomb t1,2 {3,5,3}File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png
truncated dodecahedron
File:Truncated dodecahedron.png File:Bitruncated icosahedral honeycomb verf.png
Bitruncated order-5 dodecahedral honeycomb t1,2 {5,3,5}File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 5.png File:CDel node.png
truncated icosahedron
File:Truncated icosahedron.png File:Bitruncated order-5 dodecahedral honeycomb verf.png
See also References Coxeter, H.S.M. Regular Polytopes ISBN 0-486-61480-8 (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, 1966John H. Conway , Heidi Burgiel , Chaim Goodman-Strauss , The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)External links
Polyhedron operators
Seed
Truncation Rectification Bitruncation Dual
Expansion Omnitruncation Alternations
File:CDel node 1.png File:CDel p.png File:CDel node n1.png File:CDel q.png File:CDel node n2.png File:CDel node 1.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node.png File:CDel node.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node.png File:CDel node.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node 1.png File:CDel node.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node 1.png File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node 1.png File:CDel node 1.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node 1.png File:CDel node h.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node.png File:CDel node.png File:CDel p.png File:CDel node h.png File:CDel q.png File:CDel node h.png File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel q.png File:CDel node h.png
File:Uniform polyhedron-43-t0.svg File:Uniform polyhedron-43-t01.svg File:Uniform polyhedron-43-t1.svg File:Uniform polyhedron-43-t12.svg File:Uniform polyhedron-43-t2.svg File:Uniform polyhedron-43-t02.png File:Uniform polyhedron-43-t012.png File:Uniform polyhedron-33-t0.png File:Uniform polyhedron-43-h01.svg File:Uniform polyhedron-43-s012.png
t0 {p ,q } p ,q } t01 {p ,q } p ,q } t1 {p ,q } p ,q } t12 {p ,q } p ,q } t2 {p ,q } p ,q } t02 {p ,q } p ,q } t012 {p ,q } p ,q } ht0 {p ,q } q ,p } ht12 {p ,q } q ,p } ht012 {p ,q } p ,q }
