Bitruncation
From Wikipedia, the free encyclopedia
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. 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 t1,2{p, q,...}.
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[edit] In regular polyhedra and tilings
For regular polyhedron, a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.
[edit] In regular polychora and honeycombs
For regular polychoron, a bitruncated form is a dual-symmetric operator. A bitruncated polychoron is the same as the bitruncated 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.
[edit] Self-dual {p,q,p} polychora/honeycombs
An interesting result of this operation is that self-dual polychora {p,q,p} (and honeycombs) are cell-transitive. 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.
| Polychoron or honeycomb | Schläfli symbol Coxeter-Dynkin diagram |
Cell type | Cell image |
|---|---|---|---|
| Bitruncated 5-cell | t1,2{3,3,3}
|
truncated tetrahedron |  |
| Bitruncated 24-cell | t1,2{3,4,3}
|
truncated cube |  |
| Bitruncated cubic honeycomb | t1,2{4,3,4}
|
truncated octahedron |  |
| Bitruncated order-3 icosahedral honeycomb | t1,2{3,5,3}
|
truncated dodecahedron |  |
| Bitruncated order-5 dodecahedral honeycomb | t1,2{5,3,5}
|
truncated icosahedron |  |
