Bitruncation
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Image:Bitruncated cubic honeycomb.png 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 a 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-uniform. There are 5 such forms corresponding to the five truncated regular polyhedra: t{q,p}:
| Polychoron or honeycomb | Extended Schläfli symbol | Cell type | Cell image |
|---|---|---|---|
| Bitruncated 5-cell | t1,2{3,3,3} | truncated tetrahedron | 
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| Bitruncated cubic honeycomb | t1,2{4,3,4} | truncated octahedron | 
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| Bitruncated 24-cell | t1,2{3,4,3} | truncated cube | 
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| Bitruncated order-3 icosahedral honeycomb (Hyperbolic space) | t1,2{3,5,3} | truncated dodecahedron | 
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| Bitruncated order-5 dodecahedral honeycomb (Hyperbolic space) | t1,2{5,3,5} | truncated icosahedron | 
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