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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For regular polyhedra, a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.
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.
An interesting result of this operation is that self-dual polychora {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.
| Polychoron or honeycomb | Schläfli symbol Coxeter-Dynkin diagram |
Cell type | Cell image |
Vertex figure |
|---|---|---|---|---|
| Bitruncated 5-cell (10-cell) (Uniform polychoron) |
t1,2{3,3,3} | truncated tetrahedron | ||
| Bitruncated 24-cell (48-cell) (Uniform polychoron) |
t1,2{3,4,3} | truncated cube | ||
| Bitruncated cubic honeycomb (Uniform convex honeycomb of Euclidean space) |
t1,2{4,3,4} | truncated octahedron | ||
| Bitruncated icosahedral honeycomb (Uniform convex honeycomb of hyperbolic space) |
t1,2{3,5,3} | truncated dodecahedron | ||
| Bitruncated order-5 dodecahedral honeycomb (Uniform convex honeycomb of hyperbolic space) |
t1,2{5,3,5} | truncated icosahedron |