Bitruncation

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 t_1,2{p, q,...}.

In regular polyhedra and tilings

For regular polyhedron, a "bitruncated" form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.

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.

Self-dual {p,q,p} polychora/honeycombs

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.

See also

* uniform polyhedron
* uniform polychoron
* Rectification (geometry)
* Truncation (geometry)