Bitruncation

A bitruncated cube is a truncated octahedron.
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. It represents a truncation beyond rectification.[citation needed] 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,...} or 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
image
Vertex figure
${\displaystyle \mathbb {S} ^{3}}$  Bitruncated 5-cell (10-cell)
(Uniform 4-polytope)
t1,2{3,3,3}

truncated tetrahedron
Bitruncated 24-cell (48-cell)
(Uniform 4-polytope)
t1,2{3,4,3}

truncated cube
${\displaystyle \mathbb {E} ^{3}}$  Bitruncated cubic honeycomb
(Uniform Euclidean convex honeycomb)
t1,2{4,3,4}

truncated octahedron
${\displaystyle \mathbb {H} ^{3}}$  Bitruncated icosahedral honeycomb
(Uniform hyperbolic convex honeycomb)
t1,2{3,5,3}

truncated dodecahedron
Bitruncated order-5 dodecahedral honeycomb
(Uniform hyperbolic convex honeycomb)
t1,2{5,3,5}

truncated icosahedron