Bitruncation Explained

In geometry, a bitruncation is an operation on regular polytopes. 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 or

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) will have its cells bitruncated into truncated cells, and the vertices are replaced by truncated cells.

Self-dual 4-polytope/honeycombs

An interesting result of this operation is that self-dual 4-polytope (and honeycombs) remain cell-transitive after bitruncation. There are 5 such forms corresponding to the five truncated regular polyhedra: t. Two are honeycombs on the 3-sphere, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space.

Space4-polytope or honeycombSchläfli symbol
Coxeter-Dynkin diagram
Cell typeCell
image
Vertex figure

S3

Bitruncated 5-cell (10-cell)
(Uniform 4-polytope)
t1,2
truncated tetrahedron
Bitruncated 24-cell (48-cell)
(Uniform 4-polytope)
t1,2
truncated cube

E3

Bitruncated cubic honeycomb
(Uniform Euclidean convex honeycomb)
t1,2
truncated octahedron

H3

Bitruncated icosahedral honeycomb
(Uniform hyperbolic convex honeycomb)
t1,2
truncated dodecahedron
Bitruncated order-5 dodecahedral honeycomb
(Uniform hyperbolic convex honeycomb)
t1,2
truncated icosahedron

See also

References