Bitruncation

Image:Birectified cube sequence.png|thumb|A bitruncated cube is a truncated octahedron.
Image:Bitruncated cubic honeycomb.png|thumb|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. 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, 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 will have its cells bitruncated into truncated cells, and the vertices are replaced by truncated cells.

Self-dual {p,q,p} 4-polytope/honeycombs

An interesting result of this operation is that self-dual 4-polytope 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.

Space	4-polytope or honeycomb	Schläfli symbol Coxeter-Dynkin diagram	Cell type	Cell image	Vertex figure
	Bitruncated 5-cell	t_1,2	truncated tetrahedron	60px
	Bitruncated 24-cell	t_1,2	truncated cube	60px
	Bitruncated cubic honeycomb	t_1,2	truncated octahedron	60px
	Bitruncated icosahedral honeycomb	t_1,2	truncated dodecahedron	60px
	Bitruncated order-5 dodecahedral honeycomb	t_1,2	truncated icosahedron	60px