# 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. 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, a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.

## In regular 4-polytopes and honeycombs

For regular 4-polytope, a bitruncated form is a dual-symmetric operator. A bitruncated 4-polytope 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} 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.

4-polytope or honeycomb Schläfli symbol
Coxeter-Dynkin diagram
Cell type Cell
image
Vertex figure
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
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

## References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)

Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}