Bitruncation

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

For regular polyhedra (i.e. regular 3-polytopes), a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.

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.

An interesting result of this operation is that self-dual 4-polytope {p,q,p} (and honeycombs) remain

, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space.

Space	4-polytope or honeycomb	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
${\displaystyle \mathbb {E} ^{3}}$	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

See also

• uniform polyhedron
• uniform 4-polytope
• Rectification (geometry)
• Truncation (geometry)

• 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
• ISBN 978-1-56881-220-5
  (Chapter 26)

• Weisstein, Eric W. "Truncation". MathWorld.

Polyhedron operators

• v
• t
• e

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}