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. 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,...}.

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
${𝕊}^3$	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} File:CDel 4.png	truncated cube	File:Truncated hexahedron.png	File:Bitruncated 24-cell verf.png
${𝔼}^3$	Bitruncated cubic honeycomb (Uniform Euclidean convex honeycomb)	t1,2{4,3,4} File:CDel 4.pngFile:CDel 4.png	truncated octahedron	File:Truncated octahedron.png	File:Bitruncated cubic honeycomb verf.png
${ℍ}^3$	Bitruncated icosahedral honeycomb (Uniform hyperbolic convex honeycomb)	t1,2{3,5,3} File:CDel 5.png	truncated dodecahedron	File:Truncated dodecahedron.png	File:Bitruncated icosahedral honeycomb verf.png
	Bitruncated order-5 dodecahedral honeycomb (Uniform hyperbolic convex honeycomb)	t1,2{5,3,5} File:CDel 5.pngFile:CDel 5.png	truncated icosahedron	File:Truncated icosahedron.png	File:Bitruncated order-5 dodecahedral honeycomb verf.png

See also

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

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (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, (Chapter 26)

External links

Polyhedron operators

Seed	Truncation	Rectification	Bitruncation	Dual	Expansion	Omnitruncation	Alternations
File:CDel p.pngFile:CDel node n1.pngFile:CDel q.pngFile:CDel node n2.png	File:CDel p.pngFile:CDel q.png	File:CDel p.pngFile:CDel q.png	File:CDel p.pngFile:CDel q.png	File:CDel p.pngFile:CDel q.png	File:CDel p.pngFile:CDel q.png	File:CDel p.pngFile:CDel q.png	File:CDel node h.pngFile:CDel p.pngFile:CDel q.png	File:CDel p.pngFile:CDel node h.pngFile:CDel q.pngFile:CDel node h.png	File:CDel node h.pngFile:CDel p.pngFile:CDel node h.pngFile:CDel q.pngFile:CDel node h.png
File:Uniform polyhedron-43-t0.svg	File:Uniform polyhedron-43-t01.svg	File:Uniform polyhedron-43-t1.svg	File:Uniform polyhedron-43-t12.svg	File:Uniform polyhedron-43-t2.svg	File:Uniform polyhedron-43-t02.png	File:Uniform polyhedron-43-t012.png	File:Uniform polyhedron-33-t0.svg	File:Uniform polyhedron-43-h01.svg	File:Uniform polyhedron-43-s012.svg
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}