Bitruncation

From Wikipedia, the free encyclopedia
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.[citation needed] 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 (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
Bitruncated 5-cell (10-cell)
(Uniform 4-polytope) 		t1,2{3,3,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 		truncated tetrahedron Truncated tetrahedron.png Bitruncated 5-cell verf.png
Bitruncated 24-cell (48-cell)
(Uniform 4-polytope) 		t1,2{3,4,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png 		truncated cube Truncated hexahedron.png Bitruncated 24-cell verf.png
Bitruncated cubic honeycomb
(Uniform Euclidean convex honeycomb) 		t1,2{4,3,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 		truncated octahedron Truncated octahedron.png Bitruncated cubic honeycomb verf.png
Bitruncated icosahedral honeycomb
(Uniform hyperbolic convex honeycomb) 		t1,2{3,5,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png 		truncated dodecahedron Truncated dodecahedron.png Bitruncated icosahedral honeycomb verf.png
Bitruncated order-5 dodecahedral honeycomb
(Uniform hyperbolic convex honeycomb) 		t1,2{5,3,5}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png 		truncated icosahedron Truncated icosahedron.png Bitruncated order-5 dodecahedral honeycomb verf.png

See also[]

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, , Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)

External links[]

Polyhedron operators
  • v
  • t
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
CDel node 1.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t12.svg Uniform polyhedron-43-t2.svg Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-33-t0.png Uniform polyhedron-43-h01.svg Uniform polyhedron-43-s012.png
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}
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