Open main menu
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,...}.

Contents

In regular polyhedra and tilingsEdit

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 honeycombsEdit

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/honeycombsEdit

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}
       
truncated tetrahedron    
Bitruncated 24-cell (48-cell)
(Uniform 4-polytope)
t1,2{3,4,3}
       
truncated cube    
  Bitruncated cubic honeycomb
(Uniform Euclidean convex honeycomb)
t1,2{4,3,4}
       
truncated octahedron    
  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 alsoEdit

ReferencesEdit

  • 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)

External linksEdit

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}