Bitruncation

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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[edit]

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

In regular polychora and honeycombs[edit]

For regular polychoron, a bitruncated form is a dual-symmetric operator. A bitruncated polychoron 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} polychora/honeycombs[edit]

An interesting result of this operation is that self-dual polychora {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.

Polychoron or honeycomb Schläfli symbol
Coxeter-Dynkin diagram
Cell type Cell
image
Vertex figure
Bitruncated 5-cell (10-cell)
(Uniform polychoron)
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 polychoron)
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 convex honeycomb of Euclidean space)
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 convex honeycomb of hyperbolic space)
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 convex honeycomb of hyperbolic space)
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[edit]

References[edit]

External links[edit]

Polyhedron operators
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.png Uniform polyhedron-43-t01.png Uniform polyhedron-43-t1.png Uniform polyhedron-43-t12.png Uniform polyhedron-43-t2.png Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-33-t0.png Uniform polyhedron-43-h01.png 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}