Bitruncation

<h2 id="in-regular-polyhedra-and-tilings">In regular polyhedra and tilings</h2>
For regular <a href="/facts/Polyhedron/UC0p9FTF">polyhedra</a> (i.e. regular 3-polytopes), a bitruncated form is the truncated <a href="/facts/Dual_polyhedron/rkIHbmAf">dual</a>. For example, a bitruncated <a href="/facts/Cube/65lUaAqN">cube</a> is a <a href="/facts/Truncated_octahedron/GA98b4TH">truncated octahedron</a>.

<h2 id="in-regular-4-polytopes-and-honeycombs">In regular 4-polytopes and honeycombs</h2>
For a regular <a href="/facts/4-polytope/mY8vBpec">4-polytope</a>, 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 <a href="/facts/Dual_polyhedron/rkIHbmAf">self-dual</a>.
A regular polytope (or <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a>) {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.

<h3>Self-dual {p,q,p} 4-polytope/honeycombs</h3>
An interesting result of this operation is that self-dual 4-polytope {p,q,p} (and honeycombs) remain <a href="/facts/Cell-transitive/ukLVQlK2">cell-transitive</a> after bitruncation. There are 5 such forms corresponding to the five truncated regular polyhedra: t{q,p}. Two are honeycombs on the <a href="/facts/3-sphere/u67nImpY">3-sphere</a>, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space.

<table><tbody><tr><th>Space</th><th>4-polytope or honeycomb</th><th><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a></th><th>Cell type</th><th>Cellimage</th><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></th></tr><tr><th rowspan="2"> S 3 {\displaystyle \mathbb {S} ^{3}} </th><td><a href="/facts/Bitruncated_5-cell/oqF6jkb7">Bitruncated 5-cell</a> (10-cell)(<a href="/facts/Uniform_4-polytope/NHxomMxC">Uniform 4-polytope</a>)</td><td>t1,2{3,3,3}</td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a></td><td></td><td></td></tr><tr><td><a href="/facts/Bitruncated_24-cell/1dnQysxX">Bitruncated 24-cell</a> (48-cell)(<a href="/facts/Uniform_4-polytope/NHxomMxC">Uniform 4-polytope</a>)</td><td>t1,2{3,4,3}</td><td><a href="/facts/Truncated_cube/XVAUG4fD">truncated cube</a></td><td></td><td></td></tr><tr><th> E 3 {\displaystyle \mathbb {E} ^{3}} </th><td><a href="/facts/Bitruncated_cubic_honeycomb/j6Prl5h0">Bitruncated cubic honeycomb</a>(<a href="/facts/Uniform_convex_honeycomb/78IkTl7A">Uniform Euclidean convex honeycomb</a>)</td><td>t1,2{4,3,4}</td><td><a href="/facts/Truncated_octahedron/GA98b4TH">truncated octahedron</a></td><td></td><td></td></tr><tr><th rowspan="2"> H 3 {\displaystyle \mathbb {H} ^{3}} </th><td><a href="/facts/Convex_uniform_honeycombs_in_hyperbolic_space/n1c2vP9b">Bitruncated icosahedral honeycomb</a>(Uniform hyperbolic convex honeycomb)</td><td>t1,2{3,5,3}</td><td><a href="/facts/Truncated_dodecahedron/PVTAsUWQ">truncated dodecahedron</a></td><td></td><td></td></tr><tr><td><a href="/facts/Convex_uniform_honeycombs_in_hyperbolic_space/n1c2vP9b">Bitruncated order-5 dodecahedral honeycomb</a>(Uniform hyperbolic convex honeycomb)</td><td>t1,2{5,3,5}</td><td><a href="/facts/Truncated_icosahedron/LR8ugaY9">truncated icosahedron</a></td><td></td><td></td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Uniform_polyhedron/PHnprqRC">uniform polyhedron</a></li>
<li><a href="/facts/Uniform_4-polytope/NHxomMxC">uniform 4-polytope</a></li>
<li><a href="/facts/Rectification_(geometry)/PkoUdq53">Rectification (geometry)</a></li>
<li><a href="/facts/Truncation_(geometry)/DYb589Je">Truncation (geometry)</a></li></ul>

<ul><li><a href="/facts/Coxeter/D2l5jWGG">Coxeter, H.S.M.</a> <a href="/facts/Regular_Polytopes_(book)/jomuRNha">Regular Polytopes</a>, (3rd edition, 1973), Dover edition, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)</li>
<li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a> Uniform Polytopes, Manuscript (1991)
<ul><li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">N.W. Johnson</a>: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966</li></ul></li>
<li><a href="/facts/John_Horton_Conway/g3PA1zoK">John H. Conway</a>, Heidi Burgiel, <a href="/facts/Chaim_Goodman-Strauss/4IKWbx8w">Chaim Goodman-Strauss</a>, The Symmetries of Things 2008, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-220-5 (Chapter 26)</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Truncation.html">"Truncation"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>
Polyhedron operators <ul><li>v</li><li>t</li><li>e</li></ul><table><tbody><tr><th>Seed</th><th><a href="/facts/Truncation_(geometry)/DYb589Je">Truncation</a></th><th><a href="/facts/Rectification_(geometry)/PkoUdq53">Rectification</a></th><th>Bitruncation</th><th><a href="/facts/Dual_polyhedron/rkIHbmAf">Dual</a></th><th><a href="/facts/Expansion_(geometry)/eMpoRlnm">Expansion</a></th><th><a href="/facts/Omnitruncation/hmbMrRm6">Omnitruncation</a></th><th colspan="3"><a href="/facts/Alternation_(geometry)/H2nVs4xj">Alternations</a></th></tr><tr><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><td><a href="/facts/Regular_polyhedron/HJ6oGyug"></a></td><td><a href="/facts/Truncated_polyhedron/DYb589Je"></a></td><td><a href="/facts/Quasiregular_polyhedron/1ukrFhec"></a></td><td><a href="/facts/Bitruncated_polyhedron/t0xarbyJ"></a></td><td><a href="/facts/Dual_polyhedron/rkIHbmAf"></a></td><td><a href="/facts/Cantellated_polyhedron/kS24Tde3"></a></td><td><a href="/facts/Omnitruncated_polyhedron/l43QqIur"></a></td><td><a href="/facts/Alternation_(geometry)/H2nVs4xj"></a></td><td><a href="/facts/Snub_polyhedron/DAD0qnGU"></a></td><td><a href="/facts/Snub_polyhedron/DAD0qnGU"></a></td></tr><tr><td><a href="/facts/Regular_polyhedron/HJ6oGyug">t0{p,q}</a>{p,q} </td><td><a href="/facts/Truncated_polyhedron/DYb589Je">t01{p,q}</a>t{p,q} </td><td><a href="/facts/Quasiregular_polyhedron/1ukrFhec">t1{p,q}</a>r{p,q} </td><td><a href="/facts/Bitruncated_polyhedron/t0xarbyJ">t12{p,q}</a>2t{p,q} </td><td><a href="/facts/Dual_polyhedron/rkIHbmAf">t2{p,q}</a>2r{p,q} </td><td><a href="/facts/Cantellated_polyhedron/kS24Tde3">t02{p,q}</a>rr{p,q} </td><td><a href="/facts/Omnitruncated_polyhedron/l43QqIur">t012{p,q}</a>tr{p,q} </td><td><a href="/facts/Alternation_(geometry)/H2nVs4xj">ht0{p,q}</a>h{q,p} </td><td><a href="/facts/Snub_polyhedron/DAD0qnGU">ht12{p,q}</a>s{q,p} </td><td><a href="/facts/Snub_polyhedron/DAD0qnGU">ht012{p,q}</a>sr{p,q} </td></tr></tbody></table>

Bitruncation open-in-new

Bitruncation