Bitruncated tesseractic honeycomb
Bitruncated tesseractic honeycomb  

(No image)  
Type  Uniform 4honeycomb 
Schläfli symbol  t_{1,2}{4,3,3,4} or 2t{4,3,3,4} t_{1,2}{4,3^{1,1}} or 2t{4,3^{1,1}} t_{2,3}{4,3^{1,1}} q_{2}{4,3,3,3,4} 
CoxeterDynkin diagram 

4face type  Bitruncated tesseract Truncated 16cell 
Cell type  Octahedron Truncated tetrahedron 20px Truncated octahedron 
Face type  {3}, {4}, {6} 
Vertex figure  Squarepyramidal pyramid 
Coxeter group  [math]\displaystyle{ {\tilde{C}}_4 }[/math] = [4,3,3,4] [math]\displaystyle{ {\tilde{B}}_4 }[/math] = [4,3^{1,1}] [math]\displaystyle{ {\tilde{D}}_4 }[/math] = [3^{1,1,1,1}] 
Dual  
Properties  vertextransitive 
In fourdimensional Euclidean geometry, the bitruncated tesseractic honeycomb is a uniform spacefilling tessellation (or honeycomb) in Euclidean 4space. It is constructed by a bitruncation of a tesseractic honeycomb. It is also called a cantic quarter tesseractic honeycomb from its q_{2}{4,3,3,4} construction.
Other names
 Bitruncated tesseractic tetracomb (batitit)
Related honeycombs
The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.
C4 honeycombs  

Extended symmetry 
Extended diagram 
Order  Honeycombs 
[4,3,3,4]:  ×1 
_{1},
_{2},
_{3},
_{4},  
[[4,3,3,4]]  ×2  _{(1)}, _{(2)}, _{(13)}, _{18} _{(6)}, _{19}, _{20}  
[(3,3)[1^{+},4,3,3,4,1^{+}]] ↔ [(3,3)[3^{1,1,1,1}]] ↔ [3,4,3,3] 
↔ ↔ 
×6 
_{14}, _{15}, _{16}, _{17} 
The [4,3,3^{1,1}], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16cell honeycomb and snub 24cell honeycomb respectively.
B4 honeycombs  

Extended symmetry 
Extended diagram 
Order  Honeycombs  
[4,3,3^{1,1}]:  ×1 
_{5}, _{6}, _{7}, _{8}  
<[4,3,3^{1,1}]>: ↔[4,3,3,4] 
↔ 
×2 
_{9}, _{10}, _{11}, _{12}, _{13}, _{14}, _{(10)}, _{15}, _{16}, _{(13)}, _{17}, _{18}, _{19}  
[3[1^{+},4,3,3^{1,1}]] ↔ [3[3,3^{1,1,1}]] ↔ [3,3,4,3] 
↔ ↔ 
×3 
_{1}, _{2}, _{3}, _{4}  
[(3,3)[1^{+},4,3,3^{1,1}]] ↔ [(3,3)[3^{1,1,1,1}]] ↔ [3,4,3,3] 
↔ ↔ 
×12 
_{20}, _{21}, _{22}, _{23} 
There are ten uniform honeycombs constructed by the [math]\displaystyle{ {\tilde{D}}_4 }[/math] Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)^{*}] (index 24), [3,3,4,3^{*}] (index 6), [1^{+},4,3,3,4,1^{+}] (index 4), [3^{1,1},3,4,1^{+}] (index 2) are all isomorphic to [3^{1,1,1,1}].
The ten permutations are listed with its highest extended symmetry relation:
D4 honeycombs  

Extended symmetry 
Extended diagram 
Extended group 
Honeycombs 
[3^{1,1,1,1}]  [math]\displaystyle{ {\tilde{D}}_4 }[/math]  (none)  
<[3^{1,1,1,1}]> ↔ [3^{1,1},3,4] 
↔ 
[math]\displaystyle{ {\tilde{D}}_4 }[/math]×2 = [math]\displaystyle{ {\tilde{B}}_4 }[/math]  (none) 
<2[^{1,1}3^{1,1}]> ↔ [4,3,3,4] 
↔ 
[math]\displaystyle{ {\tilde{D}}_4 }[/math]×4 = [math]\displaystyle{ {\tilde{C}}_4 }[/math]  _{1}, _{2} 
[3[3,3^{1,1,1}]] ↔ [3,3,4,3] 
↔ 
[math]\displaystyle{ {\tilde{D}}_4 }[/math]×6 = [math]\displaystyle{ {\tilde{F}}_4 }[/math]  _{3}, _{4}, _{5}, _{6} 
[4[^{1,1}3^{1,1}]] ↔ [[4,3,3,4]] 
↔ 
[math]\displaystyle{ {\tilde{D}}_4 }[/math]×8 = [math]\displaystyle{ {\tilde{C}}_4 }[/math]×2  _{7}, _{8}, _{9} 
[(3,3)[3^{1,1,1,1}]] ↔ [3,4,3,3] 
↔ 
[math]\displaystyle{ {\tilde{D}}_4 }[/math]×24 = [math]\displaystyle{ {\tilde{F}}_4 }[/math]  
[(3,3)[3^{1,1,1,1}]]^{+} ↔ [3^{+},4,3,3] 
↔ 
½[math]\displaystyle{ {\tilde{D}}_4 }[/math]×24 = ½[math]\displaystyle{ {\tilde{F}}_4 }[/math]  _{10} 
See also
Regular and uniform honeycombs in 4space:
 Tesseractic honeycomb
 Demitesseractic honeycomb
 24cell honeycomb
 Truncated 24cell honeycomb
 Snub 24cell honeycomb
 5cell honeycomb
 Truncated 5cell honeycomb
 Omnitruncated 5cell honeycomb
Notes
References
 Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN:9780471010036 [1]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345] See p318 [2]
 George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
 Klitzing, Richard. "4D Euclidean tesselations#4D". https://bendwavy.org/klitzing/dimensions/flat.htm. x3x3x *b3o *b3o, x3x3x *b3o4o, o3x3o *b3x4o, o4x3x3o4o  batitit  O92
 Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0387985859. https://archive.org/details/spherepackingsla0000conw_b8u0.
Fundamental convex regular and uniform honeycombs in dimensions 29
 

Space  Family  [math]\displaystyle{ {\tilde{A}}_{n1} }[/math]  [math]\displaystyle{ {\tilde{C}}_{n1} }[/math]  [math]\displaystyle{ {\tilde{B}}_{n1} }[/math]  [math]\displaystyle{ {\tilde{D}}_{n1} }[/math]  [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n1} }[/math] 
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 
Original source: https://en.wikipedia.org/wiki/Bitruncated tesseractic honeycomb.
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