quarter cubic honeycomb  

(No image)  
Type  Uniform 4honeycomb 
Family  Quarter hypercubic honeycomb 
Schläfli symbol  r{4,3,3,4} r{4,3^{1,1}} r{4,3^{1,1}} q{4,3,3,4} 
CoxeterDynkin diagram 

4face type  h{4,3^{2}}, h_{3}{4,3^{2}}, 
Cell type  {3,3}, t_{1}{4,3}, 
Face type  {3} {4} 
Edge figure  Square pyramid 
Vertex figure  Elongated {3,4}×{} 
Coxeter group  = [4,3,3,4] = [4,3^{1,1}] = [3^{1,1,1,1}] 
Dual  
Properties  vertextransitive 
In fourdimensional Euclidean geometry, the rectified tesseractic honeycomb is a uniform spacefilling tessellation (or honeycomb) in Euclidean 4space. It is constructed by a rectification of a tesseractic honeycomb which creates new vertices on the middle of all the original edges, rectifying the cells into rectified tesseracts, and adding new 16cell facets at the original vertices. Its vertex figure is an octahedral prism, {3,4}×{}.
It is also called a quarter tesseractic honeycomb since it has half the vertices of the 4demicubic honeycomb, and a quarter of the vertices of a tesseractic honeycomb.^{[1]}
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 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}]  (none)  
<[3^{1,1,1,1}]> ↔ [3^{1,1},3,4] 
↔ 
×2 =  (none) 
<2[^{1,1}3^{1,1}]> ↔ [4,3,3,4] 
↔ 
×4 =  _{1}, _{2} 
[3[3,3^{1,1,1}]] ↔ [3,3,4,3] 
↔ 
×6 =  _{3}, _{4}, _{5}, _{6} 
[4[^{1,1}3^{1,1}]] ↔ [[4,3,3,4]] 
↔ 
×8 = ×2  _{7}, _{8}, _{9} 
[(3,3)[3^{1,1,1,1}]] ↔ [3,4,3,3] 
↔ 
×24 =  
[(3,3)[3^{1,1,1,1}]]^{+} ↔ [3^{+},4,3,3] 
↔ 
½ ×24 = ½  _{10} 
Regular and uniform honeycombs in 4space:
Space  Family  / /  

E^{2}  Uniform tiling  0_{[3]}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  0_{[4]}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  0_{[5]}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  0_{[6]}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  0_{[7]}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  0_{[8]}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  0_{[9]}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  0_{[10]}  δ_{10}  hδ_{10}  qδ_{10}  
E^{10}  Uniform 10honeycomb  0_{[11]}  δ_{11}  hδ_{11}  qδ_{11}  
E^{n1}  Uniform (n1)honeycomb  0_{[n]}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 