Rectified tesseractic honeycomb
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}×{}.
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 
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]
Related honeycombs
The [4,3,3,4],
C4 honeycombs  

Extended symmetry 
Extended diagram 
Order  Honeycombs 
[4,3,3,4]:  ×1 
 
[[4,3,3,4]]  ×2  
[(3,3)[1^{+},4,3,3,4,1^{+}]] ↔ [(3,3)[3^{1,1,1,1}]] ↔ [3,4,3,3] 
↔ ↔ 
×6 

The [4,3,3^{1,1}],
B4 honeycombs  

Extended symmetry 
Extended diagram 
Order  Honeycombs  
[4,3,3^{1,1}]:  ×1 
 
<[4,3,3^{1,1}]>: ↔[4,3,3,4] 
↔ 
×2 
 
[3[1^{+},4,3,3^{1,1}]] ↔ [3[3,3^{1,1,1}]] ↔ [3,3,4,3] 
↔ ↔ 
×3 
 
[(3,3)[1^{+},4,3,3^{1,1}]] ↔ [(3,3)[3^{1,1,1,1}]] ↔ [3,4,3,3] 
↔ ↔ 
×12 

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 =  
[3[3,3^{1,1,1}]] ↔ [3,3,4,3] 
↔ 
×6 =  
[4[^{1,1}3^{1,1}]] ↔ [[4,3,3,4]] 
↔ 
×8 = ×2  
[(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 
See also
Regular and uniform honeycombs in 4space:
Notes
 Coxeter, Regular and SemiRegular Polytopes III, (1988), p318
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
 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". o4x3o3o4o, o3o3o *b3x4o, x3o3x *b3o4o, x3o3x *b3o *b3o  rittit  O87
 Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0387985859.
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
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} 