5cubic honeycomb
The 5cubic honeycomb or penteractic honeycomb is the only regular spacefilling tessellation (or honeycomb) in Euclidean 5space. Four 5cubes meet at each cubic cell, and it is more explicitly called an order4 penteractic honeycomb.
5cubic honeycomb  

(no image)  
Type  Regular 5space honeycomb Uniform 5honeycomb 
Family  Hypercube honeycomb 
Schläfli symbol  {4,3^{3},4} t_{0,5}{4,3^{3},4} {4,3,3,3^{1,1}} {4,3,4}x{∞} {4,3,4}x{4,4} {4,3,4}x{∞}^{2} {4,4}^{2}x{∞} {∞}^{5} 
CoxeterDynkin diagrams 

5face type  {4,3^{3}} 
4face type  {4,3,3} 
Cell type  {4,3} 
Face type  {4} 
Face figure  {4,3} (octahedron) 
Edge figure  8 {4,3,3} (16cell) 
Vertex figure  32 {4,3^{3}} (5orthoplex) 
Coxeter group  , [4,3^{3},4] 
Dual  selfdual 
Properties  vertextransitive, edgetransitive, facetransitive, celltransitive 
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3space, and the tesseractic honeycomb of 4space.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3^{3},4}. Another form has two alternating 5cube facets (like a checkerboard) with Schläfli symbol {4,3,3,3^{1,1}}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}^{5}.
Related polytopes and honeycombs
The [4,3^{3},4],
The 5cubic honeycomb can be alternated into the 5demicubic honeycomb, replacing the 5cubes with 5demicubes, and the alternated gaps are filled by 5orthoplex facets.
It is also related to the regular 6cube which exists in 6space with 3 5cubes on each cell. This could be considered as a tessellation on the 5sphere, an order3 penteractic honeycomb, {4,3^{4}}.
Tritruncated 5cubic honeycomb
A tritruncated 5cubic honeycomb,
See also
 List of regular polytopes
Regular and uniform honeycombs in 5space:
References
 Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808 p. 296, Table II: Regular honeycombs
 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
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
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} 