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}×{∞} {4,3,4}×{4,4} {4,3,4}×{∞}^{(2)} {4,4}^{(2)}×{∞} {∞}^{(5)} 
CoxeterDynkin diagrams 

5face type  {4,3^{3}} (5cube) 
4face type  {4,3,3} (tesseract) 
Cell type  {4,3} (cube) 
Face type  {4} (square) 
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 
In geometry, 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.
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3space, and the tesseractic honeycomb of 4space.
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)}.
The [4,3^{3},4], , Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5cubic honeycomb is geometrically identical to the 5cubic honeycomb.
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 three 5cubes on each cell. This could be considered as a tessellation on the 5sphere, an order3 penteractic honeycomb, {4,3^{4}}.
The Penrose tilings are 2dimensional aperiodic tilings that can be obtained as a projection of the 5cubic honeycomb along a 5fold rotational axis of symmetry. The vertices correspond to points in the 5dimensional cubic lattice, and the tiles are formed by connecting points in a predefined manner.^{[1]}
A tritruncated 5cubic honeycomb, , contains all bitruncated 5orthoplex facets and is the Voronoi tessellation of the D_{5}^{*} lattice. Facets can be identically colored from a doubled ×2, [[4,3^{3},4]] symmetry, alternately colored from , [4,3^{3},4] symmetry, three colors from , [4,3,3,3^{1,1}] symmetry, and 4 colors from , [3^{1,1},3,3^{1,1}] symmetry.
Regular and uniform honeycombs in 5space:
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} 