A regular square tiling. 1 color 
A cubic honeycomb in its regular form. 1 color 
A checkboard square tiling 2 colors 
A cubic honeycomb checkerboard. 2 colors 
Expanded square tiling 3 colors 
Expanded cubic honeycomb 4 colors 
4 colors 
8 colors 
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in ndimensional spaces with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter group R_{n} (or B^{~}_{n–1}) for n ≥ 3.
The tessellation is constructed from 4 nhypercubes per ridge. The vertex figure is a crosspolytope {3...3,4}.
The hypercubic honeycombs are selfdual.
Coxeter named this family as δ_{n+1} for an ndimensional honeycomb.
A Wythoff construction is a method for constructing a uniform polyhedron or plane tiling.
The two general forms of the hypercube honeycombs are the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard.
A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lowerdimensional elements. For example, an expanded cubic honeycomb has cubic cells centered on the original cubes, on the original faces, on the original edges, on the original vertices, creating 4 colors of cells around in vertex in 1:3:3:1 counts.
The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, in which each of the three axial directions may have different edge lengths. The facets are hyperrectangles, also called orthotopes; in 2 and 3 dimensions the orthotopes are rectangles and cuboids respectively.
δ_{n}  Name  Schläfli symbols  CoxeterDynkin diagrams  

Orthotopic {∞}^{(n)} (2^{m} colors, m < n) 
Regular (Expanded) {4,3^{n–1},4} (1 color, n colors) 
Checkerboard {4,3^{n–4},3^{1,1}} (2 colors)  
δ_{2}  Apeirogon  {∞}  
δ_{3}  Square tiling  {∞}^{(2)} {4,4} 


δ_{4}  Cubic honeycomb  {∞}^{(3)} {4,3,4} {4,3^{1,1}} 


δ_{5}  4cube honeycomb  {∞}^{(4)} {4,3^{2},4} {4,3,3^{1,1}} 


δ_{6}  5cube honeycomb  {∞}^{(5)} {4,3^{3},4} {4,3^{2},3^{1,1}} 


δ_{7}  6cube honeycomb  {∞}^{(6)} {4,3^{4},4} {4,3^{3},3^{1,1}} 


δ_{8}  7cube honeycomb  {∞}^{(7)} {4,3^{5},4} {4,3^{4},3^{1,1}} 


δ_{9}  8cube honeycomb  {∞}^{(8)} {4,3^{6},4} {4,3^{5},3^{1,1}} 


δ_{n}  nhypercubic honeycomb  {∞}^{(n)} {4,3^{n3},4} {4,3^{n4},3^{1,1}} 
... 
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} 