Tesseractic honeycomb
In fourdimensional euclidean geometry, the tesseractic honeycomb is one of the three regular spacefilling tessellations (or honeycombs), represented by Schläfli symbol {4,3,3,4}, and constructed by a 4dimensional packing of tesseract facets.
Tesseractic honeycomb  

Perspective projection of a 3x3x3x3 redblue chessboard.  
Type  Regular 4space honeycomb Uniform 4honeycomb 
Family  Hypercubic honeycomb 
Schläfli symbols  {4,3,3,4} t_{0,4}{4,3,3,4} {4,3,3^{1,1}} {4,4}^{2} {4,3,4}x{∞} {4,4}x{∞}^{2} {∞}^{4} 
CoxeterDynkin diagrams  
4face type  {4,3,3} 
Cell type  {4,3} 
Face type  {4} 
Edge figure  {3,4} (octahedron) 
Vertex figure  {3,3,4} (16cell) 
Coxeter groups  , [4,3,3,4] , [4,3,3^{1,1}] 
Dual  selfdual 
Properties  vertextransitive, edgetransitive, facetransitive, celltransitive, 4facetransitive 
Its vertex figure is a 16cell. Two tesseracts meet at each cubic cell, four meet at each square face, eight meet on each edge, and sixteen meet at each vertex.
It is an analog of the square tiling, {4,4}, of the plane and the cubic honeycomb, {4,3,4}, of 3space. These are all part of the hypercubic honeycomb family of tessellations of the form {4,3,...,3,4}. Tessellations in this family are Selfdual.
Coordinates
Vertices of this honeycomb can be positioned in 4space in all integer coordinates (i,j,k,l).
Sphere packing
Like all regular hypercubic honeycombs, the tesseractic honeycomb corresponds to a sphere packing of edgelengthdiameter spheres centered on each vertex, or (dually) inscribed in each cell instead. In the hypercubic honeycomb of 4 dimensions, vertexcentered 3spheres and cellinscribed 3spheres will both fit at once, forming the unique regular bodycentered cubic lattice of equalsized spheres (in any number of dimensions). Since the tesseract is radially equilateral, there is exactly enough space in the hole between the 16 vertexcentered 3spheres for another edgelengthdiameter 3sphere. (This 4dimensional body centered cubic lattice is actually the union of two tesseractic honeycombs, in dual positions.)
This is the same densest known regular 3sphere packing, with kissing number 24, that is also seen in the other two regular tessellations of 4space, the 16cell honeycomb and the 24cellhoneycomb. Each tesseractinscribed 3sphere kisses a surrounding shell of 24 3spheres, 16 at the vertices of the tesseract and 8 inscribed in the adjacent tesseracts. These 24 kissing points are the vertices of a 24cell of radius (and edge length) 1/2.
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 tesseract facets (like a checkerboard) with Schläfli symbol {4,3,3^{1,1}}. The lowest symmetry Wythoff construction has 16 types of facets around each vertex and a prismatic product Schläfli symbol {∞}^{4}. One can be made by stericating another.
Related polytopes and tessellations
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 

The 24cell honeycomb is similar, but in addition to the vertices at integers (i,j,k,l), it has vertices at half integers (i+1/2,j+1/2,k+1/2,l+1/2) of odd integers only. It is a halffilled body centered cubic (a checkerboard in which the red 4cubes have a central vertex but the black 4cubes do not).
The tesseract can make a regular tessellation of the 4sphere, with three tesseracts per face, with Schläfli symbol {4,3,3,3}, called an order3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5space.
The tesseract can make a regular tessellation of 4dimensional hyperbolic space, with 5 tesseracts around each face, with Schläfli symbol {4,3,3,5}, called an order5 tesseractic honeycomb.
Birectified tesseractic honeycomb
A birectified tesseractic honeycomb,
See also
Regular and uniform honeycombs in 4space:
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]
 George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)  Model 1
 Klitzing, Richard. "4D Euclidean tesselations". x∞o x∞o x∞o x∞o, x∞x x∞o x∞o x∞o, x∞x x∞x x∞o x∞o, x∞x x∞x x∞x x∞o,x∞x x∞x x∞x x∞x, x∞o x∞o x4o4o, x∞o x∞o o4x4o, x∞x x∞o x4o4o, x∞x x∞o o4x4o, x∞o x∞o x4o4x, x∞x x∞x x4o4o, x∞x x∞x o4x4o, x∞x x∞o x4o4x, x∞x x∞x x4o4x, x4o4x x4o4x, x4o4x o4x4o, x4o4x x4o4o, o4x4o o4x4o, x4o4o o4x4o, x4o4o x4o4o, x∞x o3o3o *d4x, x∞o o3o3o *d4x, x∞x x4o3o4x, x∞o x4o3o4x, x∞x x4o3o4o, x∞o x4o3o4o, o3o3o *b3o4x, x4o3o3o4x, x4o3o3o4o  test  O1
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} 