Tesseractic honeycomb

Tesseractic honeycomb
Perspective projection of a 3x3x3x3 red-blue chessboard.
Type	Regular 4-space honeycomb Uniform 4-honeycomb
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
Coxeter-Dynkin diagrams
4-face type	{4,3,3}
Cell type	{4,3}
Face type	{4}
Edge figure	{3,4} (octahedron)
Vertex figure	{3,3,4} (16-cell)
Coxeter groups	${{\tilde {C}}}_{4}$ , [4,3,3,4] ${\tilde {B}}_{4}$ , [4,3,3^1,1]
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive

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

Its vertex figure is a 16-cell. 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 3-space. These are all part of the hypercubic honeycomb family of tessellations of the form {4,3,...,3,4}. Tessellations in this family are Self-dual.

Coordinates

Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).

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], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.

C4 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[4,3,3,4]:		×1	₁, ₂, ₃, ₄, ₅, ₆, ₇, ₈, ₉, ₁₀, ₁₁, ₁₂, ₁₃
[[4,3,3,4]]		×2	₍₁₎, ₍₂₎, ₍₁₃₎, ₁₈ ₍₆₎, ₁₉, ₂₀
[(3,3)[1⁺,4,3,3,4,1⁺]] ↔ [(3,3)[3^1,1,1,1]] ↔ [3,4,3,3]	↔ ↔	×6	₁₄, ₁₅, ₁₆, ₁₇

The [4,3,3^1,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

B4 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[4,3,3^1,1]:		×1	₅, ₆, ₇, ₈
<[4,3,3^1,1]>: ↔[4,3,3,4]	↔	×2	₉, ₁₀, ₁₁, ₁₂, ₁₃, ₁₄, ₍₁₀₎, ₁₅, ₁₆, ₍₁₃₎, ₁₇, ₁₈, ₁₉
[3[1⁺,4,3,3^1,1]] ↔ [3[3,3^1,1,1]] ↔ [3,3,4,3]	↔ ↔	×3	₁, ₂, ₃, ₄
[(3,3)[1⁺,4,3,3^1,1]] ↔ [(3,3)[3^1,1,1,1]] ↔ [3,4,3,3]	↔ ↔	×12	₂₀, ₂₁, ₂₂, ₂₃

The 24-cell honeycomb is similar, but as a body centered cubic, it has vertices positioned at integers (i,j,k,l), and half integers (i+1/2,j+1/2,k+1/2,l+1/2).

The tesseract can make a regular tessellation of the 4-sphere, with three tesseracts per face, with Schläfli symbol {4,3,3,3}, called an order-3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5-space.

The tesseract can make a regular tessellation of 4-dimensional hyperbolic space, with 5 tesseracts around each face, with Schläfli symbol {4,3,3,5}, called an order-5 tesseractic honeycomb.

Birectified tesseractic honeycomb

A birectified tesseractic honeycomb, , contains all rectified 16-cell (24-cell) facets and is the Voronoi tessellation of the D₄^* lattice. Facets can be identically colored from a doubled ${{\tilde {C}}}_{4}$ ×2, [[4,3,3,4]] symmetry, alternately colored from ${{\tilde {C}}}_{4}$ , [4,3,3,4] symmetry, three colors from ${\tilde {B}}_{4}$ , [4,3,3^1,1] symmetry, and 4 colors from ${\tilde {D}}_{4}$ , [3^1,1,1,1] symmetry.

References

Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 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, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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 2-9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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