Tesseractic honeycomb

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Tesseractic honeycomb
Tesseractic tetracomb.png
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}
t0,4{4,3,3,4}
{4,3,31,1}
{4,4}2
{4,3,4}x{∞}
{4,4}x{∞}2
{∞}4
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
4-face type {4,3,3} Schlegel wireframe 8-cell.png
Cell type {4,3} Hexahedron.png
Face type {4}
Edge figure 8 {4,3}
(octahedron)
Vertex figure 16 {4,3,3}
(16-cell)
Coxeter groups {\tilde{C}}_4, [4,3,3,4]
{\tilde{B}}_4, [4,3,31,1]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-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[edit]

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

Constructions[edit]

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 {4,3,31,1}. 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[edit]

The [4,3,3,4], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, 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.

Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,3,4]: CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png ×1

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 1, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 2, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 3, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 4,
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 5, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 6, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 7, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 8,
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 9, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 10, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 11, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 12,
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 13

[[4,3,3,4]] CDel node c3.pngCDel split1.pngCDel nodeab c2.pngCDel 4a4b.pngCDel nodeab c1.png ×2 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png (1), CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png (2), CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png (13), CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 18
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png (6), CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png 19, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png 20
[(3,3)[1+,4,3,3,4,1+]]
= [(3,3)[31,1,1,1]]
= [3,4,3,3]
CDel node c2.pngCDel split1.pngCDel nodeab c1.pngCDel 4a4b.pngCDel nodes.png
= CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png
= CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
×6

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 14, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 15, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 16, CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png 17

The [4,3,31,1], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, 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.

Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,31,1]: CDel node c5.pngCDel 4.pngCDel node c4.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png ×1

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 5, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 6, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 7, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 8

<[4,3,31,1]>:
=[4,3,3,4]
CDel node c5.pngCDel 4.pngCDel node c4.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel nodeab c1.png
= CDel node c5.pngCDel 4.pngCDel node c4.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.png
×2

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png 9, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png 10, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png 11, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png 12, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png 13, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png 14,

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png (10), CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png 15, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png 16, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png (13), CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 17, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 18, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png 19

[3[1+,4,3,31,1]]
= [3[3,31,1,1]]
= [3,3,4,3]
CDel node c3.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.pngCDel 4a.pngCDel nodea.png
= CDel node c3.pngCDel 3.pngCDel node c2.pngCDel splitsplit1.pngCDel branch3 c1.pngCDel node c1.png
= CDel node c3.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
×3

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 1, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 2, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 3, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 4

[(3,3)[1+,4,3,31,1]]
= [(3,3)[31,1,1,1]]
= [3,4,3,3]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png
= CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png
= CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
×12

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png 20, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png 21, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 22, CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png 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 a 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[edit]

A birectified tesseractic honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png, 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, [4,3,31,1] symmetry, and 4 colors from {\tilde{D}}_4, [31,1,1,1] symmetry.

See also[edit]

Regular and uniform honeycombs in 4-space:

References[edit]

  • 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 [1]
    • (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
  • Richard Klitzing, 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