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{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Tesseractic honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Tesseractic tetracomb.png|280px]]<br>[[Perspective projection]] of a 3x3x3x3 red-blue chessboard.
|-
|bgcolor=#e7dcc3|Type||[[List_of_regular_polytopes#Tessellations_of_Euclidean_4-space|Regular 4-space honeycomb]]<BR>[[Uniform_polyteron#Regular_and_uniform_honeycombs|Uniform 4-honeycomb]]
|-
|bgcolor=#e7dcc3|Family||[[Hypercubic honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s|| {4,3,3,4}<BR>t<sub>0,4</sub>{4,3,3,4}<BR>{4,3,3<sup>1,1</sup>}<BR>{4,4}<sup>2</sup><BR>{4,3,4}x{&infin;}<BR>{4,4}x{&infin;}<sup>2</sup><BR>{&infin;}<sup>4</sup>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|4|node|3|node|3|node|4|node_1}}<BR>{{CDD|nodes|split2|node|3|node|4|node_1}}<BR>{{CDD|node_1|4|node|4|node|2|node_1|4|node|4|node}}<BR>{{CDD|node_1|4|node|4|node|2|node_1|infin|node|2|node_1|infin|node}}<BR>{{CDD|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}}
|-
|bgcolor=#e7dcc3|4-face type||[[tesseract|{4,3,3}]] [[File:Schlegel_wireframe_8-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell type||[[cube|{4,3}]] [[File:Hexahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Face type||[[square (geometry)|{4}]]
|-
|bgcolor=#e7dcc3|Edge figure||8 [[cube|{4,3}]]<BR>([[octahedron]])
|-
|bgcolor=#e7dcc3|Vertex figure||16 [[tesseract|{4,3,3}]]<BR>([[16-cell]])
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||<math>{\tilde{C}}_4</math>, [4,3,3,4]<BR><math>{\tilde{B}}_4</math>, [4,3,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Dual||[[Self-dual polytope|self-dual]]
|-
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]], [[edge-transitive]], [[face-transitive]], [[cell-transitive]]
|}
In [[Four-dimensional space|four-dimensional]] [[euclidean geometry]], the '''tesseractic honeycomb''' is one of the three [[List_of_regular_polytopes#Tessellations_of_Euclidean_4-space|regular]] space-filling [[tessellation]]s (or [[honeycomb (geometry)|honeycomb]]s), 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 (geometry)|cell]], four meet at each [[Square (geometry)|square]] [[Face (geometry)|face]], eight meet on each [[Edge (geometry)|edge]], and sixteen meet at each [[Vertex (geometry)|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 polytope|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 construction]]s of this honeycomb. The most symmetric form is [[Regular polytope|regular]], with [[Schläfli symbol]] {4,3,3,4}. Another form has two alternating [[tesseract]] facets (like a checkerboard) with Schläfli symbol {4,3,3<sup>1,1</sup>}. The lowest symmetry Wythoff construction has 16 types of facets around each vertex and a prismatic product Schläfli symbol {&infin;}<sup>4</sup>. One can be made by [[sterication|stericating]] another.
 
== Related polytopes and tessellations ==
{{C4_honeycombs}}
 
{{B4 honeycombs}}
 
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 [[N-sphere|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 [[List_of_regular_polytopes#Tessellations_of_hyperbolic_4-space|order-5 tesseractic honeycomb]].
 
=== Birectified tesseractic honeycomb ===
A [[birectified tesseractic honeycomb]], {{CDD|node_1|split1|nodes|4a4b|nodes}}, contains all rectified 16-cell ([[24-cell]]) facets and is the [[Voronoi tessellation]] of the [[16-cell honeycomb#D4 lattice|D<sub>4</sub><sup>*</sup> lattice]]. Facets can be identically colored from a doubled <math>{\tilde{C}}_4</math>×2, <nowiki>[[</nowiki>4,3,3,4]] symmetry, alternately colored from <math>{\tilde{C}}_4</math>, [4,3,3,4] symmetry, three colors from <math>{\tilde{B}}_4</math>, [4,3,3<sup>1,1</sup>] symmetry, and 4 colors from <math>{\tilde{D}}_4</math>, [3<sup>1,1,1,1</sup>] symmetry.
 
== See also ==
Regular and uniform honeycombs in 4-space:
*[[16-cell honeycomb]]
*[[24-cell honeycomb]]
*[[5-cell honeycomb]]
*[[Truncated 5-cell honeycomb]]
*[[Omnitruncated 5-cell honeycomb]]
 
== References ==
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.&nbsp;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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (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
* {{KlitzingPolytopes|flat.htm|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
{{Honeycombs}}
 
[[Category:Honeycombs (geometry)]]
[[Category:5-polytopes]]

Latest revision as of 17:05, 14 March 2013

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}
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 Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD
4-face type {4,3,3}
Cell type {4,3}
Face type {4}
Edge figure 8 {4,3}
(octahedron)
Vertex figure 16 {4,3,3}
(16-cell)
Coxeter groups C~4, [4,3,3,4]
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

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 {4,3,31,1}. 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

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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

A birectified tesseractic honeycomb, Template:CDD, 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 C~4×2, [[4,3,3,4]] symmetry, alternately colored from C~4, [4,3,3,4] symmetry, three colors from B~4, [4,3,31,1] symmetry, and 4 colors from D~4, [31,1,1,1] symmetry.

See also

Regular and uniform honeycombs in 4-space:

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 [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
  • Template:KlitzingPolytopes 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

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