24-cell honeycomb

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24-cell honeycomb
Icositetrachoronic tetracomb.png
A 24-cell and first layer of its adjacent 4-faces.
Type Regular 4-space honeycomb
Uniform 4-honeycomb
Schläfli symbol {3,4,3,3}
r{3,3,4,3}
2r{4,3,3,4}
2r{4,3,31,1}
{31,1,1,1}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes.png
4-face type {3,4,3} Schlegel wireframe 24-cell.png
Cell type {3,4} Uniform polyhedron-43-t2.svg
Face type {3}
Edge figure {3,3}
Vertex figure {4,3,3}
Dual {3,3,4,3}
Coxeter groups {\tilde{F}}_4, [3,4,3,3]
{\tilde{C}}_4, [4,3,3,4]
{\tilde{B}}_4, [4,3,31,1]
{\tilde{D}}_4, [31,1,1,1]
Properties regular

In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol {3,4,3,3}.

The dual tessellation by regular 16-cell honeycomb has Schläfli symbol {3,3,4,3}. Together with the tesseractic honeycomb (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space.

Kissing number[edit]

If a 3-sphere is inscribed in each hypercell of this tessellation, the resulting arrangement is the densest possible regular sphere packing in four dimensions, with the kissing number 24. The packing density of this arrangement is

\frac{\pi^2}{16}\cong0.61685.

Coordinates[edit]

The 24-cell honeycomb can be constructed as the Voronoi tessellation of the D4 or F4 root lattice. Each 24-cell is then centered at a D4 lattice point, i.e. one of

\left\{(x_i)\in\mathbb Z^4 : {\textstyle\sum_i} x_i \equiv 0\;(\mbox{mod }2)\right\}.

These points can also be described as Hurwitz quaternions with even square norm.

The vertices of the honeycomb lie at the deep holes of the D4 lattice. These are the Hurwitz quaternions with odd square norm.

It can be constructed as a birectified tesseractic honeycomb, by taking a tesseractic honeycomb and placing vertices at the centers of all the square faces. The 24-cell facets exist between these vertices as rectified 16-cells. If the coordinates of the tesseractic honeycomb are integers (i,j,k,l), the birectified tesseractic honeycomb vertices can be placed at all permutations of half-unit shifts in two of the four dimensions, thus: (i+½,j+½,k,l), (i+½,j,k+½,l), (i+½,j,k,l+½), (i,j+½,k+½,l), (i,j+½,k,l+½), (i,j,k+½,l+½).

Configuration[edit]

Each 24-cell in the 24-cell honeycomb has 24 neighboring 24-cells. With each neighbor it shares exactly one octahedral cell.

It has 24 more neighbors such that with each of these it shares a single vertex.

It has no neighbors with which it shares only an edge or only a face.

The vertex figure of the 24-cell honeycomb is a tesseract (4-dimensional cube). So there are 16 edges, 32 triangles, 24 octahedra, and 8 24-cells meeting at every vertex. The edge figure is a tetrahedron, so there are 4 triangles, 6 octahedra, and 4 24-cells surrounding every edge. Finally, the face figure is a triangle, so there are 3 octahedra and 3 24-cells meeting at every face.

Cross-sections[edit]

One way to visualize 4-dimensional figures is to consider various 3-dimensional cross-sections. Applying this technique to the 24-cell honeycomb gives rise to various 3-dimensional honeycombs with varying degrees of regularity.

Vertex-first sections
Rhombic dodecahedra.png Partial cubic honeycomb.png
Rhombic dodecahedral honeycomb Cubic honeycomb
Cell-first sections
Rectified cubic honeycomb.png Bitruncated cubic honeycomb.png
Rectified cubic honeycomb Bitruncated cubic honeycomb

A vertex-first cross-section is one orthogonal to a line joining opposite vertices of one of the 24-cells. For instance, one could take any of the coordinate hyperplanes in the coordinate system given above (i.e. the planes determined by xi = 0). The cross-section of {3,4,3,3} by one of these hyperplanes gives a rhombic dodecahedral honeycomb. Each of the rhombic dodecahedra corresponds to a maximal cross-section of one of the 24-cells intersecting the hyperplane (the center of each such 24-cell lies in the hyperplane). Accordingly, the rhombic dodecahedral honeycomb is the Voronoi tessellation of the D3 root lattice (a face-centered cubic lattice). Shifting this hyperplane halfway to one of the vertices (e.g. xi = ½) gives rise to a regular cubic honeycomb. In this case the center of each 24-cell lies off the hyperplane. Shifting again, so the hyperplane intersects the vertex, gives another rhombic dodecahedral honeycomb but with new 24-cells (the former ones having shrunk to points). In general, for any integer n, the cross-section through xi = n is a rhombic dodecahedral honeycomb, and the cross-section through xi = n + ½ is a cubic honeycomb. As the hyperplane moves through 4-space, the cross-section morphs between the two periodically.

A cell-first cross-section is one parallel to one of the octahedral cells of a 24-cell. Consider, for instance, the hyperplane orthogonal to (1,1,0,0). The cross-section of {3,4,3,3} by this hyperplane is a rectified cubic honeycomb. Each cuboctahedron in this honeycomb is a maximal cross-section of a 24-cell whose center lies in the plane. Meanwhile, each octahedron is a boundary cell of a 24-cell whose center lies off the plane. Shifting this hyperplane till it lies halfway between the center of a 24-cell and the boundary, one obtains a bitruncated cubic honeycomb. The cuboctahedra have shrunk, and the octahedra have grown until they are both truncated octahedra. Shifting again, so the hyperplane intersects the boundary of the central 24-cell gives a rectified cubic honeycomb again, the cuboctahedra and octahedra having swapped positions. As the hyperplane sweeps through 4-space, the cross-section morphs between these two honeycombs periodically.

Symmetry constructions[edit]

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 24-cell facets. In all cases, eight 24-cells meet at each vertex, but the vertex figures have different symmetry generators.

Coxeter group Schläfli symbols Coxeter diagram Facets
(24-cells)
Vertex figure
(8-cell)
Vertex
figure
symmetry
order
{\tilde{F}}_4 \begin{Bmatrix} 3, 4, 3, 3 \end{Bmatrix} {3,4,3,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 8: CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 384
\left\{\begin{array}{l}3\\3,4,3\end{array}\right\} r{3,3,4,3} CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 6: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
2: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png 96
{\tilde{C}}_4 \left\{\begin{array}{l}3,4\\3,4\end{array}\right\} 2r{4,3,3,4} CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 4,4: CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png 64
{\tilde{B}}_4 \left\{\begin{array}{l}3\\3\\3,4\end{array}\right\} 2r{4,3,31,1} CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 2,2: CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
4: CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png 32
{\tilde{D}}_4 \left\{\begin{array}{l}3\\3\\3\\3\end{array}\right\} {31,1,1,1} CDel nodes.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes.png 2,2,2,2:
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png 16

Related honeycombs[edit]

The [3,4,3,3], CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,31,1] families. The alternation (13) is also repeated in other families.

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

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

[3,4,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png ×1

CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 2, CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 4, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 7, CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 13,
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 14, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 15, CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 16, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 17,
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 18, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 19, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 20, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 21,
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 22 CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 23, CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 24, CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 25,
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 26, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 27, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png 28, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png 29

[(3,3)[3,3,4,3*]]
=[(3,3)[31,1,1,1]]
=[3,4,3,3]
CDel node c2.pngCDel split1.pngCDel nodeab c1.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.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
×4

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png (2), CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png (4), CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png (7), CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png (13)

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

This honeycomb is one of ten uniform honeycombs constructed by the {\tilde{D}}_4 Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)*] (index 24), [3,3,4,3*] (index 6), [1+,4,3,3,4,1+] (index 4), [31,1,3,4,1+] (index 2) are all isomorphic to [31,1,1,1]. The ten permutations are listed with its highest extended symmetry relation:

Extended
symmetry
Extended
diagram
Order Honeycombs
[31,1,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png ×1 (none)
<[31,1,1,1]>
= [31,1,3,4]
CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel split1.pngCDel nodeab c4.png
= CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 4.pngCDel node.png
×2 (none)
<<[1,131,1]>>
= [4,3,3,4]
CDel nodeab c1.pngCDel split2.pngCDel node c3.pngCDel split1.pngCDel nodeab c2.png
= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png
×4 CDel nodes 11.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png 1, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes.png 2
[3[3,31,1,1]]
= [3,4,3,3]
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
×6 CDel node 1.pngCDel 3.pngCDel node.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png3, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png 4, CDel node.pngCDel 3.pngCDel node.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png 5, CDel node.pngCDel 3.pngCDel node 1.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png 6
[<<[1,131,1]>>]
= [[4,3,3,4]]
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png
= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.png
×8 CDel nodes.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes.png 7, CDel nodes 11.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 11.png 8, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 9, CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel split1.pngCDel nodes hh.png 10
[(3,3)[31,1,1,1]]
= [3,3,4,3]
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
×24

See also[edit]

Other 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 88
  • Richard Klitzing, 4D, Euclidean tesselations o4o3x3o4o, o3x3o *b3o4o, o3x3o *b3o4o, o3x3o4o3o, o3o3o4o3x - icot - O88