24-cell honeycomb
24-cell honeycomb | |
---|---|
A 24-cell and first layer of its adjacent 4-faces. | |
Type | 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 |
|
4-face type | {3,4,3} |
Cell type | {3,4} |
Face type | {3} |
Edge figure |
{3,3} |
Vertex figure | {4,3,3} |
Dual | {3,3,4,3}
|
Coxeter groups | , [3,4,3,3] , [4,3,3,4] , [4,3,31,1] , [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
Coordinates
The 24-cell honeycomb can be constructed as the
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+1/2,j+1/2,k,l), (i+1/2,j,k+1/2,l), (i+1/2,j,k,l+1/2), (i,j+1/2,k+1/2,l), (i,j+1/2,k,l+1/2), (i,j,k+1/2,l+1/2).
Configuration
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
Cross-sections
One way to visualize a 4-dimensional figure is to consider various 3-dimensional cross-sections. That is, the intersection of various hyperplanes with the figure in question. Applying this technique to the 24-cell honeycomb gives rise to various 3-dimensional honeycombs with varying degrees of regularity.
Vertex-first sections | |
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Rhombic dodecahedral honeycomb | Cubic honeycomb |
Cell-first sections | |
Rectified cubic honeycomb
|
Bitruncated cubic honeycomb |
A vertex-first cross-section uses some hyperplane
A cell-first cross-section uses some hyperplane parallel to one of the octahedral cells of a 24-cell. Consider, for instance, some hyperplane orthogonal to the vector (1,1,0,0). The cross-section of {3,4,3,3} by this hyperplane is a
Kissing number
If a 3-sphere is inscribed in each hypercell of this tessellation, the resulting arrangement is the densest known[note 1] regular sphere packing in four dimensions, with the kissing number 24. The packing density of this arrangement is
Each inscribed 3-sphere kisses 24 others at the centers of the octahedral facets of its 24-cell, since each such octahedral cell is shared with an adjacent 24-cell. In a unit-edge-length tessellation, the diameter of the spheres (the distance between the centers of kissing spheres) is √2.
Just outside this surrounding shell of 24 kissing 3-spheres is another less dense shell of 24 3-spheres which do not kiss each other or the central 3-sphere; they are inscribed in 24-cells with which the central 24-cell shares only a single vertex (rather than an octahedral cell). The center-to-center distance between one of these spheres and any of its shell neighbors or the central sphere is 2.
Alternatively, the same sphere packing arrangement with kissing number 24 can be carried out with smaller 3-spheres of edge-length-diameter, by locating them at the centers and the vertices of the 24-cells. (This is equivalent to locating them at the vertices of a 16-cell honeycomb of unit-edge-length.) In this case the central 3-sphere kisses 24 others at the centers of the cubical facets of the three tesseracts inscribed in the 24-cell. (This is the unique body-centered cubic packing of edge-length spheres of the tesseractic honeycomb.)
Just outside this shell of kissing 3-spheres of diameter 1 is another less dense shell of 24 non-kissing 3-spheres of diameter 1; they are centered in the adjacent 24-cells with which the central 24-cell shares an octahedral facet. The center-to-center distance between one of these spheres and any of its shell neighbors or the central sphere is √2.
Symmetry constructions
There are five different Wythoff constructions of this tessellation as a uniform polytope. They are geometrically identical to the regular form, but the symmetry differences can be represented by 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) |
8-cell )
|
Vertex figure symmetry order | |
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= [3,4,3,3] | {3,4,3,3} | 8: | 384 | |||
r{3,3,4,3} | 6: 2: |
96 | ||||
= [4,3,3,4] | 2r{4,3,3,4} | 4,4: | 64 | |||
= [4,3,31,1] | 2r{4,3,31,1} | 2,2: 4: |
32 | |||
= [31,1,1,1] | {31,1,1,1} | 2,2,2,2: |
16 |
See also
Other uniform honeycombs in 4-space:
- Truncated 5-cell honeycomb
- Omnitruncated 5-cell honeycomb
- Truncated 24-cell honeycomb
- Rectified 24-cell honeycomb
- Snub 24-cell honeycomb
Notes
- ^ The sphere packing problem and the kissing number problem are remarkably difficult and optimal solutions are only known in 1, 2, 3, 8, and 24 dimensions (plus dimension 4 for the kissing number problem).
References
- ISBN 0-486-61480-8p. 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
- Klitzing, Richard. "4D Euclidean tesselations". o4o3x3o4o, o3x3o *b3o4o, o3x3o *b3o4o, o3x3o4o3o, o3o3o4o3x - icot - O88
Space | Family | / / | ||||
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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 | |
E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | {3[n]}
|
δn | hδn | qδn | 1k2 • 2k1 • k21 |