7-cubic honeycomb

7-cubic honeycomb
(no image)
TypeRegular 7-space honeycomb
FamilyHypercube honeycomb
Schläfli symbol {4,35,4}
{4,34,31,1}
{∞}7
Coxeter-Dynkin diagrams

7-face type{4,3,3,3,3,3}
6-face type{4,3,3,3,3}
5-face type{4,3,3,3}
4-face type{4,3,3}
Cell type{4,3}
Face type{4}
Face figure{4,3}
(octahedron)
Edge figure8 {4,3,3}
(16-cell)
Vertex figure128 {4,35}
(7-orthoplex)
Coxeter group[4,35,4]
Dualself-dual
Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive

The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,35,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,34,31,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}7.

Related honeycombs

The [4,35,4], , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb.

The 7-cubic honeycomb can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets.

Quadritruncated 7-cubic honeycomb

A quadritruncated 7-cubic honeycomb, , containins all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D7* lattice. Facets can be identically colored from a doubled {\tilde{C}}_7×2, [[4,35,4]] symmetry, alternately colored from {\tilde{C}}_7, [4,35,4] symmetry, three colors from {\tilde{B}}_7, [4,34,31,1] symmetry, and 4 colors from {\tilde{D}}_7, [31,1,33,31,1] symmetry.

See also

References

Fundamental convex regular and uniform honeycombs in dimensions 2–10
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}
Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
Uniform n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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