7-cubic honeycomb
(no image)
Type	Regular 7-honeycomb Uniform 7-honeycomb
Family	Hypercube 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 figure	8{4,3,3} (16-cell)
Vertex figure	128{4,35} (7-orthoplex)
Coxeter group	[4,35,4]
Dual	self-dual
Properties	vertex-transitive,edge-transitive,face-transitive,cell-transitive

The7-cubic honeycomb orhepteractic honeycomb is the only regular space-fillingtessellation (orhoneycomb) in Euclidean 7-space.

It is analogous to thesquare tiling of the plane and to thecubic honeycomb of 3-space.

There are many differentWythoff constructions of this honeycomb. The most symmetric form isregular, withSchläfli symbol {4,3⁵,4}. Another form has two alternating7-cube facets (like a checkerboard) with Schläfli symbol {4,3⁴,3^1,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}⁽⁷⁾.

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

The7-cubic honeycomb can bealternated into the7-demicubic honeycomb, replacing the 7-cubes with7-demicubes, and the alternated gaps are filled by7-orthoplex facets.

Aquadritruncated 7-cubic honeycomb,, contains alltritruncated 7-orthoplex facets and is theVoronoi tessellation of theD₇^* lattice. Facets can be identically colored from a doubled${\displaystyle {\tilde{C}}_7}$×2, [[4,3⁵,4]] symmetry, alternately colored from${\displaystyle {\tilde{C}}_7}$, [4,3⁵,4] symmetry, three colors from${\displaystyle {\tilde{B}}_7}$, [4,3⁴,3^1,1] symmetry, and 4 colors from${\displaystyle {\tilde{D}}_7}$, [3^1,1,3³,3^1,1] symmetry.

• List of regular polytopes

• 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 ofH. 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]

v t e Fundamental convexregular anduniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde{A}}_{n-1}}$	${\displaystyle {\tilde{C}}_{n-1}}$	${\displaystyle {\tilde{B}}_{n-1}}$	${\displaystyle {\tilde{D}}_{n-1}}$	${\displaystyle {\tilde{G}}_2}$ /${\displaystyle {\tilde{F}}_4}$ /${\displaystyle {\tilde{E}}_{n-1}}$
E2	Uniform tiling	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 •331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 •251 •521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 •2k1 •k21

• Honeycombs (geometry)
• 8-polytopes
• Regular tessellations