5-cubic honeycomb

5-cubic honeycomb
(no image)
Type	Regular 5-space honeycomb Uniform 5-honeycomb
Family	Hypercube honeycomb
Schläfli symbol	Template:Math
Coxeter-Dynkin diagrams	Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
5-face type	Template:Math (5-cube)
4-face type	Template:Math (tesseract)
Cell type	Template:Math (cube)
Face type	Template:Math (square)
Face figure	Template:Math (octahedron)
Edge figure	Template:Math (16-cell)
Vertex figure	Template:Math (5-orthoplex)
Coxeter group	${\tilde{C}}_5$ Template:Math
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive

In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.

Constructions

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3³,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,3^1,1}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}⁽⁵⁾.

Related polytopes and honeycombs

The [4,3³,4], Template:CDD, Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.

The 5-cubic honeycomb can be alternated into the 5-demicubic honeycomb, replacing the 5-cubes with 5-demicubes, and the alternated gaps are filled by 5-orthoplex facets.

It is also related to the regular 6-cube which exists in 6-space with three 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,3⁴}.

The Penrose tilings are 2-dimensional aperiodic tilings that can be obtained as a projection of the 5-cubic honeycomb along a 5-fold rotational axis of symmetry. The vertices correspond to points in the 5-dimensional cubic lattice, and the tiles are formed by connecting points in a predefined manner.

Tritruncated 5-cubic honeycomb

A tritruncated 5-cubic honeycomb, Template:CDD, contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D₅^* lattice. Facets can be identically colored from a doubled ${\tilde{C}}_5$×2, [[4,3³,4]] symmetry, alternately colored from ${\tilde{C}}_5$, [4,3³,4] symmetry, three colors from ${\tilde{B}}_5$, [4,3,3,3^1,1] symmetry, and 4 colors from ${\tilde{D}}_5$, [3^1,1,3,3^1,1] symmetry.

See also

• List of regular polytopes

Regular and uniform honeycombs in 5-space:

• 5-demicubic honeycomb
• 5-simplex honeycomb
• Truncated 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Template:ISBN 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, Template:ISBN [1]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Template:Navbar-collapsible
Space	Family	${\tilde{A}}_{n-1}$	${\tilde{C}}_{n-1}$	${\tilde{B}}_{n-1}$	${\tilde{D}}_{n-1}$	${\tilde{G}}_2$ / ${\tilde{F}}_4$ / ${\tilde{E}}_{n-1}$
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