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.
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3<sup>3</sup>,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,3<sup>1,1</sup>}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {âÂÂ}<sup>(5)</sup>.
The [4,3<sup>3</sup>,4], , 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, {4,3<sup>4</sup>}.
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.
A tritruncated 5-cubic honeycomb, , contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D<sub>5</sub><sup>*</sup> lattice. Facets can be identically colored from a doubled ÃÂ2, <nowiki>[[</nowiki>4,3<sup>3</sup>,4] symmetry, alternately colored from , [4,3<sup>3</sup>,4] symmetry, three colors from , [4,3,3,3<sup>1,1</sup>] symmetry, and 4 colors from , [3<sup>1,1</sup>,3,3<sup>1,1</sup>] symmetry.
Regular and uniform honeycombs in 5-space: