geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the simplicial honeycomb (or -simplex honeycomb) is a dimensional infinite series of

honeycomb A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...

s, based on the

_n

affine

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of nodes with one node ringed. It is composed of -

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

facets, along with all rectified -simplices. It can be thought of as an -dimensional

hypercubic honeycomb In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for . The tessellation is constructed from 4 -hypercube ...

that has been subdivided along all

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

x+y+\cdots\in\mathbb

, then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

of an -''simplex honeycomb'' is an expanded -

. In 2 dimensions, the honeycomb represents the

triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

, with Coxeter graph filling the plane with alternately colored triangles. In 3 dimensions it represents the

tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names inc ...

, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph , with

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional space, four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pe ...

and

rectified 5-cell In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In ...

facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph , filling space by

5-simplex In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The ...

, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.

By dimension

Projection by folding

The (2n−1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional

by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...

Kissing number

These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the

. This represents the highest

kissing number In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...

for 2 and 3 dimensions, but falls short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.

References

* George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' *

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentGeombinatorics Alexander Soifer is a Russian-born American mathematician and mathematics author. Soifer obtained his Ph.D. in 1973 and has been a professor of mathematics at the University of Colorado since 1979. He was visiting fellow at Princeton University ...

4(1994), 49 - 56. * Norman Johnson ''Uniform Polytopes'', Manuscript (1991) * Coxeter, H.S.M. ''

Regular Polytopes ''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...

'', (3rd edition, 1973), Dover edition, * 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,

** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10(1.9 Uniform space-fillings) ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45 {{Honeycombs Honeycombs (geometry) Uniform polytopes

By dimension

Projection by folding

Kissing number

See also

References