four-dimensional Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

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 ...

. It is composed of

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 ...

s 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 ...

s facets in a ratio of 1:1.

Structure

Cells of 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 ...

are ten

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

s and 20

triangular prism In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...

s, corresponding to the ten

s and 20

s that meet at each vertex. All the vertices lie in parallel realms in which they form alternated cubic honeycombs, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell.

Alternate names

* Cyclopentachoric tetracomb * Pentachoric-dispentachoric tetracomb

Projection by folding

The ''5-cell honeycomb'' can be projected into the 2-dimensional

square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tili ...

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 ...

: Two different

aperiodic tiling An aperiodic tiling is a non-periodic Tessellation, tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic set of prototiles, aperiodic if copie ...

s with 5-fold symmetry can be obtained by projecting two-dimensional slices of the honeycomb: the

Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and a tiling is ''aperiodic'' if it does not contain arbitrarily large Perio ...

composed of rhombi, and the

Tübingen triangle The Tübingen triangle is a form of substitution tiling. It is, apart from the Penrose tiling, Penrose rhomb tilings and their variations, a classical candidate to model 5-fold (respectively 10-fold) quasicrystals. The inflation factor is – as ...

tiling composed of isosceles triangles.

A4 lattice

The

of the ''5-cell honeycomb'' is called the A4 lattice, or 4-simplex lattice. The 20 vertices of its

, the runcinated 5-cell represent the 20 roots of the

_4

Coxeter group. It is the 4-dimensional case of a

simplectic honeycomb In geometry, the simplicial honeycomb (or -simplex honeycomb) is a dimensional infinite series of honeycombs, based on the _n affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of nodes with one node ...

. The A lattice is the union of five A₄ lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the

Voronoi cell In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...

of this lattice is an omnitruncated 5-cell : ∪ ∪ ∪ ∪ = dual of

Related polytopes and honeycombs

The ''tops'' of the 5-cells in this honeycomb adjoin the ''bases'' of the 5-cells, and vice versa, in adjacent laminae (or layers); but alternating laminae may be inverted so that the tops of the rectified 5-cells adjoin the tops of the rectified 5-cells and the bases of the 5-cells adjoin the bases of other 5-cells. This inversion results in another non-Wythoffian uniform convex honeycomb.

Octahedral prism In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms. Alternative names *Octahedral dyadic prism ( Norman W. Johnson) *Ope (Jonathan Bowers, for o ...

s and

tetrahedral prism In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedron, polyhedral cells: 2 tetrahedron, tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vert ...

s may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.

Rectified 5-cell honeycomb

The rectified 4-simplex honeycomb or rectified 5-cell honeycomb is a space-filling

Alternate names

* small cyclorhombated pentachoric tetracomb * small prismatodispentachoric tetracomb

Cyclotruncated 5-cell honeycomb

The cyclotruncated 4-simplex honeycomb or cyclotruncated 5-cell honeycomb is a space-filling

. It can also be seen as a birectified 5-cell honeycomb. It is composed of

truncated 5-cell In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the Truncation (geometry), truncation of the regular 5-cell. There are two degrees of truncations, including a bitruncation. Truncated 5-cell The ...

s, and

bitruncated 5-cell In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell. There are two degrees of truncations, including a bitruncation. Truncated 5-cell The truncated 5-cell, tru ...

s facets in a ratio of 2:2:1. Its

is a tetrahedral antiprism, with 2

regular tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

, 8

triangular pyramid In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex ...

, and 6

tetragonal disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same s ...

cells, defining 2

, 8

, and 6

facets around a vertex. It can be constructed as five sets of parallel

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 ...

s that divide space into two half-spaces. The 3-space hyperplanes contain quarter cubic honeycombs as a collection facets.

Alternate names

* Cyclotruncated pentachoric tetracomb * Small truncated-pentachoric tetracomb

Truncated 5-cell honeycomb

The truncated 4-simplex honeycomb or truncated 5-cell honeycomb is a space-filling

. It can also be called a cyclocantitruncated 5-cell honeycomb.

Alaternate names

* Great cyclorhombated pentachoric tetracomb * Great truncated-pentachoric tetracomb

Cantellated 5-cell honeycomb

The cantellated 4-simplex honeycomb or cantellated 5-cell honeycomb is a space-filling

. It can also be called a cycloruncitruncated 5-cell honeycomb.

Alternate names

* Cycloprismatorhombated pentachoric tetracomb * Great prismatodispentachoric tetracomb

Bitruncated 5-cell honeycomb

The bitruncated 4-simplex honeycomb or bitruncated 5-cell honeycomb is a space-filling

. It can also be called a cycloruncicantitruncated 5-cell honeycomb.

Alternate names

* Great cycloprismated pentachoric tetracomb * Grand prismatodispentachoric tetracomb

Omnitruncated 5-cell honeycomb

The omnitruncated 4-simplex honeycomb or omnitruncated 5-cell honeycomb is a space-filling

. It can also be seen as a cyclosteriruncicantitruncated 5-cell honeycomb. It is composed entirely of omnitruncated 5-cell (omnitruncated 4-simplex) facets.

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

calls this Hinton's honeycomb after C. H. Hinton, who described it in his book ''The Fourth Dimension'' in 1906. (The classification of Zonohededra, page 73) The facets of all omnitruncated simplectic honeycombs are called permutohedra and can be positioned in ''n+1'' space with integral coordinates, permutations of the whole numbers (0,1,...,n).

Alternate names

* Omnitruncated cyclopentachoric tetracomb * Great-prismatodecachoric tetracomb

A₄^* lattice

The A lattice is the union of five A₄ lattices, and is the dual to the omnitruncated 5-cell honeycomb, and therefore the

of this lattice is an omnitruncated 5-cell.The Lattice A4*
/ref> : ∪ ∪ ∪ ∪ = dual of

Alternated form

This honeycomb can be alternated, creating omnisnub 5-cells with irregular

s created at the deleted vertices. Although it is not uniform, the 5-cells have a symmetry of order 10.

Notes

References

* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) * ''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* George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' Model 134 * , x3o3o3o3o3*a - cypit - O134, x3x3x3x3x3*a - otcypit - 135, x3x3x3o3o3*a - gocyropit - O137, x3x3o3x3o3*a - cypropit - O138, x3x3x3x3o3*a - gocypapit - O139, x3x3x3x3x3*a - otcypit - 140 * Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals, Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013) {{Honeycombs Honeycombs (geometry) 5-polytopes

Structure

Alternate names

Projection by folding

A4 lattice

Related polytopes and honeycombs

Rectified 5-cell honeycomb

Alternate names

Cyclotruncated 5-cell honeycomb

Alternate names

Truncated 5-cell honeycomb

Alaternate names

Cantellated 5-cell honeycomb

Alternate names

Bitruncated 5-cell honeycomb

Alternate names

Omnitruncated 5-cell honeycomb

Alternate names

A4* lattice

Alternated form

See also

Notes

References

A₄^* lattice