four-dimensional A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs 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: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, 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 (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

honeycomb A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen. beekeeping, Beekee ...

. 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 object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...

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 t ...

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 polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

are ten

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

s and 20

triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...

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 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 incl ...

s, 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. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the s ...

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, equ ...

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 In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell. There are 3 unique degrees of runcinations of the 5-cell, including with per ...

represent the 20 roots of the

_4

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

simplectic honeycomb In geometry, the simplectic 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 nod ...

. The A lattice is the union of five A₄ lattices, and is the dual to the

omnitruncated 5-simplex honeycomb In Five-dimensional space, five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb (geometry), honeycomb). It is composed entirely of omnitruncat ...

, 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. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed t ...

of this lattice is an

omnitruncated 5-cell In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell. There are 3 unique degrees of runcinations of the 5-cell, including with pe ...

: ∪ ∪ ∪ ∪ = 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 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices. It is one of 18 u ...

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 of the regular 5-cell. There are two degrees of truncations, including a bitruncation. Truncated 5-cell The truncated 5-cell, tr ...

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, tr ...

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

is a tetrahedral antiprism, with 2

regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

, 8

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

, and 6

tetragonal disphenoid In geometry, a disphenoid () is a tetrahedron whose four Face (geometry), faces are Congruence (geometry), congruent acute-angled triangles. It can also be described as a tetrahedron in which every two Edge (geometry), edges that are opposite ea ...

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 subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

s that divide space into two half-spaces. The 3-space hyperplanes contain

quarter cubic honeycomb The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is c ...

s 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 4-simplex) facets.

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

calls this Hinton's honeycomb after

C. H. Hinton Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction works titled ''Scientific Romances''. He was interested in n-dimensional space, higher dimensions, particularly the Four-dimensional space, ...

, who described it in his book ''The Fourth Dimension'' in 1906. (The classification of Zonohededra, page 73) The facets of all

omnitruncated simplectic honeycomb In geometry an omnitruncated simplectic honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional Uniform honeycomb, uniform tessellation, based on the symmetry of the _n affine Coxeter group. Each is composed of omnitruncation (geometry), ...

s 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

.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