5-cell honeycomb

In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1.

4-simplex honeycomb
(No image)
TypeUniform 4-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[5]}
Coxeter diagram
4-face types{3,3,3}
t1{3,3,3}
Cell types{3,3}
t1{3,3}
Face types{3}
Vertex figure
t0,3{3,3,3}
Symmetry×2, {3[5]}
Propertiesvertex-transitive

Structure

Cells of the vertex figure are ten tetrahedrons and 20 triangular prisms, corresponding to the ten 5-cells and 20 rectified 5-cells 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.[1]

Alternate names

  • Cyclopentachoric tetracomb
  • Pentachoric-dispentachoric tetracomb

Projection by folding

The 5-cell honeycomb can be projected into the 2-dimensional square tiling by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

A4 lattice

The vertex arrangement of the 5-cell honeycomb is called the A4 lattice, or 4-simplex lattice. The 20 vertices of its vertex figure, the runcinated 5-cell represent the 20 roots of the Coxeter group.[2][3] It is the 4-dimensional case of a simplectic honeycomb.

The A*
4
lattice[4] is the union of five A4 lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell

= dual of

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 prisms and tetrahedral prisms may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.[5]

This honeycomb is one of seven unique uniform honeycombs[6] constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Rectified 5-cell honeycomb

Rectified 5-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt0,2{3[5]} or r{3[5]}
Coxeter diagram
4-face typest1{33}
t0,2{33}
t0,3{33}
Cell typesTetrahedron
Octahedron
Cuboctahedron
Triangular prism
Vertex figuretriangular elongated-antiprismatic prism
Symmetry×2, {3[5]}
Propertiesvertex-transitive

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

Alternate names

  • small cyclorhombated pentachoric tetracomb
  • small prismatodispentachoric tetracomb

Cyclotruncated 5-cell honeycomb

Cyclotruncated 5-cell honeycomb
(No image)
TypeUniform 4-honeycomb
FamilyTruncated simplectic honeycomb
Schläfli symbolt0,1{3[5]}
Coxeter diagram
4-face types{3,3,3}
t{3,3,3}
2t{3,3,3}
Cell types{3,3}
t{3,3}
Face typesTriangle {3}
Hexagon {6}
Vertex figure
Elongated tetrahedral antiprism
[3,4,2+], order 48
Symmetry×2, {3[5]}
Propertiesvertex-transitive

The cyclotruncated 4-simplex honeycomb or cyclotruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be seen as a birectified 5-cell honeycomb.

It is composed of 5-cells, truncated 5-cells, and bitruncated 5-cells facets in a ratio of 2:2:1. Its vertex figure is an Elongated tetrahedral antiprism, with 8 equilateral triangle and 24 isosceles triangle faces, defining 8 5-cell and 24 truncated 5-cell facets around a vertex.

It can be constructed as five sets of parallel hyperplanes that divide space into two half-spaces. The 3-space hyperplanes contain quarter cubic honeycombs as a collection facets.[7]

Alternate names

  • Cyclotruncated pentachoric tetracomb
  • Small truncated-pentachoric tetracomb

Truncated 5-cell honeycomb

Truncated 4-simplex honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt0,1,2{3[5]} or t{3[5]}
Coxeter diagram
4-face typest0,1{33}
t0,1,2{33}
t0,3{33}
Cell typesTetrahedron
Truncated tetrahedron
Truncated octahedron
Triangular prism
Vertex figuretriangular elongated-antiprismatic pyramid
Symmetry×2, {3[5]}
Propertiesvertex-transitive

The truncated 4-simplex honeycomb or truncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cyclocantitruncated 5-cell honeycomb.

Alaternate names

  • Great cyclorhombated pentachoric tetracomb
  • Great truncated-pentachoric tetracomb

Cantellated 5-cell honeycomb

Cantellated 5-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt0,1,3{3[5]} or rr{3[5]}
Coxeter diagram
4-face typest0,2{33}
t1,2{33}
t0,1,3{33}
Cell typesTruncated tetrahedron
Octahedron
Cuboctahedron
Triangular prism
Hexagonal prism
Vertex figuretriangular-prismatic antifastigium
Symmetry×2, {3[5]}
Propertiesvertex-transitive

The cantellated 4-simplex honeycomb or cantellated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cycloruncitruncated 5-cell honeycomb.


Alternate names

  • Cycloprismatorhombated pentachoric tetracomb
  • Great prismatodispentachoric tetracomb

Bitruncated 5-cell honeycomb

Bitruncated 5-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt0,1,2,3{3[5]} or 2t{3[5]}
Coxeter diagram
4-face typest0,1,3{33}
t0,1,2{33}
t0,1,2,3{33}
Cell typesCuboctahedron

Truncated octahedron
Truncated tetrahedron
Hexagonal prism
Triangular prism

Vertex figuretilted rectangular duopyramid
Symmetry×2, {3[5]}
Propertiesvertex-transitive

The bitruncated 4-simplex honeycomb or bitruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cycloruncicantitruncated 5-cell honeycomb.

Alternate names

  • Great cycloprismated pentachoric tetracomb
  • Grand prismatodispentachoric tetracomb

Omnitruncated 5-cell honeycomb

Omnitruncated 4-simplex honeycomb
(No image)
TypeUniform 4-honeycomb
FamilyOmnitruncated simplectic honeycomb
Schläfli symbolt0,1,2,3,4{3[5]} or tr{3[5]}
Coxeter diagram
4-face typest0,1,2,3{3,3,3}
Cell typest0,1,2{3,3}
{6}x{}
Face types{4}
{6}
Vertex figure
Irr. 5-cell
Symmetry×10, [5[3[5]}
Propertiesvertex-transitive, cell-transitive

The omnitruncated 4-simplex honeycomb or omnitruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be seen as a cantitruncated 5-cell honeycomb and also a cyclosteriruncicantitruncated 5-cell honeycomb. .

It is composed entirely of omnitruncated 5-cell (omnitruncated 4-simplex) facets.

Coxeter calls this Hinton's honeycomb after C. H. Hinton, who described it in his book The Fourth Dimension in 1906.[8]

The facets of all omnitruncated simplectic honeycombs are called permutahedra 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*
4
lattice is the union of five A4 lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell.[9]

= dual of

Alternated form

This honeycomb can be alternated, creating omnisnub 5-cells with irregular 5-cells created at the deleted vertices. Although it is not uniform, the 5-cells have a symmetry of order 10.

See also

Regular and uniform honeycombs in 4-space:

Notes

  1. Olshevsky (2006), Model 134
  2. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html
  3. https://m.wolframalpha.com/input/?i=A4+root+lattice&lk=3
  4. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html
  5. Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143
  6. mathworld: Necklace, OEIS sequence A000029 8-1 cases, skipping one with zero marks
  7. Olshevsky, (2006) Model 135
  8. The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. (The classification of Zonohededra, page 73)
  9. The Lattice A4*

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, ISBN 978-0-471-01003-6
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. 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, [Math. 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
  • Klitzing, Richard. "4D Euclidean tesselations"., 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)
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21
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