5-cell honeycomb explained

bgcolor=#e7dcc3 colspan=24-simplex honeycomb
bgcolor=#ffffff align=center colspan=2(No image)
TypeUniform 4-honeycomb
FamilySimplectic honeycomb
Schläfli symbol = 0[5]
Coxeter diagram
4-face types
Cell types
Face types
Vertex figure
t0,3
Symmetry

{\tilde{A}}4

×2
[3<sup>[5]]
Propertiesvertex-transitive

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.

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

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:

Two different aperiodic tilings with 5-fold symmetry can be obtained by projecting two-dimensional slices of the honeycomb: the Penrose tiling composed of rhombi, and the Tübingen triangle tiling composed of isosceles triangles.[2]

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

{\tilde{A}}4

Coxeter group.[3] [4] It is the 4-dimensional case of a simplectic honeycomb.

The A lattice[5] 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

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 prisms and tetrahedral prisms may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.[6]

Rectified 5-cell honeycomb

bgcolor=#e7dcc3 colspan=2Rectified 5-cell honeycomb
bgcolor=#ffffff align=center colspan=2(No image)
TypeUniform 4-honeycomb
Schläfli symbolt0,2 or r
Coxeter diagram
4-face types
Cell types
Vertex figuretriangular elongated-antiprismatic prism
Symmetry

{\tilde{A}}4

×2
[3<sup>[5]]
Propertiesvertex-transitive
The rectified 4-simplex honeycomb or rectified 5-cell honeycomb is a space-filling tessellation honeycomb.

Alternate names

Cyclotruncated 5-cell honeycomb

bgcolor=#e7dcc3 colspan=2Cyclotruncated 5-cell honeycomb
bgcolor=#ffffff align=center colspan=2(No image)
TypeUniform 4-honeycomb
FamilyTruncated simplectic honeycomb
Schläfli symbolt0,1
Coxeter diagram
4-face types
Cell types
Face types
Vertex figure
Tetrahedral antiprism
[3,4,2<sup>+</sup>], order 48
Symmetry

{\tilde{A}}4

×2
[3<sup>[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 a tetrahedral antiprism, with 2 regular tetrahedron, 8 triangular pyramid, and 6 tetragonal disphenoid cells, defining 2 5-cell, 8 truncated 5-cell, and 6 bitruncated 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

Truncated 5-cell honeycomb

bgcolor=#e7dcc3 colspan=2Truncated 4-simplex honeycomb
bgcolor=#ffffff align=center colspan=2(No image)
TypeUniform 4-honeycomb
Schläfli symbolt0,1,2 or t
Coxeter diagram
4-face types
Cell types
Vertex figuretriangular elongated-antiprismatic pyramid
Symmetry

{\tilde{A}}4

×2
[3<sup>[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

Cantellated 5-cell honeycomb

bgcolor=#e7dcc3 colspan=2Cantellated 5-cell honeycomb
bgcolor=#ffffff align=center colspan=2(No image)
TypeUniform 4-honeycomb
Schläfli symbolt0,1,3 or rr
Coxeter diagram
4-face types
Cell types
Vertex figureBidiminished rectified pentachoron
Symmetry

{\tilde{A}}4

×2
[3<sup>[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

Bitruncated 5-cell honeycomb

bgcolor=#e7dcc3 colspan=2Bitruncated 5-cell honeycomb
bgcolor=#ffffff align=center colspan=2(No image)
TypeUniform 4-honeycomb
Schläfli symbolt0,1,2,3 or 2t
Coxeter diagram
4-face types
Cell types
Vertex figuretilted rectangular duopyramid
Symmetry

{\tilde{A}}4

×2
[3<sup>[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

Omnitruncated 5-cell honeycomb

bgcolor=#e7dcc3 colspan=2Omnitruncated 4-simplex honeycomb
bgcolor=#ffffff align=center colspan=2(No image)
TypeUniform 4-honeycomb
FamilyOmnitruncated simplectic honeycomb
Schläfli symbolt0,1,2,3,4 or tr
Coxeter diagram
4-face types
Cell types
Face types
Vertex figure
Irr. 5-cell
Symmetry

{\tilde{A}}4

×10, [5[3[5]]]|-|bgcolor=#e7dcc3|Properties||vertex-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 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 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 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:

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
    • (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
  • , 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)

]

Notes and References

  1. Olshevsky (2006), Model 134
  2. Baake . M. . Kramer . P. . Schlottmann . M. . Zeidler . D. . PLANAR PATTERNS WITH FIVEFOLD SYMMETRY AS SECTIONS OF PERIODIC STRUCTURES IN 4-SPACE . International Journal of Modern Physics B . December 1990 . 04 . 15n16 . 2217–2268 . 10.1142/S0217979290001054.
  3. Web site: The Lattice A4.
  4. Web site: A4 root lattice - Wolfram|Alpha.
  5. Web site: The Lattice A4.
  6. Olshevsky (2006), Klitzing, elong(x3o3o3o3o3*a) - ecypit - O141, schmo(x3o3o3o3o3*a) - zucypit - O142, elongschmo(x3o3o3o3o3*a) - ezucypit - O143
  7. Olshevsky, (2006) Model 135
  8. Book: The Beauty of Geometry: Twelve Essays. 1999. Dover Publications. 99035678. 0-486-40919-8 . (The classification of Zonohededra, page 73)
  9. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html The Lattice A4*