stringtranslate.com

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.

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

The A*
4 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]

This honeycomb is one of seven unique uniform honeycombs[7] constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Rectified 5-cell honeycomb

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

Alternate names

Cyclotruncated 5-cell honeycomb

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.[8]

Alternate names

Truncated 5-cell honeycomb

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

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

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

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.[9]

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

A4* lattice

The A*
4 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.[10]

∪ ∪ ∪ ∪ = 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. ^ Baake, M.; Kramer, P.; Schlottmann, M.; Zeidler, D. (December 1990). "PLANAR PATTERNS WITH FIVEFOLD SYMMETRY AS SECTIONS OF PERIODIC STRUCTURES IN 4-SPACE". International Journal of Modern Physics B. 04 (15n16): 2217–2268. doi:10.1142/S0217979290001054.
  3. ^ "The Lattice A4".
  4. ^ "A4 root lattice - Wolfram|Alpha".
  5. ^ "The Lattice A4".
  6. ^ Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143
  7. ^ mathworld: Necklace, OEIS sequence A000029 8-1 cases, skipping one with zero marks
  8. ^ Olshevsky, (2006) Model 135
  9. ^ The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. (The classification of Zonohededra, page 73)
  10. ^ The Lattice A4*

References