BREAKING NEWS
5-cell honeycomb

## Summary

4-simplex honeycomb
(No image)
Type Uniform 4-honeycomb
Family Simplectic 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 ${\displaystyle {\tilde {A}}_{4}}$×2, {3[5]}
Properties vertex-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

• 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:

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 ${\displaystyle {\tilde {A}}_{4}}$  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

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 ${\displaystyle {\tilde {A}}_{4}}$  Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

A4 honeycombs
Pentagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycomb diagrams
a1 [3[5]]       ${\displaystyle {\tilde {A}}_{4}}$  (None)
i2 [[3[5]]]       ${\displaystyle {\tilde {A}}_{4}}$ ×2       1,      2,      3,

4,      5,      6

r10 [5[3[5]]]       ${\displaystyle {\tilde {A}}_{4}}$ ×10       7

### Rectified 5-cell honeycomb

Rectified 5-cell honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol t0,2{3[5]} or r{3[5]}
Coxeter diagram
4-face types t1{33}
t0,2{33}
t0,3{33}
Cell types Tetrahedron
Octahedron
Cuboctahedron
Triangular prism
Vertex figure triangular elongated-antiprismatic prism
Symmetry ${\displaystyle {\tilde {A}}_{4}}$ ×2, {3[5]}
Properties vertex-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)
Type Uniform 4-honeycomb
Family Truncated simplectic honeycomb
Schläfli symbol t0,1{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 types Triangle {3}
Hexagon {6}
Vertex figure
Tetrahedral antiprism
[3,4,2+], order 48
Symmetry ${\displaystyle {\tilde {A}}_{4}}$ ×2, {3[5]}
Properties vertex-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.[8]

#### Alternate names

• Cyclotruncated pentachoric tetracomb
• Small truncated-pentachoric tetracomb

### Truncated 5-cell honeycomb

Truncated 4-simplex honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol t0,1,2{3[5]} or t{3[5]}
Coxeter diagram
4-face types t0,1{33}
t0,1,2{33}
t0,3{33}
Cell types Tetrahedron
Truncated tetrahedron
Truncated octahedron
Triangular prism
Vertex figure triangular elongated-antiprismatic pyramid
Symmetry ${\displaystyle {\tilde {A}}_{4}}$ ×2, {3[5]}
Properties vertex-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)
Type Uniform 4-honeycomb
Schläfli symbol t0,1,3{3[5]} or rr{3[5]}
Coxeter diagram
4-face types t0,2{33}
t1,2{33}
t0,1,3{33}
Cell types Truncated tetrahedron
Octahedron
Cuboctahedron
Triangular prism
Hexagonal prism
Vertex figure Bidiminished rectified pentachoron
Symmetry ${\displaystyle {\tilde {A}}_{4}}$ ×2, {3[5]}
Properties vertex-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)
Type Uniform 4-honeycomb
Schläfli symbol t0,1,2,3{3[5]} or 2t{3[5]}
Coxeter diagram
4-face types t0,1,3{33}
t0,1,2{33}
t0,1,2,3{33}
Cell types Cuboctahedron
Vertex figure tilted rectangular duopyramid
Symmetry ${\displaystyle {\tilde {A}}_{4}}$ ×2, {3[5]}
Properties vertex-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)
Type Uniform 4-honeycomb
Family Omnitruncated simplectic honeycomb
Schläfli symbol t0,1,2,3,4{3[5]} or tr{3[5]}
Coxeter diagram
4-face types t0,1,2,3{3,3,3}
Cell types t0,1,2{3,3}
{6}x{}
Face types {4}
{6}
Vertex figure
Irr. 5-cell
Symmetry ${\displaystyle {\tilde {A}}_{4}}$ ×10, [5[3[5]]]
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.[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

• 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-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.

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

• 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 [1]
• (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) arXiv:1209.1878
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
E10 Uniform 10-honeycomb {3[11]} δ11 11 11
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21