5-cell honeycomb

From HandWiki
Short description: Geometric figure
4-simplex honeycomb
(No image)
Type Uniform 4-honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[5]}
Coxeter diagram CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
4-face types {3,3,3}Schlegel wireframe 5-cell.png
t1{3,3,3} Schlegel half-solid rectified 5-cell.png
Cell types {3,3} Uniform polyhedron-33-t0.png
t1{3,3} Uniform polyhedron-33-t1.png
Face types {3}
Vertex figure 4-simplex honeycomb verf.png
t0,3{3,3,3}
Symmetry [math]\displaystyle{ {\tilde{A}}_4 }[/math]×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:

[math]\displaystyle{ {\tilde{A}}_3 }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
[math]\displaystyle{ {\tilde{C}}_2 }[/math] CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png

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 [math]\displaystyle{ {\tilde{A}}_4 }[/math] 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

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png ∪ CDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel branch.png ∪ CDel node.pngCDel split1.pngCDel nodes 01lr.pngCDel 3ab.pngCDel branch.png ∪ CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.png ∪ CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch 01l.png = dual of CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

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.[5]


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 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png
4-face types t1{33} Schlegel half-solid rectified 5-cell.png
t0,2{33} 40px
t0,3{33} Schlegel half-solid runcinated 5-cell.png
Cell types Tetrahedron Tetrahedron.png
Octahedron 20px
Cuboctahedron 20px
Triangular prism Triangular prism.png
Vertex figure triangular elongated-antiprismatic prism
Symmetry [math]\displaystyle{ {\tilde{A}}_4 }[/math]×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 CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
4-face types {3,3,3} Schlegel wireframe 5-cell.png
t{3,3,3} 40px
2t{3,3,3} Schlegel half-solid bitruncated 5-cell.png
Cell types {3,3} Tetrahedron.png
t{3,3} Truncated tetrahedron.png
Face types Triangle {3}
Hexagon {6}
Vertex figure Truncated 5-cell honeycomb verf.png
Tetrahedral antiprism
[3,4,2+], order 48
Symmetry [math]\displaystyle{ {\tilde{A}}_4 }[/math]×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.[6]

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 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png
4-face types t0,1{33} Schlegel half-solid truncated pentachoron.png
t0,1,2{33} 40px
t0,3{33} Schlegel half-solid runcinated 5-cell.png
Cell types Tetrahedron Tetrahedron.png
Truncated tetrahedron 20px
Truncated octahedron 20px
Triangular prism Triangular prism.png
Vertex figure triangular elongated-antiprismatic pyramid
Symmetry [math]\displaystyle{ {\tilde{A}}_4 }[/math]×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 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png
4-face types t0,2{33} Schlegel half-solid cantellated 5-cell.png
t1,2{33} 40px
t0,1,3{33} Schlegel half-solid runcitruncated 5-cell.png
Cell types Truncated tetrahedron Truncated tetrahedron.png
Octahedron 20px
Cuboctahedron 20px
Triangular prism 20px
Hexagonal prism Hexagonal prism.png
Vertex figure Bidiminished rectified pentachoron
Symmetry [math]\displaystyle{ {\tilde{A}}_4 }[/math]×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 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png
4-face types t0,1,3{33} Schlegel half-solid runcitruncated 5-cell.png
t0,1,2{33} 40px
t0,1,2,3{33} Schlegel half-solid omnitruncated 5-cell.png
Cell types Cuboctahedron Cuboctahedron.png

Truncated octahedron Truncated octahedron.png
Truncated tetrahedron Truncated tetrahedron.png
Hexagonal prism Hexagonal prism.png
Triangular prism Triangular prism.png

Vertex figure tilted rectangular duopyramid
Symmetry [math]\displaystyle{ {\tilde{A}}_4 }[/math]×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 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png
4-face types t0,1,2,3{3,3,3} Schlegel half-solid omnitruncated 5-cell.png
Cell types t0,1,2{3,3} Uniform polyhedron-33-t012.png
{6}x{} Hexagonal prism.png
Face types {4}
{6}
Vertex figure Omnitruncated 4-simplex honeycomb verf.png
Irr. 5-cell
Symmetry [math]\displaystyle{ {\tilde{A}}_4 }[/math]×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.[7]

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.[8]

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png ∪ CDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel branch.png ∪ CDel node.pngCDel split1.pngCDel nodes 01lr.pngCDel 3ab.pngCDel branch.png ∪ CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch 10l.png ∪ CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch 01l.png = dual of CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png

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. ↑ "The Lattice A4". http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html. 
  3. ↑ "A4 root lattice - Wolfram|Alpha". https://m.wolframalpha.com/input/?i=A4+root+lattice&lk=3. 
  4. ↑ "The Lattice A4". 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. ↑ Olshevsky, (2006) Model 135
  7. ↑ The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8.  (The classification of Zonohededra, page 73)
  8. ↑ 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". https://bendwavy.org/klitzing/dimensions/flat.htm. , 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
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
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 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21



Retrieved from "https://handwiki.org/wiki/index.php?title=5-cell_honeycomb&oldid=2879441"