5-cell honeycomb

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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 {\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.

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[edit]

  • Cyclopentachoric tetracomb
  • Pentachoric-dispentachoric tetracomb

Projection by folding[edit]

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:

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

A4 lattice[edit]

This vertex arrangement 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.[2] It is the 4-dimensional case of a simplectic honeycomb.

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

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png + CDel node.pngCDel split1.pngCDel nodes 10lr.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[edit]

The tops of the 5-cells in this honeycomb adjoin the bases of the 5-cells, and vice versa, in adjacent laminae; 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.[3]

This honeycomb is one of seven unique uniform honeycombs[4] constructed by the {\tilde{A}}_4 Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Pentagon
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
a1 [3[5]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png ×1 (None)
i2 [[3[5]]] CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel branch c3.png ×2 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png 1,CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png 2,CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png 3,

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png 4,CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png 5,CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png 6

r10 [5[3[5]]] CDel node c1.pngCDel split1.pngCDel nodeab c1.pngCDel 3ab.pngCDel branch c1.png ×10 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png 7

Rectified 5-cell honeycomb[edit]

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} Schlegel half-solid cantellated 5-cell.png
t0,3{33} Schlegel half-solid runcinated 5-cell.png
Cell types Tetrahedron Tetrahedron.png
Octahedron Octahedron.png
Cuboctahedron Cuboctahedron.png
Triangular prism Triangular prism.png
Vertex figure triangular elongated-antiprismatic prism
Symmetry {\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[edit]

  • small cyclorhombated pentachoric tetracomb
  • small prismatodispentachoric tetracomb


Cyclotruncated 5-cell honeycomb[edit]

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} Schlegel half-solid truncated pentachoron.png
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
Elongated tetrahedral antiprism
[3,4,2+], order 48
Symmetry {\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 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.[1]

Alternate names[edit]

  • Cyclotruncated pentachoric tetracomb
  • Small truncated-pentachoric tetracomb


Truncated 5-cell honeycomb[edit]

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} Schlegel half-solid cantitruncated 5-cell.png
t0,3{33} Schlegel half-solid runcinated 5-cell.png
Cell types Tetrahedron Tetrahedron.png
Truncated tetrahedron Truncated tetrahedron.png
Truncated octahedron Truncated octahedron.png
Triangular prism Triangular prism.png
Vertex figure triangular elongated-antiprismatic pyramid
Symmetry {\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[edit]

  • Great cyclorhombated pentachoric tetracomb
  • Great truncated-pentachoric tetracomb


Cantellated 5-cell honeycomb[edit]

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} Schlegel half-solid bitruncated 5-cell.png
t0,1,3{33} Schlegel half-solid runcitruncated 5-cell.png
Cell types Truncated tetrahedron Truncated tetrahedron.png
Octahedron Octahedron.png
Cuboctahedron Cuboctahedron.png
Triangular prism Triangular prism.png
Hexagonal prism Hexagonal prism.png
Vertex figure triangular-prismatic antifastigium
Symmetry {\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[edit]

  • Cycloprismatorhombated pentachoric tetracomb
  • Great prismatodispentachoric tetracomb


Bitruncated 5-cell honeycomb[edit]

Bitruncated 5-cell honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol t0,1,2,3{3[5]} or 2t{3[5]}
Coxeter–Dynkin 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} Schlegel half-solid cantitruncated 5-cell.png
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 {\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[edit]

  • Great cycloprismated pentachoric tetracomb
  • Grand prismatodispentachoric tetracomb


Omnitruncated 5-cell honeycomb[edit]

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 {\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 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.[5]

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).

Alernate names[edit]

  • Omnitruncated cyclopentachoric tetracomb
  • Great-prismatodecachoric tetracomb

A4* lattice[edit]

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

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


See also[edit]

Regular and uniform honeycombs in 4-space:

Notes[edit]

  1. ^ a b Olshevsky (2006), Model 134
  2. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html
  3. ^ Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143
  4. ^ [1], A000029 8-1 cases, skipping one with zero marks
  5. ^ The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.  (The classification of Zonohededra, page 73)

References[edit]

  • 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 [2]
    • (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
  • Richard Klitzing, 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) [3]