Witting polytope

Last updated
Witting polytope
Witting polytope.png
Schläfli symbol 3{3}3{3}3{3}3
Coxeter diagram CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png
Cells240 3{3}3{3}3 Complex polyhedron 3-3-3-3-3.png
Faces2160 3{3}3 Complex polygon 3-3-3.png
Edges2160 3{} Complex trion.png
Vertices240
Petrie polygon 30-gon
van Oss polygon 90 3{4}3 Complex polygon 3-4-3.png
Shephard group L4 = 3[3]3[3]3[3]3, order 155,520
Dual polyhedron Self-dual
PropertiesRegular

In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3{3}3{3}3{3}3, and Coxeter diagram CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png. It has 240 vertices, 2160 3{} edges, 2160 3{3}3 faces, and 240 3{3}3{3}3 cells. It is self-dual. Each vertex belongs to 27 edges, 72 faces, and 27 cells, corresponding to the Hessian polyhedron vertex figure.

Contents

Symmetry

Its symmetry by 3[3]3[3]3[3]3 or CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png, order 155,520. [1] It has 240 copies of CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png, order 648 at each cell. [2]

Structure

The configuration matrix is: [3]

The number of vertices, edges, faces, and cells are seen in the diagonal of the matrix. These are computed by the order of the group divided by the order of the subgroup, by removing certain complex reflections, shown with X below. The number of elements of the k-faces are seen in rows below the diagonal. The number of elements in the vertex figure, etc., are given in rows above the digonal.

L4CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png k-face fkf0f1f2f3 k-figure Notes
L3CDel node x.pngCDel 2.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png( )f0240277227 3{3}3{3}3 L4/L3 = 216*6!/27/4! = 240
L2L1CDel 3node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel 3node.pngCDel 3.pngCDel 3node.png3{ }f13216088 3{3}3 L4/L2L1 = 216*6!/4!/3 = 2160
CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel 3node.png 3{3}3 f288216033{ }
L3CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 2.pngCDel node x.png 3{3}3{3}3 f3277227240( )L4/L3 = 216*6!/27/4! = 240

Coordinates

Its 240 vertices are given coordinates in :

(0, ±ωμ, -±ων, ±ωλ)
(-±ωμ, 0, ±ων, ±ωλ)
ωμ, -±ων, 0, ±ωλ)
(-±ωλ, -±ωμ, -±ων, 0)
(±iωλ√3, 0, 0, 0)
(0, ±iωλ√3, 0, 0)
(0, 0, ±iωλ√3, 0)
(0, 0, 0, ±iωλ√3)

where .

The last 6 points form hexagonal holes on one of its 40 diameters. There are 40 hyperplanes contain central 3{3}3{4}2, CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png figures, with 72 vertices.

Witting configuration

Coxeter named it after Alexander Witting for being a Witting configuration in complex projective 3-space: [4]

or

The Witting configuration is related to the finite space PG(3,22), consisting of 85 points, 357 lines, and 85 planes. [5]

Its 240 vertices are shared with the real 8-dimensional polytope 421, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png. Its 2160 3-edges are sometimes drawn as 6480 simple edges, slightly less than the 6720 edges of 421. The 240 difference is accounted by 40 central hexagons in 421 whose edges are not included in 3{3}3{3}3{3}3. [6]

The honeycomb of Witting polytopes

The regular Witting polytope has one further stage as a 4-dimensional honeycomb, CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png. It has the Witting polytope as both its facets, and vertex figure. It is self-dual, and its dual coincides with itself. [7]

Hyperplane sections of this honeycomb include 3-dimensional honeycombs CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png.

The honeycomb of Witting polytopes has a real representation as the 8-dimensional polytope 521, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

Its f-vector element counts are in proportion: 1, 80, 270, 80, 1. [8] The configuration matrix for the honeycomb is:

L5CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png k-face fkf0f1f2f3f4 k-figure Notes
L4CDel node x.pngCDel 2.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png( )f0N240216021602403{3}3{3}3{3}3L5/L4 = N
L3L1CDel 3node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png3{ }f1380N277227 3{3}3{3}3 L5/L3L1 = 80N
L2L2CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel 3node.pngCDel 3.pngCDel 3node.png 3{3}3 f288270N88 3{3}3 L5/L2L2 = 270N
L3L1CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel 3node.png 3{3}3{3}3 f327722780N33{}L5/L3L1 = 80N
L4CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 2.pngCDel node x.png3{3}3{3}3{3}3f424021602160240N( )L5/L4 = N

Notes

  1. Coxeter Regular Convex Polytopes, 12.5 The Witting polytope
  2. Coxeter, Complex Regular Polytopes, p.134
  3. Coxeter, Complex Regular polytopes, p.132
  4. Alexander Witting, Ueber Jacobi'sche Functionen kter Ordnung Zweier Variabler, Mathemematische Annalen 29 (1887), 157-70, see especially p.169
  5. Coxeter, Complex regular polytopes, p.133
  6. Coxeter, Complex Regular Polytopes, p.134
  7. Coxeter, Complex Regular Polytopes, p.135
  8. Coxeter Regular Convex Polytopes, 12.5 The Witting polytope

Related Research Articles

<span class="mw-page-title-main">Tesseract</span> Four-dimensional analogue of the cube

In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

<span class="mw-page-title-main">24-cell</span> Regular object in four dimensional geometry

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

<span class="mw-page-title-main">5-cell</span> Four-dimensional analogue of the tetrahedron

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

<span class="mw-page-title-main">16-cell</span> Four-dimensional analog of the octahedron

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid [sic?].

<span class="mw-page-title-main">Runcinated 5-cell</span>

In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

<span class="mw-page-title-main">Truncated 5-cell</span>

In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.

<span class="mw-page-title-main">6-orthoplex</span>

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

<span class="mw-page-title-main">Rectified 5-simplexes</span>

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

1<sub> 22</sub> polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).

1 <sub>42</sub> polytope

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

2<sub> 41</sub> polytope

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

4<sub> 21</sub> polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

<span class="mw-page-title-main">Simplectic honeycomb</span> Tiling of n-dimensional space

In geometry, the simplectic honeycomb is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of n + 1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

<span class="mw-page-title-main">Hessian polyhedron</span>

In geometry, the Hessian polyhedron is a regular complex polyhedron 3{3}3{3}3, , in . It has 27 vertices, 72 3{} edges, and 27 3{3}3 faces. It is self-dual.

<span class="mw-page-title-main">Möbius–Kantor polygon</span>

In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in . 3{3}3 has 8 vertices, and 8 edges. It is self-dual. Every vertex is shared by 3 triangular edges. Coxeter named it a Möbius–Kantor polygon for sharing the complex configuration structure as the Möbius–Kantor configuration, (83).

References