6-polytope

Last updated
Graphs of three regular and five Uniform 6-polytopes
6-simplex t0.svg
6-simplex
6-cube t5.svg
6-orthoplex, 311
6-cube t0.svg
6-cube (Hexeract)
Up 2 21 t0 E6.svg
221
6-simplex t05.svg
Expanded 6-simplex
6-cube t4.svg
Rectified 6-orthoplex
6-demicube t0 D6.svg
6-demicube 131
(Demihexeract)
Up 1 22 t0 E6.svg
122

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

Contents

Definition

A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A vertex is a point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron. A 4-face is a polychoron, and a 5-face is a 5-polytope. Furthermore, the following requirements must be met:

Characteristics

The topology of any given 6-polytope is defined by its Betti numbers and torsion coefficients. [1]

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 6-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. [1]

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients. [1]

Classification

6-polytopes may be classified by properties like "convexity" and "symmetry".

Regular 6-polytopes

Regular 6-polytopes can be generated from Coxeter groups represented by the Schläfli symbol {p,q,r,s,t} with t {p,q,r,s} 5-polytope facets around each cell.

There are only three such convex regular 6-polytopes:

There are no nonconvex regular polytopes of 5 or more dimensions.

For the three convex regular 6-polytopes, their elements are:

Name Schläfli
symbol
Coxeter
diagram
VerticesEdgesFacesCells4-faces5-faces Symmetry (order)
6-simplex {3,3,3,3,3}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png7213535217A6 (720)
6-orthoplex {3,3,3,3,4}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png126016024019264 B6 (46080)
6-cube {4,3,3,3,3}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png641922401606012B6 (46080)

Uniform 6-polytopes

Here are six simple uniform convex 6-polytopes, including the 6-orthoplex repeated with its alternate construction.

Name Schläfli
symbol(s)
Coxeter
diagram(s)
VerticesEdgesFacesCells4-faces5-faces Symmetry (order)
Expanded 6-simplex t0,5{3,3,3,3,3}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png422104906304341262×A6 (1440)
6-orthoplex, 311
(alternate construction)
{3,3,3,31,1}CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png126016024019264D6 (23040)
6-demicube {3,33,1}
h{4,3,3,3,3}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3224064064025244D6 (23040)
½B6
Rectified 6-orthoplex t1{3,3,3,3,4}
t1{3,3,3,31,1}
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
604801120120057676B6 (46080)
2×D6
221 polytope {3,3,32,1}CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png27216720108064899E6 (51840)
122 polytope {3,32,2}CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
or CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
7272021602160702542×E6 (103680)

The expanded 6-simplex is the vertex figure of the uniform 6-simplex honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png. The 6-demicube honeycomb, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, vertex figure is a rectified 6-orthoplex and facets are the 6-orthoplex and 6-demicube. The uniform 222 honeycomb,CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png, has 122 polytope is the vertex figure and 221 facets.

References

  1. 1 2 3 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compoundsPolytope operations