6-polytope

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Graphs of three regular and five Uniform 6-polytopes

6-simplex
6-orthoplex, 311
6-cube (Hexeract)
221
Expanded 6-simplex
Rectified 6-orthoplex
6-demicube 131
(Demihexeract)
122

In

facets
.

Definition

A 6-polytope is a closed six-dimensional figure with

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

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".

  • A 6-polytope is
    Kepler-Poinsot polyhedra
    .
  • A regular 6-polytope has all identical regular 5-polytope facets. All regular 6-polytope are convex.
Main article: List of regular polytopes § Convex_5
Main article: Uniform 6-polytope

Regular 6-polytopes

Regular 6-polytopes can be generated from

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
 Vertices Edges Faces Cells 4-faces 5-faces
order
)
6-simplex {3,3,3,3,3} 7 21 35 35 21 7 A6 (720)
6-orthoplex {3,3,3,3,4} 12 60 160 240 192 64 B6 (46080)
6-cube {4,3,3,3,3} 64 192 240 160 60 12 B6 (46080)

Uniform 6-polytopes

Main article: Uniform 6-polytope

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

Name Schläfli
symbol(s)
Coxeter
diagram(s)
 Vertices Edges Faces Cells 4-faces 5-faces
order
)
Expanded 6-simplex
 t0,5{3,3,3,3,3} 42 210 490 630 434 126 2×A6 (1440)
6-orthoplex, 311
(alternate construction)		 {3,3,3,31,1} 12 60 160 240 192 64 D6 (23040)
6-demicube {3,33,1}
h{4,3,3,3,3}
32 240 640 640 252 44 D6 (23040)
½B6
Rectified 6-orthoplex
 t1{3,3,3,3,4}
t1{3,3,3,31,1}
60 480 1120 1200 576 76 B6 (46080)
2×D6
221 polytope {3,3,32,1} 27 216 720 1080 648 99 E6 (51840)
122 polytope {3,32,2}
or 		72 720 2160 2160 702 54 2×E6 (103680)

The expanded 6-simplex is the

6-demicube honeycomb, , vertex figure is a rectified 6-orthoplex and facets are the 6-orthoplex and 6-demicube. The uniform 222 honeycomb
,, has 122 polytope is the vertex figure and 221 facets.

References

  1. ^ a b c Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2
Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron
 Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics:
List of regular polytopes and compounds
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