Uniform 7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes.

Regular 7-polytopes

Regular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6-polytopes facets around each 4-face.

There are exactly three such convex regular 7-polytopes:

1. {3,3,3,3,3,3} - 7-simplex
2. {4,3,3,3,3,3} - 7-cube
3. {3,3,3,3,3,4} - 7-orthoplex

There are no nonconvex regular 7-polytopes.

Characteristics

The topology of any given 7-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, 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]

Uniform 7-polytopes by fundamental Coxeter groups

Uniform 7-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

# Coxeter group Regular and semiregular forms Uniform count
1A7[36]
71
2B7[4,35]
127 + 32
3D7[33,1,1]
95 (0 unique)
4E7[33,2,1]
127

The A7 family

The A7 family has symmetry of order 40320 (8 factorial).

There are 71 (64+8-1) forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. All 71 are enumerated below. Norman Johnson's truncation names are given. Bowers names and acronym are also given for cross-referencing.

See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes.

The B7 family

The B7 family has symmetry of order 645120 (7 factorial x 27).

There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Johnson and Bowers names.

See also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes.

The D7 family

The D7 family has symmetry of order 322560 (7 factorial x 26).

This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 Coxeter-Dynkin diagram. Of these, 63 (2×32−1) are repeated from the B7 family and 32 are unique to this family, listed below. Bowers names and acronym are given for cross-referencing.

See also list of D7 polytopes for Coxeter plane graphs of these polytopes.

The E7 family

The E7 Coxeter group has order 2,903,040.

There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

See also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes.

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups and sixteen prismatic groups that generate regular and uniform tessellations in 6-space:

# Coxeter group Coxeter diagram Forms
1${\displaystyle {\tilde {A}}_{6}}$[3[7]]17
2${\displaystyle {\tilde {C}}_{6}}$[4,34,4]71
3${\displaystyle {\tilde {B}}_{6}}$h[4,34,4]
[4,33,31,1]
95 (32 new)
4${\displaystyle {\tilde {D}}_{6}}$q[4,34,4]
[31,1,32,31,1]
41 (6 new)
5${\displaystyle {\tilde {E}}_{6}}$[32,2,2]39

Regular and uniform tessellations include:

• ${\displaystyle {\tilde {A}}_{6}}$, 17 forms
• ${\displaystyle {\tilde {C}}_{6}}$, [4,34,4], 71 forms
• ${\displaystyle {\tilde {B}}_{6}}$, [31,1,33,4], 95 forms, 64 shared with ${\displaystyle {\tilde {C}}_{6}}$, 32 new
• ${\displaystyle {\tilde {D}}_{6}}$, [31,1,32,31,1], 41 unique ringed permutations, most shared with ${\displaystyle {\tilde {B}}_{6}}$ and ${\displaystyle {\tilde {C}}_{6}}$, and 6 are new. Coxeter calls the first one a quarter 6-cubic honeycomb.
• =
• =
• =
• =
• =
• =
• ${\displaystyle {\tilde {E}}_{6}}$: [32,2,2], 39 forms
• Uniform 222 honeycomb: represented by symbols {3,3,32,2},
• Uniform t4(222) honeycomb: 4r{3,3,32,2},
• Uniform 0222 honeycomb: {32,2,2},
• Uniform t2(0222) honeycomb: 2r{32,2,2},
Prismatic groups
# Coxeter group Coxeter-Dynkin diagram
1${\displaystyle {\tilde {A}}_{5}}$x${\displaystyle {\tilde {I}}_{1}}$[3[6],2,∞]
2${\displaystyle {\tilde {B}}_{5}}$x${\displaystyle {\tilde {I}}_{1}}$[4,3,31,1,2,∞]
3${\displaystyle {\tilde {C}}_{5}}$x${\displaystyle {\tilde {I}}_{1}}$[4,33,4,2,∞]
4${\displaystyle {\tilde {D}}_{5}}$x${\displaystyle {\tilde {I}}_{1}}$[31,1,3,31,1,2,∞]
5${\displaystyle {\tilde {A}}_{4}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[3[5],2,∞,2,∞,2,∞]
6${\displaystyle {\tilde {B}}_{4}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[4,3,31,1,2,∞,2,∞]
7${\displaystyle {\tilde {C}}_{4}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[4,3,3,4,2,∞,2,∞]
8${\displaystyle {\tilde {D}}_{4}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[31,1,1,1,2,∞,2,∞]
9${\displaystyle {\tilde {F}}_{4}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[3,4,3,3,2,∞,2,∞]
10${\displaystyle {\tilde {C}}_{3}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[4,3,4,2,∞,2,∞,2,∞]
11${\displaystyle {\tilde {B}}_{3}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[4,31,1,2,∞,2,∞,2,∞]
12${\displaystyle {\tilde {A}}_{3}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[3[4],2,∞,2,∞,2,∞]
13${\displaystyle {\tilde {C}}_{2}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[4,4,2,∞,2,∞,2,∞,2,∞]
14${\displaystyle {\tilde {H}}_{2}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[6,3,2,∞,2,∞,2,∞,2,∞]
15${\displaystyle {\tilde {A}}_{2}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[3[3],2,∞,2,∞,2,∞,2,∞]
16${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$x${\displaystyle {\tilde {I}}_{1}}$[∞,2,∞,2,∞,2,∞,2,∞]

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 7, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 paracompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.

 ${\displaystyle {\bar {P}}_{6}}$ = [3,3[6]]: ${\displaystyle {\bar {Q}}_{6}}$ = [31,1,3,32,1]: ${\displaystyle {\bar {S}}_{6}}$ = [4,3,3,32,1]:

Notes on the Wythoff construction for the uniform 7-polytopes

The reflective 7-dimensional uniform polytopes are constructed through a Wythoff construction process, and represented by a Coxeter-Dynkin diagram, where each node represents a mirror. An active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope. Uniform polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may be named in two equally valid ways.

Here are the primary operators available for constructing and naming the uniform 7-polytopes.

The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.

Operation Extended
Schläfli symbol
Coxeter-
Dynkin
diagram
Description
Parent t0{p,q,r,s,t,u} Any regular 7-polytope
Rectified t1{p,q,r,s,t,u} The edges are fully truncated into single points. The 7-polytope now has the combined faces of the parent and dual.
Birectified t2{p,q,r,s,t,u} Birectification reduces cells to their duals.
Truncated t0,1{p,q,r,s,t,u} Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 7-polytope. The 7-polytope has its original faces doubled in sides, and contains the faces of the dual.
Bitruncated t1,2{p,q,r,s,t,u} Bitrunction transforms cells to their dual truncation.
Tritruncated t2,3{p,q,r,s,t,u} Tritruncation transforms 4-faces to their dual truncation.
Cantellated t0,2{p,q,r,s,t,u} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Bicantellated t1,3{p,q,r,s,t,u} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Runcinated t0,3{p,q,r,s,t,u} Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated t1,4{p,q,r,s,t,u} Runcination reduces cells and creates new cells at the vertices and edges.
Stericated t0,4{p,q,r,s,t,u} Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
Pentellated t0,5{p,q,r,s,t,u} Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps.
Hexicated t0,6{p,q,r,s,t,u} Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. (expansion operation for 7-polytopes)
Omnitruncated t0,1,2,3,4,5,6{p,q,r,s,t,u} All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.

References

1. Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
• H.S.M. Coxeter:
• H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• 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 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Klitzing, Richard. "7D uniform polytopes (polyexa)".