In ten-dimensional
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a 10-polytope is a 10-dimensional
polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
whose boundary consists of
9-polytope
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.
A uniform 9-polytope is one which is vertex-transitive, and ...
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
, exactly two such facets meeting at each
8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transitive, ...
ridge.
A uniform 10-polytope is one which is
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
, and constructed from
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
.
Regular 10-polytopes
Regular 10-polytopes can be represented by the
Schläfli symbol , with x 9-polytope
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
around each
peak Peak or The Peak may refer to:
Basic meanings Geology
* Mountain peak
** Pyramidal peak, a mountaintop that has been sculpted by erosion to form a point Mathematics
* Peak hour or rush hour, in traffic congestion
* Peak (geometry), an (''n''-3)-di ...
.
There are exactly three such
convex regular 10-polytopes:
# -
10-simplex
In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, ...
# -
10-cube
# -
10-orthoplex
There are no nonconvex regular 10-polytopes.
Euler characteristic
The topology of any given 10-polytope is defined by its
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplici ...
s and
torsion coefficient
A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportiona ...
s.
[Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008.]
The value of the
Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 10-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.
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.
Uniform 10-polytopes by fundamental Coxeter groups
Uniform 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the
Coxeter-Dynkin diagrams:
Selected regular and uniform 10-polytopes from each family include:
#
Simplex family: A
10 9">9-
#* 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular:
#*# -
10-simplex
In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, ...
-
#
Hypercube/
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
family: B
10 8">,38-
#* 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones:
#*# -
10-cube or dekeract -
#*# -
10-orthoplex or decacross -
#*# h -
10-demicube
In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified ...
.
#
Demihypercube
In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''- polytopes constructed from alternation of an ''n''- hypercube, labeled as ''hγn'' for being ''half'' of the hy ...
D
10 family:
7,1,1">7,1,1-
#* 767 uniform 10-polytopes as permutations of rings in the group diagram, including:
#*# 1
7,1 -
10-demicube
In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified ...
or demidekeract -
#*# 7
1,1 -
10-orthoplex -
The A10 family
The A
10 family has symmetry of order 39,916,800 (11
factorial).
There are 512+16-1=527 forms based on all permutations of the
Coxeter-Dynkin diagrams with one or more rings. 31 are shown below: all one and two ringed forms, and the final omnitruncated form. Bowers-style acronym names are given in parentheses for cross-referencing.
The B10 family
There are 1023 forms based on all permutations of the
Coxeter-Dynkin diagrams with one or more rings.
Twelve cases are shown below: ten single-ring (
rectified) forms, and two truncations. Bowers-style acronym names are given in parentheses for cross-referencing.
The D10 family
The D
10 family has symmetry of order 1,857,945,600 (10
factorial × 2
9).
This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D
10 Coxeter-Dynkin diagram. Of these, 511 (2×256−1) are repeated from the B
10 family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing.
Regular and uniform honeycombs
There are four fundamental affine
Coxeter groups
In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, H. S. M. Coxeter, is an group (mathematics), abstract group that admits a group presentation, formal description in terms of Reflection (mathematics), reflections (or Kal ...
that generate regular and uniform tessellations in 9-space:
Regular and uniform tessellations include:
*
Regular 9-hypercubic honeycomb, with symbols ,
* Uniform
alternated 9-hypercubic honeycomb with symbols h,
Regular and uniform hyperbolic honeycombs
There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
. However, there are
3 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.
Three honeycombs from the
family, generated by end-ringed Coxeter diagrams are:
*
621 honeycomb:
*
261 honeycomb:
*
162 honeycomb:
References
*
T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'',
Messenger of Mathematics
The ''Messenger of Mathematics'' is a defunct British mathematics journal. The founding editor-in-chief was William Allen Whitworth with Charles Taylor and volumes 1–58 were published between 1872 and 1929. James Whitbread Lee Glaisher was th ...
, 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,
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. Zeit. 46 (1940) 380-407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'',
ath. Zeit. 188 (1985) 559-591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45*
N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
*
External links
Polytope names Jonathan Bowers
*
{{Polytopes
10-polytopes