In six-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 c ...
, a pentellated 6-simplex is a convex
uniform 6-polytope
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
The complete set of convex uniform 6-polytopes has not been determined, ...
with 5th order
truncations of the regular
6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.
Alt ...
.
There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-simplex is also called an expanded 6-simplex, constructed by an
expansion
Expansion may refer to:
Arts, entertainment and media
* ''L'Expansion'', a French monthly business magazine
* ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004
* ''Expansions'' (McCoy Tyner album), 1970
* ''Expansio ...
operation applied to the regular
6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.
Alt ...
. The highest form, the ''pentisteriruncicantitruncated 6-simplex'', is called an ''omnitruncated 6-simplex'' with all of the nodes ringed.
Pentellated 6-simplex
Alternate names
* Expanded 6-simplex
* Small terated tetradecapeton (Acronym: staf) (Jonathan Bowers)
Coordinates
The vertices of the ''pentellated 6-simplex'' can be positioned in 7-space as permutations of (0,1,1,1,1,1,2). This construction is based on
facets of the
pentellated 7-orthoplex.
A second construction in 7-space, from the center of a
rectified 7-orthoplex
In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.
There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being th ...
is given by coordinate permutations of:
: (1,-1,0,0,0,0,0)
Root vectors
Its 42 vertices represent the root vectors of the
simple Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...
A
6. It is the
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 lines ...
of the
6-simplex honeycomb.
Images
Pentitruncated 6-simplex
Alternate names
* Teracellated heptapeton (Acronym: tocal) (Jonathan Bowers)
Coordinates
The vertices of the ''runcitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on
facets of the
runcitruncated 7-orthoplex
In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex.
There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, a ...
.
Images
Penticantellated 6-simplex
Alternate names
* Teriprismated heptapeton (Acronym: topal) (Jonathan Bowers)
Coordinates
The vertices of the ''runcicantellated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on
facets of the
penticantellated 7-orthoplex
In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations ( pentellation) of the regular 7-orthoplex.
There are 32 unique of the 7-orthoplex with permutations of truncations, cantellati ...
.
Images
Penticantitruncated 6-simplex
Alternate names
* Terigreatorhombated heptapeton (Acronym: togral) (Jonathan Bowers)
Coordinates
The vertices of the ''penticantitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on
facets of the
penticantitruncated 7-orthoplex.
Images
Pentiruncitruncated 6-simplex
Alternate names
* Tericellirhombated heptapeton (Acronym: tocral) (Jonathan Bowers)
Coordinates
The vertices of the ''pentiruncitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on
facets of the
pentiruncitruncated 7-orthoplex.
Images
Pentiruncicantellated 6-simplex
Alternate names
* Teriprismatorhombated tetradecapeton (Acronym: taporf) (Jonathan Bowers)
Coordinates
The vertices of the ''pentiruncicantellated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,1,2,3,3,4). This construction is based on
facets of the
pentiruncicantellated 7-orthoplex
In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations ( pentellation) of the regular 7-orthoplex.
There are 32 unique of the 7-orthoplex with permutations of truncations, cantellati ...
.
Images
Pentiruncicantitruncated 6-simplex
Alternate names
* Terigreatoprismated heptapeton (Acronym: tagopal) (Jonathan Bowers)
Coordinates
The vertices of the ''pentiruncicantitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,1,2,3,4,5). This construction is based on
facets of the
pentiruncicantitruncated 7-orthoplex.
Images
Pentisteritruncated 6-simplex
Alternate names
* Tericellitruncated tetradecapeton (Acronym: tactaf) (Jonathan Bowers)
Coordinates
The vertices of the ''pentisteritruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,2,2,2,3,4). This construction is based on
facets of the
pentisteritruncated 7-orthoplex.
Images
Pentistericantitruncated 6-simplex
Alternate names
* Great teracellirhombated heptapeton (Acronym: gatocral) (Jonathan Bowers)
Coordinates
The vertices of the ''pentistericantittruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,2,2,3,4,5). This construction is based on
facets of the
pentistericantitruncated 7-orthoplex.
Images
Omnitruncated 6-simplex
The omnitruncated 6-simplex has 5040
vertices, 15120
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed b ...
s, 16800
faces (4200
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A ''regular hexagon'' h ...
s and 1260
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ad ...
), 8400
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35
uniform 6-polytope
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
The complete set of convex uniform 6-polytopes has not been determined, ...
s generated from the regular
6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.
Alt ...
.
Alternate names
* Pentisteriruncicantitruncated 6-simplex (Johnson's
omnitruncation
In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed.
It is a ''short ...
for 6-polytopes)
* Omnitruncated heptapeton
* Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)
[Klitzing, (x3x3x3x3x3x - gotaf)]
Permutohedron and related tessellation
The omnitruncated 6-simplex is the
permutohedron of order 7. The omnitruncated 6-simplex is a
zonotope
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
, the
Minkowski sum
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set
: A + B = \.
Analogously, the Minkowsk ...
of seven line segments parallel to the seven lines through the origin and the seven vertices of the 6-simplex.
Like all uniform omnitruncated n-simplices, the omnitruncated 6-simplex can
tessellate
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety o ...
space by itself, in this case 6-dimensional space with three facets around each hypercell. It has
Coxeter-Dynkin diagram of .
Coordinates
The vertices of the ''omnitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,1,2,3,4,5,6). This construction is based on
facets of the
pentisteriruncicantitruncated 7-orthoplex
In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations ( pentellation) of the regular 7-orthoplex.
There are 32 unique of the 7-orthoplex with permutations of truncations, cantellati ...
, t
0,1,2,3,4,5, .
Images
Full snub 6-simplex
The full snub 6-simplex or omnisnub 6-simplex, defined as an
alternation of the omnitruncated 6-simplex is not uniform, but it can be given Coxeter diagram and
symmetry +, and constructed from 14
snub 5-simplexes, 42 snub 5-cell antiprisms, 70 3-s duoantiprisms, and 2520 irregular
5-simplex
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(), or approximately 78.46°.
The 5-s ...
es filling the gaps at the deleted vertices.
Related uniform 6-polytopes
The pentellated 6-simplex is one of 35
uniform 6-polytopes based on the
,3,3,3,3Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
, all shown here in A
6 Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal ...
s.
Notes
References
*
H.S.M. Coxeter:
** 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*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* x3o3o3o3o3x - staf, x3x3o3o3o3x - tocal, x3o3x3o3o3x - topal, x3x3x3o3o3x - togral, x3x3o3x3o3x - tocral, x3x3x3x3o3x - tagopal, x3x3o3o3x3x - tactaf, x3x3x3o3x3x - tacogral, x3x3x3x3x3x - gotaf
External links
*
Polytopes of Various Dimensions
{{Polytopes
6-polytopes