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, bu ...

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°. Alter ...

. 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

. 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 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 ...

of the

pentellated 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, cantellations, ...

. 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 ...

A₆. 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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

of the

6-simplex honeycomb In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportion ...

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

of the runcitruncated 7-orthoplex.

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

of the penticantellated 7-orthoplex.

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

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

of the

pentiruncitruncated 7-orthoplex In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order Truncation (geometry), truncations (pentellation) of the regular 7-orthoplex. There are 32 unique of the 7-orthoplex with permutations of trunc ...

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

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, cantellation ...

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

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

of the

pentisteritruncated 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, cantellations, ...

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

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 by ...

s, 16800

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

(4200

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

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 adj ...

), 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

s generated from the regular

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 ''shortc ...

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 In mathematics, the permutohedron of order ''n'' is an (''n'' − 1)-dimensional polytope embedded in an ''n''-dimensional space. Its vertex coordinates (labels) are the permutations of the first ''n'' natural numbers. The edges ident ...

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 Minkowski ...

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 of ...

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

of the pentisteriruncicantitruncated 7-orthoplex, 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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

⁺, 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 ,3,3,3,3

Coxeter 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 refl ...

, all shown here in A₆

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 space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...

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