In eight-dimensional

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a heptellated 8-simplex is a

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

uniform 8-polytope, including 7th-order truncations (heptellation) from the regular

8-simplex In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angl ...

. There are 35 unique heptellations for the 8-simplex, including all

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

s of truncations, cantellations, runcinations,

sterication In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...

s, pentellations, and hexications. The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular

. The highest form, the ''heptihexipentisteriruncicantitruncated 8-simplex'' is more simply called an ''omnitruncated 8-simplex'' with all of the nodes ringed.

Heptellated 8-simplex

Alternate names

* Expanded 8-simplex * Small exated enneazetton (Acronym: soxeb) (Jonathan Bowers)

Coordinates

The vertices of the ''heptellated 8-simplex'' can be positioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on facets of the heptellated 9-orthoplex. A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of: : (1,-1,0,0,0,0,0,0,0)

Root vectors

Its 72 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 symm ...

A₈.

Images

Omnitruncated 8-simplex

The symmetry order of an omnitruncated 8-simplex is 725760. The symmetry of a family of a uniform polytopes is equal to the number of vertices of the

omnitruncation In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each face of any dimension of the original polytope. Omnitruncation is the dual ...

, being 362880 (9

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

) in the case of the omnitruncated 8-simplex; but when the CD symbol is palindromic, the symmetry order is doubled, 725760 here, because the element corresponding to any element of the underlying 8-simplex can be exchanged with one of those corresponding to an element of its dual.

Alternate names

* Heptihexipentisteriruncicantitruncated 8-simplex * Great exated enneazetton (Acronym: goxeb) (Jonathan Bowers)

Coordinates

The

Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

s of the vertices of the ''omnitruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,1,2,3,4,5,6,7,8). This construction is based on facets of the heptihexipentisteriruncicantitruncated 9-orthoplex, t_{0,1,2,3,4,5,6,7}

Images

Permutohedron and related tessellation

The ''omnitruncated 8-simplex'' is the

permutohedron In mathematics, the permutohedron (also spelled permutahedron) of order is an -dimensional polytope embedded in an -dimensional space. Its vertex (geometry), vertex coordinates (labels) are the permutations of the first natural numbers. The edg ...

of order 9. The omnitruncated 8-simplex is a

zonotope In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

, the

Minkowski sum In geometry, the Minkowski sum 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'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...

of nine line segments parallel to the nine lines through the origin and the nine vertices of the 8-simplex. Like all uniform omnitruncated n-simplices, the omnitruncated 8-simplex can tessellate space by itself, in this case 8-dimensional space with three facets around each

ridge A ridge is a long, narrow, elevated geomorphologic landform, structural feature, or a combination of both separated from the surrounding terrain by steep sides. The sides of a ridge slope away from a narrow top, the crest or ridgecrest, wi ...

. It has Coxeter-Dynkin diagram of .

Related polytopes

The two presented polytopes are selected from 135 uniform 8-polytopes with A₈ symmetry, shown in the table below.

Notes

Explanatory 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
wiley.com
*** (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. * x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb

External links

Polytopes of Various Dimensions

{{Polytopes 8-polytopes