In eight-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 truncated 8-simplex is a convex

uniform 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 ...

, being a

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

of 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 angle is c ...

. There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the

tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

cells of the 8-simplex.

Truncated 8-simplex

Alternate names

* Truncated enneazetton (Acronym: tene) (Jonathan Bowers)

Coordinates

The

Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

s of the vertices of the ''truncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex.

Images

Bitruncated 8-simplex

Alternate names

* Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)

Coordinates

The

s of the vertices of the ''bitruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,2). This construction is based on facets of the

bitruncated 9-orthoplex In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves. Bitruncated regular polyt ...

Images

Tritruncated 8-simplex

Alternate names

* Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)

Coordinates

The

s of the vertices of the ''tritruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 9-orthoplex.

Images

Quadritruncated 8-simplex

The quadritruncated 8-simplex an isotopic polytope, constructed from 18

tritruncated 7-simplex In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of t ...

facets.

Alternate names

* Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)Klitizing, (o3o3o3x3x3o3o3o - be)

Coordinates

The

s of the vertices of the ''quadritruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,2,2,2). This construction is based on facets of the quadritruncated 9-orthoplex.

Images

Related polytopes

This polytope is one of 135 uniform 8-polytopes with A₈ symmetry.

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. * x3x3o3o3o3o3o3o - tene, o3x3x3o3o3o3o3o - batene, o3o3x3x3o3o3o3o - tatene, o3o3o3x3x3o3o3o - be

External links

Polytopes of Various Dimensions

{{Polytopes 8-polytopes