In seven-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 7-orthoplex is a convex

uniform 7-polytope In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose ...

, 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

7-orthoplex In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 Vertex (geometry), vertices, 84 Edge (geometry), edges, 280 triangle Face (geometry), faces, 560 tetrahedron Cell (mathematics), cells, 672 5-cells ''4-faces'', 448 ...

. There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex 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 o ...

cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.

Truncated 7-orthoplex

Alternate names

* Truncated heptacross * Truncated hecatonicosoctaexon (Jonathan Bowers)

Coordinates

Cartesian coordinates 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 in t ...

for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42)

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

s of : (±2,±1,0,0,0,0,0)

Images

Construction

There are two

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 ...

s associated with the ''truncated 7-orthoplex'', one with the C₇ or ,3⁵Coxeter group, and a lower symmetry with the D₇ or ^4,1,1Coxeter group.

Bitruncated 7-orthoplex

Alternate names

* Bitruncated heptacross * Bitruncated hecatonicosoctaexon (Jonathan Bowers)

Coordinates

for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate

s of : (±2,±2,±1,0,0,0,0)

Images

Tritruncated 7-orthoplex

The tritruncated 7-orthoplex can tessellation space in the

quadritruncated 7-cubic honeycomb The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. There are many different W ...

Alternate names

* Tritruncated heptacross * Tritruncated hecatonicosoctaexon (Jonathan Bowers)Klitzing, (o3o3x3x3o3o4o - totaz)

Coordinates

for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate

s of : (±2,±2,±2,±1,0,0,0)

Images

Notes

References

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

: ** 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. * x3x3o3o3o3o4o - tez, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz

External links

Polytopes of Various Dimensions

{{polytopes 7-polytopes