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 rectified 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 rectification 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 unique 7 degrees of rectifications, the zeroth being the

, and the 6th and last being the

7-cube In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being c ...

. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in 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 ...

cell centers of the 7-orthoplex.

Rectified 7-orthoplex

The ''rectified 7-orthoplex'' 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 ...

for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the

kissing number In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of ...

of a sphere-packing constructed from this honeycomb. : or

Alternate names

* rectified heptacross * rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon

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 ''rectified heptacross'', one with the C₇ or ,3,3,3,3,3Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D₇ or ^4,1,1Coxeter group.

Cartesian 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 rectified heptacross, centered at the origin, edge length

\sqrt\

are all permutations of: : (±1,±1,0,0,0,0,0)

Root vectors

Its 84 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 ...

D₇. The vertices can be seen in 3

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

s, with the 21 vertices

rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a Rectification (geometry), rectification of the regular 6-simplex. There are three unique degrees of rectifications, including the zeroth, the 6-simplex itse ...

s cells on opposite sides, and 42 vertices of an

expanded 6-simplex In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations ...

passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B₇ and C₇ simple Lie groups.

Birectified 7-orthoplex

Alternate names

* Birectified heptacross * Birectified hecatonicosoctaexon (Acronym barz) (Jonathan Bowers) - birectified 128-faceted polyexonKlitzing, (o3o3x3o3o3o4o - barz)

Images

Cartesian coordinates

for the vertices of a birectified 7-orthoplex, centered at the origin, edge length

\sqrt\

are all permutations of: : (±1,±1,±1,0,0,0,0)

Trirectified 7-orthoplex

trirectified 7-orthoplex In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a Rectification (geometry), rectification of the regular 7-orthoplex. There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and ...

is the same as a trirectified 7-cube.

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. * o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz

External links

Polytopes of Various Dimensions

{{polytopes 7-polytopes