7-orthoplex

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces. ^[1]

36 relations: Cartesian coordinate system, Convex polytope, Coxeter group, Coxeter notation, Coxeter–Dynkin diagram, Cross-polytope, Dual polyhedron, Edge (geometry), Face (geometry), Facet (geometry), Geometry, Gosset–Elte figures, Greek language, Hanner polytope, Harold Scott MacDonald Coxeter, Hypercube, Hyperrectangle, Norman Johnson (mathematician), Petrie polygon, Projection (linear algebra), Quasiregular polyhedron, Rectified 7-orthoplexes, Regular polytope, Schläfli symbol, Tetradecagon, Tetrahedron, Triangle, Truncated 7-orthoplexes, Uniform 7-polytope, Vertex (geometry), Vertex figure, 5-cell, 5-simplex, 6-orthoplex, 6-simplex, 7-cube.
7-polytopes

Cartesian coordinate system

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 oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system.

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Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n.

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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).

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Coxeter notation

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups.

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Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

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Cross-polytope

In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space.

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Dual polyhedron

In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.

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Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.

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Face (geometry)

In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron.

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Facet (geometry)

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

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Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

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Gosset–Elte figures

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles.

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Greek language

Greek (Elliniká,; Hellēnikḗ) is an independent branch of the Indo-European family of languages, native to Greece, Cyprus, Italy (in Calabria and Salento), southern Albania, and other regions of the Balkans, the Black Sea coast, Asia Minor, and the Eastern Mediterranean.

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Hanner polytope

In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.

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Hypercube

In geometry, a hypercube is an ''n''-dimensional analogue of a square and a cube.

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Hyperrectangle

In geometry, a hyperrectangle (also called a box, hyperbox, or orthotopeCoxeter, 1973), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions.

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Norman Johnson (mathematician)

Norman Woodason Johnson was a mathematician at Wheaton College, Norton, Massachusetts.

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Petrie polygon

In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no) belongs to one of the facets.

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Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P.

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Quasiregular polyhedron

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex.

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Rectified 7-orthoplexes

In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex. 7-orthoplex and rectified 7-orthoplexes are 7-polytopes.

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Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.

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Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.

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Tetradecagon

In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.

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Tetrahedron

In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.

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Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.

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Truncated 7-orthoplexes

In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex. 7-orthoplex and truncated 7-orthoplexes are 7-polytopes.

See 7-orthoplex and Truncated 7-orthoplexes

Uniform 7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. 7-orthoplex and Uniform 7-polytope are 7-polytopes.

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Vertex (geometry)

In geometry, a vertex (vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect.

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Vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

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5-cell

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol.

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5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

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6-orthoplex

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

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6-simplex

In geometry, a 6-simplex is a self-dual regular 6-polytope.

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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. 7-orthoplex and 7-cube are 7-polytopes.

See 7-orthoplex and 7-cube

References

[1] https://en.wikipedia.org/wiki/7-orthoplex

Also known as 4 11 polytope, 7-cross, 7-cross polytope, 7-cross-polytope, Hecatonicosoctaexon, Heptacross.

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7-orthoplex

Table of Contents

See also

7-polytopes

References