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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. [1]

Cartesian coordinate system

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

Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

Coxeter element

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.

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).

Coxeter notation

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

Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).

Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions.

Dodecagon

In geometry, a dodecagon is any polygon with twelve sides and twelve angles.

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other.

Edge (geometry)

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

Face (geometry)

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

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.

Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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.

Greek language

Greek or Hellenic (Modern Greek: ελληνικά, elliniká, "Greek", ελληνική γλώσσα, ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to the southern Balkans, the Aegean Islands, western Asia Minor, parts of northern and Eastern Anatolia and the South Caucasus, southern Italy, Albania and Cyprus.

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 &ndash; March 31, 2003) was a British-born Canadian geometer.

Hosohedron

In geometry, an ''n''-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.

Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n.

Hyperrectangle

In geometry, an n-orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.

Icosahedron

In geometry, an icosahedron is a polyhedron with 20 faces.

Norman Johnson (mathematician)

Norman W. Johnson (born November 12, 1930) is a mathematician, previously at Wheaton College, Norton, Massachusetts.

Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every (n-1) consecutive sides (but no n) belong to one of the facets.

Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

Regular polytope

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

Schläfli symbol

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

Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, three of which meet at each corner or vertex.

Triangle

A triangle is a polygon with three edges and three vertices.

Uniform 6-polytope

In six-dimensional geometry, a uniform polypeton (or uniform 6-polytope) is a six-dimensional uniform polytope.

Vertex (geometry)

In geometry, a vertex (plural vertices) is a special kind of point that describes the corners or intersections of geometric shapes.

Vertex figure

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

5-cell

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.

5-orthoplex

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

5-simplex

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

6-cube

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

6-polytope

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

References

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