46 relations: Cartesian coordinate system, Convex polytope, Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, Dual polyhedron, Face (geometry), Four-dimensional space, Geometry, Harold Scott MacDonald Coxeter, Hyperplane, Hypersimplex, Isogonal figure, Isotoxal figure, John Horton Conway, Messenger of Mathematics, Michael Guy, Net (polyhedron), Norman Johnson (mathematician), Octahedron, Orthant, Pentagon, Petrie polygon, Platonic solid, Rectification (geometry), Rectified 5-cubes, Rectified 5-orthoplexes, Regular polytope, Schläfli symbol, Schlegel diagram, Simplex, Stereographic projection, Tetrahedron, Thorold Gosset, Triangle, Triangular bipyramid, Triangular prism, Truncated 5-cell, Uniform 4-polytope, Uniform k 21 polytope, Uniform polytope, Vertex figure, 2 21 polytope, 24-cell, 5-cell, 5-demicube.

## 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.

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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 of points in the n-dimensional space Rn.

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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–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).

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## Cross-polytope

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

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## Dual polyhedron

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

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## 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.

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## Four-dimensional space

In mathematics, four-dimensional space ("4D") is a geometric space with four dimensions.

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## 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.

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## Harold Scott MacDonald Coxeter

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

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## Hyperplane

In geometry a hyperplane is a subspace of one dimension less than its ambient space.

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## Hypersimplex

In polyhedral combinatorics, a hypersimplex, Δd,k, is a convex polytope that generalizes the simplex.

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## Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same.

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## Isotoxal figure

In geometry, a polytope (for example, a polygon or a polyhedron), or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges.

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## John Horton Conway

John Horton Conway FRS (born 26 December 1937) is a British mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

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## Messenger of Mathematics

The Messenger of Mathematics is a defunct mathematics journal.

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## Michael Guy

Michael J. T. Guy (born c.1942) is a British computer scientist and mathematician.

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## Net (polyhedron)

In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron.

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## Norman Johnson (mathematician)

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

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## Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces.

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## Orthant

In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.

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## Pentagon

In geometry, a pentagon (from the Greek πέντε pente and γωνία gonia, meaning five and angle) is any five-sided polygon.

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## 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.

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## Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

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## Rectification (geometry)

In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.

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## Rectified 5-cubes

In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.

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## Rectified 5-orthoplexes

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

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## 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.

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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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## Schlegel diagram

In geometry, a Schlegel diagram is a projection of a polytope from R^d into R^ through a point beyond one of its facets or faces.

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## Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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## Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

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## 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.

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## Thorold Gosset

John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.

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## Triangle

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

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## Triangular bipyramid

In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids.

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## Triangular prism

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.

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## Truncated 5-cell

In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell.

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## Uniform 4-polytope

In geometry, a uniform 4-polytope is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

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## Uniform k 21 polytope

In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the ''E''''n'' Coxeter group, and having only regular polytope facets.

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## Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.

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## Vertex figure

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

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## 2 21 polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group.

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## 24-cell

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

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## 5-cell

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

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## 5-demicube

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices truncated.

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