60 relations: Apex (geometry), Boerdijk–Coxeter helix, Cantellated 5-cubes, Cantellated 5-simplexes, Cartesian coordinate system, Convex polytope, Coxeter group, Coxeter–Dynkin diagram, Dihedral angle, Dual polyhedron, Facet (geometry), Four-dimensional space, Geometry, Harold Scott MacDonald Coxeter, Hexagonal tiling honeycomb, Hyperplane, Isogonal figure, Isohedral figure, Isotoxal figure, John Horton Conway, Norman Johnson (mathematician), Order-6 tetrahedral honeycomb, Pentagon, Pentagram, Platonic solid, Pyramid (geometry), Rectified 5-cell, Rectified 5-cubes, Regular 4-polytope, Regular Polytopes (book), Runcinated 5-cubes, Runcinated 5-orthoplexes, Runcinated 5-simplexes, Schläfli symbol, Schlegel diagram, Simplex, Stereographic projection, Stericated 5-cubes, Stericated 5-simplexes, Tesseract, Tetrahedron, Thorold Gosset, Triangle, Triangular bipyramid, Triangular prism, Triangular tiling, Truncated 24-cell honeycomb, Truncated 5-cubes, Truncated 5-simplexes, Uniform 4-polytope, ..., Uniform 5-polytope, Uniform polytope, Vertex figure, 120-cell, 3-sphere, 5-cell honeycomb, 5-cube, 5-orthoplex, 5-simplex, 600-cell. Expand index (10 more) »

## Apex (geometry)

In geometry, an apex (Latin for 'summit, peak, tip, top, extreme end;') is the vertex which is in some sense the highest of the figure to which it belongs.

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## Boerdijk–Coxeter helix

The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to a single tetrahedron form three intertwined helices.

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## Cantellated 5-cubes

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

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## Cantellated 5-simplexes

In five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex.

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## 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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## Dihedral angle

In geometry, a dihedral or torsion angle is the angle between two hyperplanes.

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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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## 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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## 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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## Hexagonal tiling honeycomb

In the field of hyperbolic geometry, the hexagonal tiling honeycomb arises one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space.

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## Hyperplane

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

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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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## Isohedral figure

In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when all its faces 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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## Norman Johnson (mathematician)

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

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## Order-6 tetrahedral honeycomb

In the geometry of hyperbolic 3-space, the order-6 tetrahedral honeycomb a paracompact regular space-filling tessellation (or honeycomb).

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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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## Pentagram

A pentagram (sometimes known as a pentalpha or pentangle or a star pentagon) is the shape of a five-pointed star drawn with five straight strokes.

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## Platonic solid

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

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## Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex.

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## Rectified 5-cell

In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells.

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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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## Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope.

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## Regular Polytopes (book)

Regular Polytopes is a mathematical geometry book written by Canadian mathematician H.S.M. Coxeter.

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## Runcinated 5-cubes

In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.

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## Runcinated 5-orthoplexes

In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.

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## Runcinated 5-simplexes

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

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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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## Stericated 5-cubes

In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.

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## Stericated 5-simplexes

In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.

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## Tesseract

In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square.

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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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## Triangular tiling

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane.

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## Truncated 24-cell honeycomb

In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb.

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## Truncated 5-cubes

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

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## Truncated 5-simplexes

In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

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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 5-polytope

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope.

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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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## 120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol.

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## 3-sphere

In mathematics, a 3-sphere (also called a glome) is a higher-dimensional analogue of a sphere.

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## 5-cell honeycomb

In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb.

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## 5-cube

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

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## 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.

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## 5-simplex

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

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## 600-cell

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

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## References

[1] https://en.wikipedia.org/wiki/5-cell