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# 16-cell

In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. 

72 relations: Boerdijk–Coxeter helix, Complete bipartite graph, Complex polytope, Configuration (polytope), Convex polytope, Coxeter element, Coxeter notation, Coxeter–Dynkin diagram, Cross-polytope, Cube, Cubic honeycomb, Demihypercube, Dihedral angle, Dual polyhedron, Duoprism, Duopyramid, Edge (geometry), Euclidean space, Face (geometry), Four-dimensional space, Geometry, Graphical projection, Harold Scott MacDonald Coxeter, Hexagonal bipyramid, Hypercube, Hyperrectangle, Isogonal figure, Isohedral figure, Isotoxal figure, John Horton Conway, Ludwig Schläfli, Möbius–Kantor polygon, Net (polyhedron), Norman Johnson (mathematician), Octagon, Octahedral pyramid, Octahedron, Order-4 dodecahedral honeycomb, Order-4 hexagonal tiling honeycomb, Order-6 tetrahedral honeycomb, Orthant, Orthographic projection, Polyhedral combinatorics, Projective cover, Quasiregular polyhedron, Rectified tesseract, Regular 4-polytope, Regular Polytopes (book), Schläfli symbol, Schlegel diagram, ... Expand index (22 more) »

## 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 only one tetrahedron form three intertwined helices.

No description.

## Complex polytope

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

## Configuration (polytope)

In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.

## 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 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 expressing the structure of a Coxeter-Dynkin diagram, 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 n-dimensions.

## Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

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## Cubic honeycomb

The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells.

## Demihypercube

In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as h&gamma;n for being half of the hypercube family, &gamma;n.

## Dihedral angle

A dihedral angle is the angle between two intersecting planes.

## Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, 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.

## Duoprism

In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher.

## Duopyramid

In geometry of 4 dimensions or higher, a duopyramid or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two.

## Edge (geometry)

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

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## 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.

## Four-dimensional space

A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space.

## 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.

## Graphical projection

Graphical projection is a protocol, used in technical drawing, by which an image of a three-dimensional object is projected onto a planar surface without the aid of numerical calculation.

## Harold Scott MacDonald Coxeter

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

## Hexagonal bipyramid

A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases.

## Hypercube

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

## 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.

## Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.

## 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.

## 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.

## John Horton Conway

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

## Ludwig Schläfli

Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces.

## Möbius–Kantor polygon

In geometry, the Möbius–Kantor polygon is a regular complex polygon 33,, in \mathbb^2.

## Net (polyhedron)

In geometry a 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.

## Norman Johnson (mathematician)

Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.

## Octagon

In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon.

## Octahedral pyramid

In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex.

## Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

## Order-4 dodecahedral honeycomb

In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs).

## Order-4 hexagonal tiling honeycomb

In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space.

## Order-6 tetrahedral honeycomb

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

## 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.

## Orthographic projection

Orthographic projection (sometimes orthogonal projection), is a means of representing three-dimensional objects in two dimensions.

## Polyhedral combinatorics

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.

## Projective cover

In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.

## Quasiregular polyhedron

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

## Rectified tesseract

In geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra.

## Regular 4-polytope

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

## Regular Polytopes (book)

Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter.

## Schläfli symbol

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

## 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.

## Stereographic projection

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

## Tessellation

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.

## Tesseract

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

## Tesseractic honeycomb

In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol, and constructed by a 4-dimensional packing of tesseract facets.

## Tetrahedron

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.

## Thorold Gosset

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

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Totally bounded space

In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of every fixed "size" (where the meaning of "size" depends on the given context).

## Triakis tetrahedron

In geometry, a triakis tetrahedron (or kistetrahedron) is an Archimedean dual solid, or a Catalan solid.

## Triangle

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

## Triangular tiling

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

## Truncated tesseract

In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.

## Venn diagram

A Venn diagram (also called primary diagram, set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets.

## Vertex (geometry)

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

## Vertex figure

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

## Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.

## 16-cell honeycomb

In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs) in Euclidean 4-space.

## 24-cell

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

## 24-cell honeycomb

In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells.

## 4-polytope

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.

## 5-cell

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

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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.

## References

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