2 22 honeycomb

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. ^[1]

41 relations: Coxeter group, Coxeter notation, Coxeter–Dynkin diagram, Duoprism, E6 (mathematics), Face (geometry), Facet, Facet (geometry), Factorial, Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, Isogonal figure, Isohedral figure, Kissing number, Lie group, Octahedron, Proprism, Rectified 5-cell, Rectified 5-cubes, Rectified 5-simplexes, Root system, Schläfli symbol, Sphere packing, Tetrahedron, Triangle, Uniform 8-polytope, Uniform honeycomb, Uniform polytope, Vertex arrangement, Vertex figure, Voronoi diagram, Wythoff construction, 1 22 polytope, 16-cell honeycomb, 2 21 polytope, 24-cell, 5-cell, 5-orthoplex, 5-simplex.
7-polytopes

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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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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Duoprism

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

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E6 (mathematics)

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6.

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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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Facets are flat faces on geometric shapes.

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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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Factorial

In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &.

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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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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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Hyperplane

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension.

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Isogonal figure

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

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Isohedral figure

In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same.

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Kissing number

In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere.

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Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

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Octahedron

In geometry, an octahedron (octahedra or octahedrons) is a polyhedron with eight faces.

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Proprism

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

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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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Rectified 5-simplexes

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

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Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.

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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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Sphere packing

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space.

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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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Uniform 8-polytope

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets.

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Uniform honeycomb

In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets.

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Uniform polytope

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

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Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions.

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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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Voronoi diagram

In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects.

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Wythoff construction

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

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1 22 polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.

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16-cell honeycomb

In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol, and constructed by a 4-dimensional packing of 16-cell facets, three around every face.

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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 four-dimensional 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 the convex 4-polytope with Schläfli symbol.

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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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References

[1] https://en.wikipedia.org/wiki/2_22_honeycomb

Also known as Birectified 2 22 honeycomb, Bitruncated 2 22 honeycomb, E6 honeycomb, E6 lattice, Gosset 2 22 honeycomb.

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2 22 honeycomb

Table of Contents

See also

7-polytopes

References