Table of Contents
46 relations: Configuration (polytope), Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, Disphenoid, E7 (mathematics), E7 polytope, Emanuel Lodewijk Elte, Equilateral triangle, Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Honeycomb (geometry), Hyperplane, Isosceles triangle, Octadecagon, Octahedron, Petrie polygon, Projection (linear algebra), Rectification (geometry), Rectified 5-cell, Rectified 5-cubes, Rectified 5-simplexes, Rectified 6-cubes, Rectified 6-simplexes, Schläfli symbol, Tetrahedron, Triangle, Uniform 1 k2 polytope, Uniform 7-polytope, Uniform polytope, Vertex figure, Voronoi diagram, Wythoff construction, 1 22 polytope, 1 32 polytope, 1 33 honeycomb, 16-cell, 2 31 polytope, 3 21 polytope, 3 31 honeycomb, 5-cell, 5-demicube, 5-simplex, 6-demicube.
- 7-polytopes
Configuration (polytope)
In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.
See 1 32 polytope and Configuration (polytope)
Convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n.
See 1 32 polytope and Convex polytope
Coxeter element
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections.
See 1 32 polytope and Coxeter element
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).
See 1 32 polytope and Coxeter group
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.
See 1 32 polytope and Coxeter–Dynkin diagram
Disphenoid
In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles.
See 1 32 polytope and Disphenoid
E7 (mathematics)
In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.
See 1 32 polytope and E7 (mathematics)
E7 polytope
In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry. 1 32 polytope and E7 polytope are 7-polytopes.
See 1 32 polytope and E7 polytope
Emanuel Lodewijk Elte
Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) at joodsmonument.nl was a Dutch mathematician.
See 1 32 polytope and Emanuel Lodewijk Elte
Equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length.
See 1 32 polytope and Equilateral triangle
Geometry
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.
See 1 32 polytope and Geometry
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.
See 1 32 polytope and Gosset–Elte figures
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.
See 1 32 polytope and Harold Scott MacDonald Coxeter
Honeycomb (geometry)
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.
See 1 32 polytope and Honeycomb (geometry)
Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension.
See 1 32 polytope and Hyperplane
Isosceles triangle
In geometry, an isosceles triangle is a triangle that has two sides of equal length.
See 1 32 polytope and Isosceles triangle
Octadecagon
In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon.
See 1 32 polytope and Octadecagon
Octahedron
In geometry, an octahedron (octahedra or octahedrons) is a polyhedron with eight faces.
See 1 32 polytope and Octahedron
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no) belongs to one of the facets.
See 1 32 polytope and Petrie polygon
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P.
See 1 32 polytope and Projection (linear algebra)
Rectification (geometry)
In Euclidean geometry, rectification, also known as critical truncation 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.
See 1 32 polytope and Rectification (geometry)
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.
See 1 32 polytope and Rectified 5-cell
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.
See 1 32 polytope and Rectified 5-cubes
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.
See 1 32 polytope and Rectified 5-simplexes
Rectified 6-cubes
In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.
See 1 32 polytope and Rectified 6-cubes
Rectified 6-simplexes
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
See 1 32 polytope and Rectified 6-simplexes
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
See 1 32 polytope and Schläfli symbol
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.
See 1 32 polytope and Tetrahedron
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.
See 1 32 polytope and Triangle
Uniform 1 k2 polytope
In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n.
See 1 32 polytope and Uniform 1 k2 polytope
Uniform 7-polytope
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. 1 32 polytope and Uniform 7-polytope are 7-polytopes.
See 1 32 polytope and Uniform 7-polytope
Uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
See 1 32 polytope and Uniform polytope
Vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
See 1 32 polytope and Vertex figure
Voronoi diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects.
See 1 32 polytope and Voronoi diagram
Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.
See 1 32 polytope and Wythoff construction
1 22 polytope
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.
See 1 32 polytope and 1 22 polytope
1 32 polytope
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. 1 32 polytope and 1 32 polytope are 7-polytopes.
See 1 32 polytope and 1 32 polytope
1 33 honeycomb
In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol, and is composed of 132 facets.
See 1 32 polytope and 1 33 honeycomb
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
2 31 polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. 1 32 polytope and 2 31 polytope are 7-polytopes.
See 1 32 polytope and 2 31 polytope
3 21 polytope
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. 1 32 polytope and 3 21 polytope are 7-polytopes.
See 1 32 polytope and 3 21 polytope
3 31 honeycomb
In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.
See 1 32 polytope and 3 31 honeycomb
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol.
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 removed.
See 1 32 polytope and 5-demicube
5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.
See 1 32 polytope and 5-simplex
6-demicube
In geometry, a 6-demicube or demihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed.
See 1 32 polytope and 6-demicube
See also
7-polytopes
- 1 32 polytope
- 2 22 honeycomb
- 2 31 polytope
- 3 21 polytope
- 6-cubic honeycomb
- 6-demicubic honeycomb
- 6-simplex honeycomb
- 7-cube
- 7-demicube
- 7-orthoplex
- 7-simplex
- A7 polytope
- B7 polytope
- Cantellated 7-cubes
- Cantellated 7-orthoplexes
- Cantellated 7-simplexes
- Cantic 7-cube
- Cyclotruncated 6-simplex honeycomb
- D7 polytope
- E7 polytope
- Hexic 7-cubes
- Hexicated 7-cubes
- Hexicated 7-orthoplexes
- Hexicated 7-simplexes
- Omnitruncated 6-simplex honeycomb
- Pentellated 7-cubes
- Pentellated 7-orthoplexes
- Pentellated 7-simplexes
- Pentic 7-cubes
- Quarter 6-cubic honeycomb
- Rectified 7-cubes
- Rectified 7-orthoplexes
- Rectified 7-simplexes
- Runcic 7-cubes
- Runcinated 7-cubes
- Runcinated 7-orthoplexes
- Runcinated 7-simplexes
- Steric 7-cubes
- Stericated 7-cubes
- Stericated 7-orthoplexes
- Stericated 7-simplexes
- Truncated 7-cubes
- Truncated 7-orthoplexes
- Truncated 7-simplexes
- Uniform 7-polytope
References
Also known as 0 321 polytope, Birectified 2 31 polytope, Gosset 1 32 polytope, Rectified 1 32 polytope, Trirectified 1 32 polytope.