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6-dimensional geometric object
six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. A 6-polytope is a closed six-dimensional figure
6-polytope
Uniform 6-dimensional polytope
a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete
Uniform_6-polytope
Geometric object with flat sides
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any
Polytope
Four-dimensional analogues of the regular polyhedra in three dimensions
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular
Regular_4-polytope
6-dimensional hypercube
dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. Acronym: ax It is a part of an infinite family of polytopes
6-cube
Four-dimensional geometric object with flat sides
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure
4-polytope
Seven-dimensional geometric object
7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is
Uniform_7-polytope
Regular polytope dual to the hypercube in any number of dimensions
In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean
Cross-polytope
A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Uniform 6-polytope
In geometry, a 6-demicube, demihexeract or hemihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed
6-demicube
Polytope in 8-dimensional geometry
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset
4_21_polytope
Regular 6 dimensional polytope
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell
6-orthoplex
Isogonal polytope with uniform facets
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means
Uniform_polytope
5-dimensional geometric object
geometry, a five-dimensional polytope (or 5-polytope or polyteron) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which
5-polytope
Uniform 6-polytope
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of
1_22_polytope
Polyhedron with four faces
tetrahedron of the cube is an example of a Heronian tetrahedron. Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme
Tetrahedron
Uniform 6-polytope
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset
2_21_polytope
regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank. There is only one polytope of
List_of_regular_polytopes
needed] 142 polytope, 241 polytope, 421 polytope, Truncated 421 polytope, Truncated 241 polytope, Truncated 142 polytope, Cantellated 421 polytope, Cantellated
List_of_mathematical_shapes
Polytope contained by 7-polytope facets
eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets, each 6-polytope ridge being shared by exactly two 7-polytope facets. A
Uniform_8-polytope
six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex. There are three unique degrees
Rectified_6-simplexes
Four-dimensional analogue of the cube
labels it the γ4 polytope. The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope. The construction
Tesseract
Geometric space with six dimensions
Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and
Six-dimensional_space
Isogonal polytope with regular facets
definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled
Semiregular_polytope
geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of 240, 2160 and 17280
E8_polytope
In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72
E6_polytope
Polytope with highest degree of symmetry
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In
Regular_polytope
geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-cube. There are 8 unique
Stericated_6-cubes
Uniform 6-polytope
6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of pentellations of the 6-simplex
Pentellated_6-simplexes
Four-dimensional analog of the octahedron
convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described
16-cell
Convex hull of a finite set of points in a Euclidean space
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 {\displaystyle n} -dimensional
Convex_polytope
uniform 7-polytope, being a truncation of the 7-demicube. A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope facets, and
Cantic_7-cube
geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex. There are 8
Runcinated_6-simplexes
Convex polytope, the n-dimensional analogue of a square and a cube
measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The
Hypercube
6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex. There are 8 unique sterications for the 6-simplex
Stericated_6-simplexes
truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube. There are 5 truncations for the 6-cube.
Truncated_6-cubes
Five-dimensional geometric shape
5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets
Uniform_5-polytope
Four-dimensional analogue of the tetrahedron
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells
5-cell
Geometric object
geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family
Uniform_k_21_polytope
Shape in six-dimensional geometry
In six-dimensional geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope. Truncated 6-demicube Truncaced demihexeract Truncated
Cantic_6-cube
runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube. There are 12 unique runcinations of the 6-cube
Runcinated_6-cubes
Generalization of a polytope in real space
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
Complex_polytope
six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees
Truncated_6-simplexes
pentic 6-cube is a convex uniform 6-polytope. There are 8 pentic forms of the 6-cube. The pentic 6-cube, , has half of the vertices of a pentellated 6-cube
Pentic_6-cubes
10-dimensional hypercube
as a 10 dimensional polytope, constructed from 20 regular facets. Acronym: deker It is a part of an infinite family of polytopes, called hypercubes. The
10-cube
Class of 4-dimensional polytopes
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra
Uniform_4-polytope
Uniform 7-dimensional polytope
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset
3_21_polytope
cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex. There are unique 4 degrees of cantellation for the 6-simplex
Cantellated_6-simplexes
six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube. Cantellated 6-demicube Cantellated demihexeract
Runcic_6-cubes
7-dimensional hypercube
called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets. This configuration matrix represents
7-cube
Uniform 6-polytope
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and
6-simplex
six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-orthoplex. There are 12 unique
Runcinated_6-orthoplexes
Regular 5-polytope
5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral
5-simplex
Isogonal honeycomb of uniform polytope facets
terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex
Uniform_honeycomb
In 5-dimensional geometry, there are 31 uniform polytopes with B5 symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices
B5_polytope
8-dimensional hypercube
hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets. It is a part of an infinite family of polytopes, called hypercubes. The dual
8-cube
In 4-dimensional geometry, there are 9 uniform 4-polytopes with F4 symmetry, and one chiral half symmetry, the snub 24-cell. There is one self-dual regular
List_of_F4_polytopes
Uniform polytope
In geometry, 1k2 polytope is a uniform polytope in n dimensions (n = k + 4) constructed from the En Coxeter group. The family was named by their Coxeter
Uniform_1_k2_polytope
In 6-dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices. Each can
A6_polytope
9-dimensional hypercube
nine-dimensional polytope constructed with 18 regular facets. It was given acronym enne by J. Bowers. It is a part of an infinite family of polytopes, called hypercubes
9-cube
Shape with six sides
for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections: A principal diagonal of a
Hexagon
Equiangular and equilateral polygon
Polyhedra? Branko Grünbaum (2003), Fig. 3 Regular polytopes, p.95 Coxeter, The Densities of the Regular Polytopes II, 1932, p.53 Lee, Hwa Young; "Origami-Constructible
Regular_polygon
Natural number
6 edges. In four dimensions, there are a total of six convex regular polytopes. In the classification of finite simple groups, twenty of twenty-six sporadic
6
5-dimensional hypercube
deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the
5-cube
In 4-dimensional geometry, there are 15 uniform 4-polytopes with B4 symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices
B4_polytope
Poset representing certain properties of a polytope
mathematics, an abstract polytope is an algebraic partially ordered set which captures certain combinatorial properties of a traditional polytope without specifying
Abstract_polytope
geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex. There are
Stericated_6-orthoplexes
Convex regular 5-polytope in geometry
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
5-orthoplex
geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex. There are unique 6 degrees of rectifications
Rectified_6-orthoplexes
Plane figure bounded by line segments
single plane. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons
Polygon
Convex uniform 7-polytope in seven-dimensional geometry
seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four unique
Rectified_7-simplexes
In 6-dimensional geometry, there are 64 uniform polytopes with B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-cube with 12 and 64 vertices
B6_polytope
Uniform 8 dimensional polytope
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its
1_42_polytope
geometry, there are 127 uniform polytopes with E7 symmetry. The three simplest forms are the 321, 231, and 132 polytopes, composed of 56, 126, and 576 vertices
E7_polytope
Polytope constructed from alternation of a hypercube
(also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled
Demihypercube
cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex. There are 8 cantellation for the 6-orthoplex including
Cantellated_6-orthoplexes
In 4-dimensional geometry, there are 9 uniform polytopes with A4 symmetry. There is one self-dual regular form, the 5-cell with 5 vertices. A4 symmetry
A4_polytope
In 5-dimensional geometry, there are 23 uniform polytopes with D5 symmetry, 8 are unique, and 15 are shared with the B5 symmetry. There are two special
D5_polytope
Type of geometric object
nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets. A
Uniform_9-polytope
Uniform polytope
In geometry, 2k1 polytope is a uniform polytope in n dimensions (n = k + 4) constructed from the En Coxeter group. The family was named by their Coxeter
Uniform_2_k1_polytope
Geometrical Shape
six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube. There are unique 6 degrees of rectifications
Rectified_6-cubes
Four-dimensional analog of the icosahedron
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known
600-cell
Four-dimensional geometric objects
In 4-dimensional geometry, there are 15 uniform polytopes with H4 symmetry. Two of these, the 120-cell and 600-cell, are regular. Each can be visualized
H4_polytope
Regular polytope whose 2D form is a pentagon
In geometry, a pentagonal polytope is a regular polytope in n dimensions constructed from the Hn Coxeter group. The family was named by H. S. M. Coxeter
Pentagonal_polytope
Uniform Polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin
2_31_polytope
Polyhedron with 12 faces
angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show
Dodecahedron
a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube. There are 8 cantellations for the 6-cube, including
Cantellated_6-cubes
Uniform polytope
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin
1_32_polytope
truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex. There are 5 degrees of truncation for the 6-orthoplex
Truncated_6-orthoplexes
geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices. Each can be visualized
A5_polytope
Four-dimensional analog of the dodecahedron
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called
120-cell
Convex regular 9 dimensional polytope
In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032
9-orthoplex
In polytope theory, the edge graph (also known as vertex-edge graph or just graph) of a polytope is a combinatorial graph whose vertices and edges correspond
Graph_of_a_polytope
Multi-dimensional generalization of triangle
dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point
Simplex
geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube. Runcinated demihexeract Runcinated 6-demicube Small
Steric_6-cubes
Shape with five sides
i = 1 5 d i 6 = 5 ( ( R 2 + L 2 ) 3 + 6 R 2 L 2 ( R 2 + L 2 ) ) , ∑ i = 1 5 d i 8 = 5 ( ( R 2 + L 2 ) 4 + 12 R 2 L 2 ( R 2 + L 2 ) 2 + 6 R 4 L 4 ) . {\displaystyle
Pentagon
Solid with twenty equal triangular faces
background in the comparison mensuration. It is analogous to a four-dimensional polytope, the 600-cell. Regular icosahedra occur both in natural and human-made
Regular_icosahedron
Regular 5-polytope
five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed
5-demicube
Regular object in four dimensional geometry
In four-dimensional geometry, the 24-cell is a convex regular 4-polytope, a four-dimensional analogue of a Platonic solid. It is named for the 24 octahedra
24-cell
In 6-dimensional geometry, there are 47 uniform polytopes with D6 symmetry, of which 16 are unique and 31 are shared with the B6 symmetry. There are two
D6_polytope
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