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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
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
Uniform 8 dimensional polytope
quadrirectified 421. These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8 dimensions, made of uniform polytope facets and vertex
1_42_polytope
Uniform 6-polytope
122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named
1_22_polytope
5-dimensional hypercube
5-cube, with its 32 vertices, 80 edges, and 40 square faces, and the other 40 square faces of the 5-cube become square holes. This polytope is one of 31 uniform
5-cube
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
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
6-dimensional hypercube
creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets.
6-cube
7-dimensional hypercube
Coxeter, Regular Polytopes, p. 12, Sec. 1.8 Configurations Coxeter (1991), p. 117. H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition
7-cube
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
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
8-dimensional hypercube
of cross-polytopes. Cartesian coordinates for the vertices of an 8-cube centered at the origin and edge length 2 are (±1,±1,±1,±1,±1,±1,±1,±1) while the
8-cube
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
one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the
List_of_regular_polytopes
Convex regular 5-polytope in geometry
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. It
5-orthoplex
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
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
convex polytopes, a distributive polytope is a convex polytope for which coordinatewise minima and maxima of pairs of points remain within the polytope. For
Distributive_polytope
mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the monotonic
Order_polytope
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
Uniform 6-dimensional polytope
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
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
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
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
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
In the study of abstract polytopes, a chiral polytope is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms
Chiral_polytope
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
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
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
Convex regular 8-polytope
In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cell
8-orthoplex
Solid with 2 parallel n-gonal bases connected by n parallelograms
n-polytope elements are doubled from the (n − 1)-polytope elements and then creating new elements from the next lower element. Take an n-polytope with
Prism_(geometry)
Multi-dimensional generalization of triangle
represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point, a 1-dimensional simplex is a line segment
Simplex
Natural number between 89 and 91
UC55) contain 90 edges or vertices. The self-dual Witting polytope contains ninety van Oss polytopes such that sections by the common plane of two non-orthogonal
90_(number)
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
Uniform polychoron
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge
Rectified_5-cell
Regular 7- polytope
In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cell
7-orthoplex
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
geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra
Snub_24-cell
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
In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. T ¯ 9 {\displaystyle {\bar {T}}_{9}} , also (E10)
E9_honeycomb
Geometric space with five dimensions
32 vertices, 80 edges, 80 faces (each a square), 40 cells (each a cube), and 10 hypercells (each a tesseract). The 5-orthoplex of the cross polytope family
Five-dimensional_space
Maths conjecture
symmetric polytope have at least 3 d {\displaystyle 3^{d}} nonempty faces? More unsolved problems in mathematics In geometry, more specifically in polytope theory
Kalai's_3^d_conjecture
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
4-D shape
geometry, a uniform antiprismatic prism or antiduoprism is a uniform 4-polytope with two uniform antiprism cells in two parallel 3-space hyperplanes, connected
Uniform_antiprismatic_prism
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
In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube. There
Stericated_5-cubes
economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions to an instance of the
Stable_matching_polytope
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
Flat-sided three-dimensional shape
two-dimensional polygons and to be the three-dimensional specialization of polytopes (a more general concept in any number of dimensions). Polyhedra have several
Polyhedron
Method to solve optimization problems
affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or
Linear_programming
In 8-dimensional geometry, there are 135 uniform polytopes with A8 symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices. Each
A8_polytope
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 respectively. The
B5_polytope
Group of polytopes
In 8-dimensional geometry, there are 255 uniform polytopes with B8 symmetry (to which this article adds for illustration the 8-demicube as an alternation
B8_polytope
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
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
Polytope whose vertices represent permutations
permutohedron (also spelled permutahedron) of order n is an (n − 1)-dimensional polytope embedded in an n-dimensional space. Its vertex coordinates (labels)
Permutohedron
Uniform 7-polytope
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is
7-demicube
Uniform polytopes with D8 symmetry
In 8-dimensional geometry, there are 191 uniform polytopes with D8 symmetry, of which 64 are unique and 127 are shared with the B8 symmetry. There is
D8_polytope
Uniform Polytope
the 231. These polytopes are part of a family of 127 (or 27−1) convex uniform polytopes in seven dimensions, made of uniform polytope facets and vertex
2_31_polytope
these 32 polytopes can be made in the B5, B4, B3, B2, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry. These 32 polytopes are each
B4_polytope
Type of tesseract
In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a
Truncated_tesseract
In 7-dimensional geometry, there are 95 uniform polytopes with D7 symmetry; 32 are unique, and 63 are shared with the B7 symmetry. There are two regular
D7_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
a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract
Runcinated_tesseracts
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
Number, approximately 1.618
1 ; 1 , 1 , 1 , … ] = 1 + 1 1 + 1 1 + 1 1 + 1 ⋱ {\displaystyle \varphi =[1;1,1,1,\dots ]=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{{\vphantom {1}}
Golden_ratio
In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube. There are four unique truncations
Truncated_5-cubes
Convex uniform 4-polytope
is a convex uniform 4-polytope or 4-dimensional polytope bounded by 56 cells: 8 small rhombicuboctahedra, 16 octahedra, and 32 triangular prisms. In the
Cantellated_tesseract
Uniform 6-polytope
6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called
6-demicube
Two raised to an integer power
number of (n − 1)-faces of an n-dimensional cross-polytope is also 2n and the formula for the number of x-faces an n-dimensional cross-polytope has is 2 x
Power_of_two
In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices. Each can
A7_polytope
hypercube family. There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC7 symmetry, and 32 are unique: Klitzing, (x3o3o *b3o3o3o3x
Hexic_7-cubes
Greek-French composer, architect and engineer (1922–2001)
Xenakis's UPIC system; and the massive multimedia performances Xenakis called polytopes, that were a summa of his interests and skills. Among the numerous theoretical
Iannis_Xenakis
Four-dimensional geometrical object
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the
Runcinated_5-cell
Solid with eight equal triangular faces
characteristic angles 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the golden ratio constant ≈ 1.618, for which Coxeter uses 𝝉 (tau)
Regular_octahedron
On the number of common zeros of Laurent polynomials
polynomial equations f 1 = ⋯ = f n = 0 {\displaystyle f_{1}=\cdots =f_{n}=0} is equal to the mixed volume of the Newton polytopes of such polynomials, assuming
Bernstein–Kushnirenko_theorem
is a convex uniform 5-polytope, being a rectification of the regular 5-cube. There are 5 degrees of rectifications of a 5-polytope, the zeroth here being
Rectified_5-cubes
Uniform polytope in 8 dimensional geometry
rectified 142. These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets, defined by
2_41_polytope
Polytope associated with combinatorial problems
In geometry, the cyclohedron is a d-dimensional polytope where d can be any non-negative integer. It was first introduced as a combinatorial object by
Cyclohedron
Kepler–Poinsot polyhedron with 20 faces
(3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR 0676126. (1st Edn University of Toronto (1938)) H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover
Great_icosahedron
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
24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra
Rectified_24-cell
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
Polyhedron formed by joining mirroring pyramids base-to-base
\end{aligned}}} A generalized n-dimensional "bipyramid" is any n-polytope constructed from an (n − 1)-polytope base lying in a hyperplane, with every base vertex connected
Bipyramid
Dutch mathematician (1881–1943)
mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher. Elte's father Hartog Elte was headmaster
Emanuel_Lodewijk_Elte
Infinite regular skew polyhedron
MR 1965665 McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF). Discrete and Computational Geometry. 32: 1–35. doi:10.1007/s00454-004-0848-5. Dress
Regular_skew_apeirohedron
Compound polyhedron
five tetrahedra is related to the regular 5-cell, the 4-simplex regular 4-polytope, which is also composed of 5 regular tetrahedra. In the 5-cell the tetrahedra
Compound_of_five_tetrahedra
In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube. There are 6 unique cantellation
Cantellated_5-cubes
Polytope
a polytope compound composed of a regular 5-cube and dual regular 5-orthoplex. A compound polytope is a figure that is composed of several polytopes sharing
Compound of 5-cube and 5-orthoplex
Compound_of_5-cube_and_5-orthoplex
strings (1964) Akrata, for wind orchestra (1964–65) Terretektorh (1966) Polytope [de Montréal], for 4 orchestras (1967) Nomos gamma (1967–8) Synaphaï, for
List of compositions by Iannis Xenakis
List_of_compositions_by_Iannis_Xenakis
Number of windings of a polytope around its center of symmetry
ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets
Density_(polytope)
Natural number
The triangular prism is the root polytope in the k21 family of polytopes, which is the simplest semiregular polytope, with k31 rooted in the analogous
72_(number)
2007 (1): 004. arXiv:hep-th/0610199. Bibcode:2007JCAP...01..004P. doi:10.1088/1475-7516/2007/01/004. S2CID 17403084. "Infinity Scrapers". www.polytope.net
Orders_of_magnitude_(numbers)
Generalization of the Rubik's Cube in n-dimensions
higher-dimension figures meet. n-Polytope. A n-dimensional figure continuing as above. A specific geometric shape may replace polytope where this is appropriate
N-dimensional sequential move puzzle
N-dimensional_sequential_move_puzzle
In 7-dimensional geometry, there are 128 uniform polytopes with B7 symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128
B7_polytope
Isogonal polyhedron with regular faces
polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying
Uniform_polyhedron
Canadian geometer (1907–2003)
author of 12 books, including The Fifty-Nine Icosahedra (1938) and Regular Polytopes (1947). Many concepts in geometry and group theory are named after him
H.S.M._Coxeter
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
Concept in geometry
ISBN 978-0-8218-3722-1 Coxeter, H.S.M., Regular Polytopes, §12.6 The number of reflections, equation 12.61 Regular polytopes, p. 233 George Lusztig
Coxeter_element
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. There
Runcinated_5-cubes
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