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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
Geometric object with flat sides
is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist
Polytope
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
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 polytope dual to the hypercube in any number of dimensions
2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets
Cross-polytope
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
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
Polyhedron with four faces
tetrahedron. Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme. There is a 3-orthoscheme, which is the "characteristic
Tetrahedron
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
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
In discrete geometry, a polytope is projectively unique (or projectively stable) if it has a unique convex realization up to projective transformations
Projectively_unique_polytope
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 vertices respectively
E6_polytope
8-dimensional hypercube
Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and
8-cube
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
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
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
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
Polytope
The Birkhoff polytope B n {\displaystyle B_{n}} is the convex polytope in R n 2 {\displaystyle \mathbb {R} ^{n^{2}}} whose points are the doubly stochastic
Birkhoff_polytope
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 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
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
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
Regular 5-polytope
Schläfli symbol { 3 3 , 3 3 } {\displaystyle \left\{3{\begin{array}{l}3,3\\3\end{array}}\right\}} or {3,32,1}. It exists in the k21 polytope family as 121
5-demicube
Multi-dimensional generalization of triangle
triangle, a 3-dimensional simplex is a tetrahedron, and a 4-dimensional simplex is a 5-cell. Specifically, a k-simplex is a k-dimensional polytope that is
Simplex
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
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
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
Type of 7-polytope
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell
7-simplex
7-dimensional hypercube
Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and
7-cube
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
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
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 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
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
Type of convex polytope
A 0/1-polytope is a convex polytope generated by the convex hull of a subset of d coordinates value 0 or 1, {0,1}d. The full domain is the unit hypercube
0/1-polytope
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
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
Convex polytope of parenthesizations
In mathematics, an associahedron Kn is an (n − 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening
Associahedron
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
Type of geometrical object
geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A
Uniform_10-polytope
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
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 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
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
truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation. Vertices of the
Truncated_6-simplexes
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
Polytope or tiling whose vertices are identical
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
Isogonal_figure
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
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
Uniform 6-polytope
six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10
Pentellated_6-simplexes
Convex regular 8-polytope
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces
8-simplex
Removal of alternate vertices
is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed
Alternation_(geometry)
Convex regular 10-polytope
In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces
10-simplex
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
Generalisation of dice with identical faces
a tessellation of dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive if all
Isohedral_figure
Shape with three equal sides
self-replicate itself. Equilateral triangles may also form a three-dimensional polytope, called a polyhedron. A polyhedron whose faces are all equilateral triangles
Equilateral_triangle
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)
Uniform 7-polytope
polytope. Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol { 3 3 , 3 , 3
7-demicube
Uniform 6-polytope
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 7 5-simplex 5-faces
6-simplex
Distance-regular graph with 56 vertices
vertices and valency 27. It is the 1-skeleton of the 7-dimensional 321 polytope. The Gosset graph can be explicitly constructed as follows: the 56 vertices
Gosset_graph
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
Space-filling tessellation
the Wayback Machine (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
Bitruncated_cubic_honeycomb
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
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
Rectified_6-simplexes
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
Quasiregular space-filling tesselation
non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
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
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
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
Polyhedron associated with another by swapping vertices for faces
of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the regular and uniform polytopes, the
Dual_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
Solid with eight equal triangular faces
Ziegler, Günter M. (1995). "Chapter 4: Steinitz' Theorem for 3-Polytopes". Lectures on Polytopes. Graduate Texts in Mathematics. Vol. 152. Springer-Verlag
Regular_octahedron
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
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
seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex. There are unique 7 degrees
Rectified_7-orthoplexes
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
regular polytope a special kind of configuration.[citation needed] Other configurations in geometry are something different. These polytope configurations
Configuration_(polytope)
Shape with four equal sides and angles
truncated square is an octagon. The square belongs to a family of regular polytopes that includes the cube in three dimensions and the hypercubes in higher
Square
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
The icosahedral pyramid is a four-dimensional convex polytope, bounded by one icosahedron as its base and by 20 triangular pyramid cells which meet at
Icosahedral_pyramid
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
Theorem about projections of coadjoint orbits of a connected compact Lie group
self-adjoint matrices with given eigenvalues Λ = (λ1, ..., λn) is the convex polytope with vertices all permutations of the coordinates of Λ. Let K be a connected
Kostant's_convexity_theorem
Plane tiling corresponding to a polyhedron
vertex figure 3.3.3. In the context of abstract polytopes, one instead refers to "locally projective polytopes" – see Abstract polytope: Local topology
Projective_polyhedron
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)
Uniform polytope in 8 dimensional geometry
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 241, describing its
2_41_polytope
Solid with six equal square faces
quadrilateral faces are squares. It is a three-dimensional hypercube, a family of polytopes that also includes the two-dimensional square and four-dimensional tesseract
Cube
Polyhedron with 12 faces
{\sqrt {5}}{2}}\cdot {\text{Long side}}} Width = 4 3 ⋅ Long side {\displaystyle {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}} Short sides = 7 12 ⋅
Dodecahedron
doi:10.1088/1475-7516/2007/01/004. S2CID 17403084. "Infinity Scrapers". www.polytope.net. Retrieved 2025-12-07. "Forcal - Aarex's Large Numbers". sites.google
Orders_of_magnitude_(numbers)
Graph theory model
stacked polytopes, polytopes formed by starting from a simplex and then repeatedly gluing simplices onto the faces of the polytope, are k-trees when k ≥ 3. This
K-tree
Pictorial representation of symmetry
(called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group. A class of closely related
Coxeter–Dynkin_diagram
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
Type of uniform tessellation
faces, 80 3{3}3{3}3 cells and 1 3{3}3{3}3{3}3 Witting polytope cells. The 521 is seventh in a dimensional series of semiregular polytopes, identified
5_21_honeycomb
Natural number
Vinberg polytopes of rank n + 4 mirrors, where there is one unique figure with eleven facets. On the other hand, such figures with rank n + 3 mirrors
7
Closed volume that completely contains the union of a set of objects
is a convex polytope containing the object (in 2-D a polygon; in 3-D a polyhedron). A 2-D rectangle is a special case of a 2-DOP, and a 3-D box is a special
Bounding_volume
Only regular space-filling tessellation of the cube
non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical
Cubic_honeycomb
Concept in geometry
5-polytope, {p, q, r, s}, with one directed Petrie polygon marked, is represented by the composite of 5 reflections. In dimensions 6 to 8 there are 3 exceptional
Coxeter_element
Shape with five sides
⋅ t tan ( 3 π 10 ) 2 = 5 t 2 tan ( 3 π 10 ) 4 {\displaystyle A={\frac {1}{2}}\cdot 5t\cdot {\frac {t\tan {\mathord {\left({\frac {3\pi }{10}}\right)}}}{2}}={\frac
Pentagon
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
Polyhedron with 8 triangles and 6 squares
equilateral polytopes are those that can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing
Cuboctahedron
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