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discrete geometry, a polytope is projectively unique (or projectively stable) if it has a unique convex realization up to projective transformations. The
Projectively_unique_polytope
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
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
Convex polytope whose vertices all have integer Cartesian coordinates
combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex
Integral_polytope
Geometric space with seven dimensions
Coxeter-Dynkin diagram. The 7-demicube is a unique polytope from the D7 family, and 321, 231, and 132 polytopes from the E7 family. The 6-sphere or hypersphere
Seven-dimensional_space
Geometric space with eight dimensions
Coxeter-Dynkin diagram. The 8-demicube is a unique polytope from the D8 family, and 421, 241, and 142 polytopes from the E8 family. The 7-sphere or hypersphere
Eight-dimensional_space
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
each other. Projectively unique polytopes, which have a unique realization up to projective transformation, and more generally, polytopes with a specified
Gale_diagram
Irrational system of points and lines
João; Macchia, Antonio; Thomas, Rekha R.; Wiebe, Amy (2020), "Projectively unique polytopes and toric slack ideals", Journal of Pure and Applied Algebra
Perles_configuration
duoprism or triangular duoprism is a four-dimensional convex polytope. The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons
3-3_duoprism
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
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
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
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 one
D8_polytope
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
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
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
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
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
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
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
the directions and measures of its facets. The theorem that every polytope is uniquely determined up to translation by this information was proven by Hermann
Minkowski problem for polytopes
Minkowski_problem_for_polytopes
Set of points described by relative position in space
described by their relative positions. They can be described by their use in polytopes. For example, a square vertex arrangement is understood to mean four points
Vertex_arrangement
Completion of the usual space with "points at infinity"
S of points is projectively independent if its span is not the span of any proper subset of S. If S is a spanning set of a projective space P, then there
Projective_space
Class of eight-dimensional polytopes
8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex
Stericated_8-simplexes
Geometric space with six dimensions
Coxeter–Dynkin diagram. The 6-demicube is a unique polytope from the D6 family, and 221 and 122 polytopes from the E6 family. The 5-sphere, or hypersphere
Six-dimensional_space
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
German mathematician
of Perles and Shephard (going back to Legendre and Steinitz) on projectively unique polyhedra." In joint work with June Huh and Eric Katz, he resolved
Karim_Adiprasito
bound matrix (DBM) is a data structure used to represent some convex polytopes called zones. This structure can be used to efficiently implement some
Difference_bound_matrix
Uniform 4-polytope
In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms
Tetrahedral_prism
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
Manifold or algebraic variety of dimension n in a space of dimension n+1
{\displaystyle x^{2}+y^{2}-1=0} in the affine space of dimension three has a unique singular point, which is at infinity, in the direction x = 0, y = 0. Affine
Hypersurface
Geometric structure used in certain particle interactions
algebraic geometry analogous to a convex polytope, that generalizes the idea of a simplex in projective space. A polytope is the n-dimensional analogue of a
Amplituhedron
Solid with eight equal triangular faces
segments. More generally, every cross-polytope and its dual, hypercube, in any higher-dimensional space are Hanner polytope. The polyhedral compounds, in which
Regular_octahedron
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
the unique distance-regular graph with intersection array {6,5,2;1,1,3}, discovered by Manley Perkel (1979). 11-cell – abstract regular polytope with
57-cell
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
Four-dimensional shape
https://www.bendwavy.org/klitzing/incmats/tedpy.htm Klitzing, Richard, "Johnson solids, Blind polytopes, and CRFs", Polytopes, retrieved 2022-11-14 v t e
Tetrahedral_bipyramid
Type of geometry
simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and
Projective_geometry
rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex. There are unique 6 degrees of rectifications, the zeroth
Rectified_6-orthoplexes
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
Geometric model of the planar projection of the physical universe
{\displaystyle \mathbb {R} ^{3}} . In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below: The Schläfli
Euclidean_plane
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
Space formed by the ''n''-tuples of real numbers
1\\\vdots \\|x_{n}|\leq 1\end{matrix}}} for [−1,1]. Each vertex of the cross-polytope has, for some k, the xk coordinate equal to ±1 and all other coordinates
Real_coordinate_space
between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their
Dehn–Sommerville_equations
Geometric model of the physical space
open subset of 3-D space. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler–Poinsot
Three-dimensional_space
Fundamental space of geometry
both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean
Euclidean_space
In mathematics, dimension of a ring
local ring is an example of such a ring. A Noetherian integral domain is a unique factorization domain if and only if every height 1 prime ideal is principal
Krull_dimension
Invariant measure of fractal dimension
object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead
Hausdorff_dimension
Algebraic variety containing an algebraic torus
polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces
Toric_variety
Five dimensional space-filling tessellation
Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings) (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III
Omnitruncated 5-simplex honeycomb
Omnitruncated_5-simplex_honeycomb
Number of vectors in any basis of the vector space
have equal cardinality; as a result, the dimension of a vector space is uniquely defined. V {\displaystyle V} is said to be finite-dimensional if the dimension
Dimension_(vector_space)
Euclidean geometry without distance and angles
numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two
Affine_geometry
algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions
H-vector
Cycle graph with all opposite nodes linked
configurations within these problems can be used to define facets of the polytope describing a linear programming relaxation of the problem; these facets
Möbius_ladder
Compact non-orientable two-dimensional manifold
polytopes – hemi-cube, hemi-dodecahedron, and hemi-icosahedron – can be constructed as regular figures in the projective plane; see also projective polyhedra
Real_projective_plane
Straight figure with zero width and depth
lines and points. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one
Line_(geometry)
Line constructed from a triangle
simplicial polytope is a polytope whose facets are all simplices (plural of simplex). For example, every polygon is a simplicial polytope. The Euler line
Euler_line
Concept in euclidean geometry
order-3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5-space. The tesseract can make a regular tessellation of
Tesseractic_honeycomb
Branch of mathematics
approach leads to a theory of non-commutative projective geometry. A non-commutative smooth projective curve turns out to be a smooth commutative curve
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
General concept and operation in mathematics
generally any convex polytope, corresponds to a dual polyhedron or dual polytope, with an i-dimensional feature of an n-dimensional polytope corresponding to
Duality_(mathematics)
Tiling of euclidean or hyperbolic space of three or more dimensions
non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space
Honeycomb_(geometry)
Algebro-geometric stability condition
} On a Riemann surface such a connection is projectively flat, and its holonomy gives rise to a projective unitary representation of the fundamental group
K-stability
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
Polygonal chain whose vertices are not all coplanar
Regular complex polytopes, p. 6 Abstract Regular Polytopes, p.217 McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge
Skew_polygon
Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings) (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III
7-simplex_honeycomb
Prism with a 3-sided base
Schönhardt polyhedron. It has a relationship with the honeycombs and polytopes. It can be found in many real-life applications as in architecture and
Triangular_prism
Generalized sphere of dimension n (mathematics)
^{n+1}:\left\|x\right\|_{1}=1\right\}.} In general, it takes the shape of a cross-polytope. The octahedral 1 {\displaystyle 1} -sphere is a square (without its
N-sphere
constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex. Its vertex arrangement is the E6 lattice, and the
2_22_honeycomb
Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings) (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III
8-simplex_honeycomb
In mathematics, a module that has a basis
second half of the definition is that the coefficients in the first half are unique for each element of M. If R {\displaystyle R} has invariant basis number
Free_module
Concept in geometry
higher-dimensional polytopes and root systems – the vertices and edges of the polytope, or roots (and some edges connecting these) are orthogonally projected onto the
Coxeter_element
Graph that encodes local operations in mathematics
geometric graphs. Among notable flip graphs, one finds the 1-skeleton of polytopes such as associahedra or cyclohedra. A prototypical flip graph is that
Flip_graph
Graph in which every two vertices are adjacent
of a torus, has the complete graph K7 as its skeleton. Every neighborly polytope in four or more dimensions also has a complete skeleton. K1 through K4
Complete_graph
Infinitely detailed mathematical structure
analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4. This number is called the fractal dimension
Fractal
Branch of mathematics
Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied
Geometry
Property of a mathematical space
Volume 4 dimensions Spacetime Fourth spatial dimension Convex regular 4-polytope Quaternion 4-manifold Polychoron Rotations in 4-dimensional Euclidean space
Dimension
Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings) (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III
5-simplex_honeycomb
Mathematical space with two coordinates
a chalkboard. On the Euclidean plane, any two points can be joined by a unique straight line along which the distance can be measured. The space is flat
Two-dimensional_space
Coordinate system that is defined by points instead of vectors
can be used to define unique barycentric coordinates. More abstractly, generalized barycentric coordinates express a convex polytope with n {\displaystyle
Barycentric_coordinate_system
Smallest convex set containing a given set
Krein–Milman theorem) every convex polytope is the convex hull of its vertices. It is the unique convex polytope whose vertices belong to S {\displaystyle
Convex_hull
Natural number
S2CID 122060727. Zbl 1004.20003. Hartley, Michael I.; Hulpke, Alexander (2010). "Polytopes Derived from Sporadic Simple Groups". Contributions to Discrete Mathematics
27_(number)
torus, with orbit space an n {\displaystyle n} -dimensional simple convex polytope. Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz
Quasitoric_manifold
Natural number
dimension for non-simplex hypercompact Vinberg polytopes of rank n + 4 mirrors, where there is one unique figure with eleven facets. On the other hand,
7
Geometric figure
Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings) (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III
5-cell_honeycomb
Mathematical set with some added structure
transformations; they all are projectively equivalent figures. The relation between the two geometries, Euclidean and projective, shows that mathematical objects
Space_(mathematics)
Relationship between two lines that meet at a right angle
be a point and m a line. If B is the point of intersection of m and the unique line through A that is perpendicular to m, then B is called the foot of
Perpendicular
Set on which a group acts freely and transitively
acting as the additive group of translations. The flags of any regular polytope form a torsor for its symmetry group. Given a vector space V we can take
Principal_homogeneous_space
Geometric object used to describe rotation in any number of dimensions
intersection. In more than three dimensions planes of rotation are not always unique. For example the negative of the identity matrix in four dimensions (the
Plane_of_rotation
Branch of mathematics
ideal defining the variety. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties. The only regular functions
Algebraic_geometry
Archimedean solid with 26 faces
octahedron – truncated tetratetrahedron Snub cube "Great rhombicuboctahedron". Polytope Wiki. 2025-10-08. Retrieved 2026-09-10. Wenninger, Magnus (1974), Polyhedron
Truncated_cuboctahedron
Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings) (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III
6-simplex_honeycomb
Group of rotations in 3 dimensions
^{n}} expressed in its standard basis. Coxeter, H. S. M. (1973). Regular polytopes (Third ed.). New York: Dover Publications, Inc. p. 53. ISBN 0-486-61480-8
3D_rotation_group
Swiss geometer (1814–1895)
the six regular convex 4-polytopes, the dimensional analogues of the five Platonic solids. Among these he discovered the unique 24-cell, 600-cell and 120-cell
Ludwig_Schläfli
Field of mathematics which studies incidence structures
produces the projective plane of order three, a (134) configuration. Conversely, starting with the projective plane of order three (it is unique) and removing
Incidence_geometry
Perimeter of a circle or ellipse
B} . Therefore the circumference can be defined, and calculated, as the unique number x {\displaystyle x} such that a < x < b {\displaystyle a<x<b} for
Circumference
degree polynomials on general compact polytopes, we have Handelman's theorem: If K {\displaystyle K} is a compact polytope in Euclidean d {\displaystyle d}
Positive_polynomial
Rational numbers with root 5 added
0\right),\left(\pm \varphi ,0,\pm 1\right).} The 600-cell is a regular 4-polytope with 120 vertices, 720 edges, 1200 triangular faces, and 600 tetrahedral
Golden_field
Notation for a polyhedron's vertex figure
{\frac {4}{2-b(1-2/a)}}} Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However, not all configurations are possible
Vertex_configuration
Mathematical set closed under positive linear combinations
Theorem for polytopes which shows that every polytope is a polyhedron and every bounded polyhedron is a polytope. The two representations of a polyhedral
Convex_cone
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