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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
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
Framework for computer program optimization
The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large to
Polytope_model
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
In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial that can be used in the asymptotic analysis of those
Newton_polytope
Relation of an integral polytope's volume to how many integer points it encloses
mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer
Ehrhart_polynomial
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 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
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
Classification of symplectic toric manifolds
It is therefore a convex polytope, also called the moment polytope. A Delzant polytope is a special kind of convex polytope in R n {\displaystyle \mathbb
Delzant's_theorem
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 polytope a special kind of configuration.[citation needed] Other configurations in geometry are something different. These polytope configurations
Configuration_(polytope)
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)
Edge-joined polygons which fold into a polyhedron
shortest path between two points on a cuboid. A net of a 4-polytope, a four-dimensional polytope, is composed of polyhedral cells that are connected by their
Net_(polyhedron)
steric 5-cube, steric 5-demicube or sterihalf 5-cube, is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half
Steric_5-cubes
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
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
Generalization of a rectangle for higher dimensions
Regular Polytopes (3rd ed.). New York: Dover. pp. 122–123. ISBN 0-486-61480-8. Weisstein, Eric W. "Orthotope". MathWorld. Foran, James (1991-01-07). Fundamentals
Hyperrectangle
Polytope made by turning a polytope's facets into pyramids
Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing each facet of P with a pyramid
Kleetope
different blockchains. Hyperbridge was launched on Polkadot mainnet in 2024 by Polytope Labs. The protocol is designed with a decentralised verification system
Hyperbridge
Algorithm for linear programming
neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function
Simplex_algorithm
Geometric problems involving the partition of a figure
dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of
Dissection_problem
Mathematical subject
convex polytopes. The study of regular polytopes, Archimedean solids, and kissing numbers is also a part of geometric combinatorics. Special polytopes are
Geometric_combinatorics
Plane tiling corresponding to a polyhedron
3. In the context of abstract polytopes, one instead refers to "locally projective polytopes" – see Abstract polytope: Local topology. For example, the
Projective_polyhedron
matching polytope of G {\displaystyle G} is a convex polytope that represents all possible fractional matchings of G {\displaystyle G} . It is a polytope in
Fractional_matching
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
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
Software for the algorithmic treatment of convex polyhedra
primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is by now also capable of dealing with simplicial complexes
Polymake
Minkowsi sum of line segments
A zonotope is a convex polytope that can be described as the Minkowski sum of a finite set of line segments in R d {\displaystyle \mathbb {R} ^{d}} or
Zonotope
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
E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York
3_31_honeycomb
Geometric inequality or concentration inequality in mathematics and probability theory
the above two conditions is a closed convex polytope defined by linear inequalities. This is the BL polytope. Note that while there are infinitely many
Brascamp–Lieb_inequality
Branch of discrete mathematics
convex polytope can have. Metric properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special
Combinatorics
Natural number
Wiley & Sons. ISBN 978-0-471-50458-0. H. S. M. Coxeter (1973). Regular Polytopes (3rd ed.). New York: Dover Publications, Inc. pp. 1–368. ISBN 978-0-486-61480-9
5
Matching which covers every node of the graph
and computational complexity theory. The perfect matching polytope of a graph is a polytope in R|E| in which each corner is an incidence vector of a perfect
Perfect_matching
contains all permutation matrices and its convex hull is the Birkhoff polytope of all doubly stochastic matrices for n ≥ 3 {\displaystyle n\geq 3} this
Unistochastic_matrix
Natural number
space, 7 is the highest dimension for non-simplex hypercompact Vinberg polytopes of rank n + 4 mirrors, where there is one unique figure with eleven facets
7
Mathematical object
In mathematics, a random polytope is a structure commonly used in convex analysis and the analysis of linear programs in d-dimensional Euclidean space
Random_polytope
Polyhedron with 12 faces
Elsevier BV: 155–161. Bibcode:2001JCrGr.225..155C. doi:10.1016/s0022-0248(01)00827-2. ISSN 0022-0248. Dutch, Steve. The 48 Special Crystal Forms Archived
Dodecahedron
Geometric space with four dimensions
both synthetic and algebraic methods. He discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean
Four-dimensional_space
Archived 2016-01-30 at the Wayback Machine, Edward Pervin Klitzing, (o3o3o3x3o3o3o *d3o - lanquoh). H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition
1_33_honeycomb
German geometer (1860–1934)
Elemente der vierdimensionalen Geometrie mit besonderer Berücksichtigung der Polytope, Jahresbericht des Vereins für Naturkunde zu Zwickau 1893 Vielecke und
Max_Brückner
Method for linear optimization
It gets stuck at a basic feasible solution (a corner of the feasible polytope) and changes bases in a cyclic way without decreasing the minimization
Bland's_rule
lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial. Formulas for
Integer points in convex polyhedra
Integer_points_in_convex_polyhedra
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
Topological invariant in mathematics
of hypercubes and simplices of dimensions 1 to 4. In the 1D cases, the edges are not counted because they are not part of the polytopes’ boundaries.
Euler_characteristic
Triangular array of the binomial coefficients
Scott Macdonald (1973-01-01). "Chapter VII: ordinary polytopes in higher space, 7.2: Pyramids, dipyramids and prisms". Regular Polytopes (3rd ed.). Courier
Pascal's_triangle
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
Four-dimensional number system
geometry Quaternionic matrix – Concept in linear algebra Quaternionic polytope – Concept in geometry Quaternionic projective space – Concept in mathematics
Quaternion
arXiv:hep-th/0610199. Bibcode:2007JCAP...01..004P. doi:10.1088/1475-7516/2007/01/004. S2CID 17403084. "Infinity Scrapers". www.polytope.net. Retrieved 2025-12-07. "Forcal
Orders_of_magnitude_(numbers)
Maximal independent set of the matroid
of J {\displaystyle J} that equals x {\displaystyle x} . Matroid polytope - a polytope in Rn (where n is the number of elements in the matroid), whose
Basis_of_a_matroid
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
Boxes and Polytopes". Mathematika. 63 (3): 1091–1113. arXiv:1701.05532. doi:10.1112/S0025579317000250. ISSN 0025-5793. Alexander, R. (1990-06-01). "Geometric
Geometric_discrepancy
honeycomb is a square tiling, {4,4}. It is a part of a series of regular polytopes and honeycombs with {p,4,4} Schläfli symbol, and square tiling vertex
Order-4-4 pentagonal honeycomb
Order-4-4_pentagonal_honeycomb
parallelohedron? Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram? Ropelength
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Topological space that locally resembles Euclidean space
manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners),
Manifold
Regular space-filling tessellation with Schläfli symbol (7,3,7)
honeycombs in hyperbolic space List of regular polytopes Infinite-order dodecahedral honeycomb Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8
Order-3-7 heptagonal honeycomb
Order-3-7_heptagonal_honeycomb
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
Theorem on triangulation graph colorings
extended the theorem from polytopes to polytopal bodies, which need not be convex or simply-connected. In particular, if P is a polytope, then the set of its
Sperner's_lemma
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)
Type of metric geometry
symmetric, under the taxicab distance, the shape of a sphere is a cross-polytope, the n-dimensional generalization of a regular octahedron, whose points
Taxicab_geometry
Casino in Montreal, Quebec, Canada
Jean-Drapeau Polytope de Montréal "25th Anniversary of the Montreal Casino: Come celebrate in style!". TASTET. 2019-01-07. Retrieved 2019-12-01. ICI.Radio-Canada
Montreal_Casino
with these prefixes. Platonic solid Dice List of polygons, polyhedra and polytopes Circle Ellipse Shape In Greek usage, τετράγωνον (romanised: tetragōnon)
List_of_polygons
this honeycomb is a cube, {4,3}. It is a part of a series of regular polytopes and honeycombs with {p,4,3} Schläfli symbol, and tetrahedral vertex figures:
Order-4-3 pentagonal honeycomb
Order-4-3_pentagonal_honeycomb
honeycomb is an icosahedron, {3,5}. It is a part of a series of regular polytopes and honeycombs with {p,3,5} Schläfli symbol, and icosahedral vertex figures
Order-3-5 heptagonal honeycomb
Order-3-5_heptagonal_honeycomb
honeycomb is a hexagonal tiling, {6,3}. It is a part of a series of regular polytopes and honeycombs with {p,6,3} Schläfli symbol, and dodecahedral vertex figures:
Order-6-3_square_honeycomb
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
Upper bound on the volume of a convex body containing one lattice point
Paffenholz confirmed the conjecture for several important classes of rational polytopes, establishing that equality holds if and only if K {\displaystyle K} is
Ehrhart's_volume_conjecture
Study of mathematical algorithms for optimization problems
equalities and inequalities. Such a constraint set is called a polyhedron or a polytope if it is bounded. Second-order cone programming (SOCP) is a convex program
Mathematical_optimization
Geometrical figure in a Euclidean space
on polytopes". Trans. Roy. Soc. Canada. Sect. III. (3). 34: 29–34. MR 0002869.. "Archived copy" (PDF). Archived from the original (PDF) on 2014-01-12
Eutactic_star
Tesselation in regular space
of regular polytopes Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs
Order-5_octahedral_honeycomb
honeycombs in hyperbolic space List of regular polytopes Infinite-order dodecahedral honeycomb Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8
Order-3-7_hexagonal_honeycomb
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
NP-hard problem in combinatorial optimization
Mark (2017). "Short combinatorial proof that the DFJ polytope is contained in the MTZ polytope for the Asymmetric Traveling Salesman Problem". Operations
Travelling_salesman_problem
distinguishes between two families of such algorithms: the first family, called polytope norm methods, construct the extremal norm by computing long trajectories
Joint_spectral_radius
Mathematician
1007/BF02834753 Birth year from Library of Congress catalog entry, retrieved 2018-12-01. Univ. of Oklahoma faculty listing, retrieved 2014-12-21. Marilyn Breen at
Marilyn_Breen
Prime number of the form 2^n – 1
geometry, the number of polytopes that are part of the family of polytopes formed by a truncation operation of a base regular polytope and its dual (excluding
Mersenne_prime
triangular tiling vertex arrangement. It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells, {5,3,p}. It a part of a sequence
Order-7 dodecahedral honeycomb
Order-7_dodecahedral_honeycomb
1971 national event in Iran
editions of the Guinness Book of World Records. A son et lumière show, the Polytope of Persepolis designed by Iannis Xenakis and accompanied by the specially-commissioned
2,500-year celebration of the Persian Empire
2,500-year_celebration_of_the_Persian_Empire
honeycomb is an octahedron, {3,4}. It is a part of a series of regular polytopes and honeycombs with {p,3,4} Schläfli symbol, and octahedral vertex figures:
Order-3-4 heptagonal honeycomb
Order-3-4_heptagonal_honeycomb
Natural number
polytopes in the form of complex n {\displaystyle n} -orthoplexes. There are also twelve paracompact hyperbolic Coxeter groups of uniform polytopes in
12_(number)
face are seen from the same side). In the 1948 first edition of Regular Polytopes, H. S. M. Coxeter describes the stellation process as the reciprocal action
List of polyhedral stellations
List_of_polyhedral_stellations
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
Mexican mathematician
p. 13. Retrieved 2017-01-24. Isabel A. Hubard (2007). From Geometry to Groups and Back: The Study of Highly Symmetric Polytopes. York University (Canada)
Isabel_Hubard_Escalera
in dimension 4 not projectively equivalent to the vertices of a convex polytope", Combinatorial geometries (Luminy, 1999), European Journal of Combinatorics
Convex_position
Two raised to an integer power
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 ( n x ) . {\displaystyle
Power_of_two
of regular polytopes Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs
Order-4_icosahedral_honeycomb
seen as alternately colored cells. It is a part of a series of regular polytopes and honeycombs with {p,3,6} Schläfli symbol, and triangular tiling vertex
Order-3-6 heptagonal honeycomb
Order-3-6_heptagonal_honeycomb
Regular paracompact honeycomb
Table III Coxeter, Regular Polytopes, 3rd ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
Hexagonal_tiling_honeycomb
Area of discrete mathematics
theory studies planar graphs, relationship to higher-dimensional convex polytopes, intersection of geometrical shaped sets, and other geometries' subareas
Graph_theory
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
Geometric shape
Science & Business Media. pp. 74–75. ISBN 9780412990410. Grünbaum, Convex Polytopes, second edition, p. 23. Weisstein, Eric W. "Cone". MathWorld. Bartol,
Cone
Formula for the "volume" of an n-simplex
an n {\displaystyle n} -simplex is an n {\displaystyle n} -dimensional polytope and the convex hull of n + 1 {\displaystyle n+1} points which do not lie
Cayley–Menger_determinant
Catalan solid with 12 faces
1007/s00283-010-9138-7, hdl:1773/15593, MR 2747698 Coxeter, Harold (1973), Regular polytopes, Dover Publications Economic Mineralogy: A Practical Guide to the Study
Rhombic_dodecahedron
78-dimensional exceptional simple Lie group
maximal subgroups of E6 up to dimension 78 are shown to the right. The E6 polytope is the convex hull of the roots of E6. It therefore exists in 6 dimensions;
E6_(mathematics)
Natural number
wolfram.com. Retrieved 2022-07-02. Coxeter, H.S.M. (1991), Regular Complex Polytopes, Cambridge University Press, p. 140, ISBN 0-521-39490-2 Sharp, Damian
22_(number)
honeycombs in hyperbolic space List of regular polytopes Infinite-order dodecahedral honeycomb Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8
Order-4-5 pentagonal honeycomb
Order-4-5_pentagonal_honeycomb
Black-box description of a convex set
K=\{x|Ax\leq b\}} . Such a set is called a convex polytope. A strong separation oracle for a convex polytope can be implemented, but its run-time depends on
Separation_oracle
Length in a vector space
vectors whose 1-norm is a given constant forms the surface of a cross polytope, which has dimension equal to the dimension of the vector space minus 1
Norm_(mathematics)
Edge-joined polygon with multiple principle shapes
unfoldings. 3D Simplicial polytope Demaine, Erik D.; Demaine, Martin L.; Itoh, Jin-ichi; Lubiw, Anna; Nara, Chie; OʼRourke, Joseph (2013-10-01). "Refold rigidity
Common_net
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