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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 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
5-dimensional hypercube
holes. This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex. Coxeter, Regular Polytopes, sec 1.8 Configurations
5-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
Polyhedron with four 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)
Tetrahedron
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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)
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
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
Solid with eight equal triangular faces
Spirals, Fractals, and More!. Tuttle. p. 42. ISBN 978-1-4629-2398-4. Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover Publications. Chapter V:
Regular_octahedron
Concept in geometry
runcic 5-cube, runcic 5-demicube or runcihalf 5-cube, is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the
Runcic_5-cubes
Branch of geometry that studies combinatorial properties and constructive methods
discrete geometry. A polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions
Discrete_geometry
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 with eight triangular faces
the three-dimensional case of an infinite family of regular polytopes, the cross polytopes. Although it does not tile space by itself, it can tile space
Octahedron
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
Newton polytopes of toric varieties, several polytopes appearing in representation theory (such as the Gelfand–Zetlin polytopes and the string polytopes of
Newton–Okounkov_body
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
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
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
Object in 4-dimensional geometry
In 4-dimensional geometry, the octahedral cupola is a 4-polytope bounded by one octahedron and a parallel rhombicuboctahedron, connected by 20 triangular
Octahedral_cupola
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
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
4D geometry item
four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 120-cell
Cantellated_120-cell
Theorem about projections of coadjoint orbits of a connected compact Lie group
segment joining Xn+1 and its reflection in the root α. Thus Xn lies in the Weyl group polytope defined by Xn+1. These convex polytopes are thus increasing
Kostant's_convexity_theorem
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
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
Any of 4 regular star polyhedra
Regular polytope Regular polyhedron List of regular polytopes Uniform polyhedron Uniform star polyhedron Polyhedral compound Regular star 4-polytope – the
Kepler–Poinsot_polyhedron
± 1 , ± 1 , ± 1 , ± 1 ) {\displaystyle (0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)} E. L. Elte identified it in 1912 as a semiregular polytope, identifying
Rectified_5-cubes
Linear stacking of regular tetrahedra that form helices
4. This helix exists as finite ring of 30 pyramids in a 4-dimensional polytope. And equilateral pentagonal pyramids can be chained with 3 vertex configurations
Boerdijk–Coxeter_helix
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
Cycle graph with all opposite nodes linked
M.; Reinelt, G. (1985a). "On the acyclic subgraph polytope". Mathematical Programming. 33: 28–42. doi:10.1007/BF01582009. MR 0809747. S2CID 206798683
Möbius_ladder
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
Uniform 4-polytope bounded by 320 cells
antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300
Grand_antiprism
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
Award for advancements in discrete mathematics
characterization of the weakly bipartite graphs (graphs whose bipartite subgraph polytope is 0-1). Satoru Iwata, Lisa Fleischer, Satoru Fujishige, and Alexander Schrijver
Fulkerson_Prize
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
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
Irrational system of points and lines
eight-dimensional convex polytope that cannot be given rational number coordinates and that has the fewest vertices (twelve) of any known irrational polytope. The Perles
Perles_configuration
Type of square matrix
convex polytope known as the Birkhoff polytope B n {\displaystyle B_{n}} . Using the matrix entries as Cartesian coordinates, it lies in an ( n − 1 ) 2 {\displaystyle
Doubly_stochastic_matrix
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
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
Shape with seven sides
46 (1), January 1973, 7–19. Sycamore916, ed. "Heptagon." Polytope Wiki. Last modified November 2023. Accessed January 20, 2024. https://polytope.miraheze
Heptagon
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
Polyhedron with non-planar faces
ISBN 978-0-521-81496-6. Chapter I Classical Regular Polytopes (Sample text) Coxeter, Regular and Semi-Regular Polytopes II, 2.34 Coxeter and Moser, Generators and
Regular_skew_polyhedron
Geometric space with four dimensions
Springer, London. doi:10.1007/978-1-84628-633-9_13. ISBN 978-1-84628-633-9. Coxeter 1973, pp. 141–144, §7. Ordinary Polytopes in Higher Space; §7.x. Historical
Four-dimensional_space
General concept and operation in mathematics
n-dimensional polytope corresponding to an (n − i − 1)-dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected
Duality_(mathematics)
Curved triangle with constant width
asymmetry for domains of constant width", Beiträge zur Algebra und Geometrie, 42 (2): 517–521, MR 1865537. Gruber (1983, p. 78) Sallee, G. T. (1969), "The
Reuleaux_triangle
Graph that encodes local operations in mathematics
of geometric graphs. Among notable flip graphs, one finds the 1-skeleton of polytopes such as associahedra or cyclohedra. A prototypical flip graph is
Flip_graph
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
Kepler–Poinsot polyhedron
its two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect. The small stellated
Small_stellated_dodecahedron
Triangular array of the binomial coefficients
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 {\displaystyle {\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad
Pascal's_triangle
Three raised to an integer power
and 1 face, and 4 + 4 + 1 = 32. Kalai's 3d conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope. In
Power_of_three
Fundamentalnaya i prikladnaya matematika. 2 (1): 205–231. Guglielmi, N.; Wirth, F.; Zennaro, M. (2005). "Complex polytope extremality results for families of matrices"
Joint_spectral_radius
Graph made from vertices and edges of a convex polyhedron
graphs are the 3-vertex-connected, planar graphs. The analogue concept for polytopes of general dimension are the polytopal graphs. The Schlegel diagram of
Polyhedral_graph
Conic solid with a polygonal base
construction gets generalized to n dimensions. The base becomes a (n − 1)-polytope in a (n − 1)-dimensional hyperplane. A point called the apex is located outside
Pyramid_(geometry)
Study of complex manifolds and several complex variables
combinatorially by its toric fan, and at least when it is non-singular, by a moment polytope. This is a polygon in R n {\displaystyle \mathbb {R} ^{n}} with the property
Complex_geometry
Uniform 5-polytope
a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated
Cantic_5-cube
Manifold or algebraic variety of dimension n in a space of dimension n+1
22(1): 301,2 Lima, Elon L. (1988). "The Jordan-Brouwer separation theorem for smooth hypersurfaces". The American Mathematical Monthly. 95 (1): 39–42. doi:10
Hypersurface
Geometric operation which truncates the edges of polyhedra
Near-miss Johnson solid Uniform 4-polytope Uniform polyhedron Spencer 1911, p. 575, or p. 597 on Wikisource, Crystallography, 1. Cubic System, Tetrahedral Class
Chamfer_(geometry)
Goldberg polyhedron with 42 faces
cell-centered orthogonal projection of the 120-cell, one of six convex regular 4-polytopes. The chamfered dodecahedron is the shape of the fullerene C80. Occasionally
Chamfered_dodecahedron
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
Invariant measure of fractal dimension
measure". Mathematical Proceedings of the Cambridge Philosophical Society. 42 (1): 15–23. doi:10.1017/S030500410002272X. Dodson, M. Maurice; Kristensen,
Hausdorff_dimension
Chemical compound
water solution of indium(I) sulfate and selenium dioxide. There are three polytopes or crystal forms. β, ε are hexagonal with unit cells spanning two layers
Indium(II)_selenide
Outermost stellation of the icosahedron
(in German). Leipzig: B.G. Treubner. Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover. 3.6 6.2 Stellating the Platonic solids, pp. 96–104.
Final stellation of the icosahedron
Final_stellation_of_the_icosahedron
torus topology, with Euler characteristic of zero. Density: the Density (polytope) represents the number of windings of a polyhedron around its center. This
List_of_uniform_polyhedra
Group of symmetries of an n-dimensional hypercube
well as the corresponding dual polytopes (the regular octahedron and its higher-dimensional counterparts, the cross-polytopes). There is one hyperoctahedral
Hyperoctahedral_group
convex uniform 5-polytope, being a rectification of the regular 5-orthoplex. There are 5 degrees of rectifications for any 5-polytope, the zeroth here
Rectified_5-orthoplexes
all of these vertices. The edges of the polytope connect pairs of vertices that have d − 1 {\displaystyle d-1} hyperplanes in common, so the vertices
Reverse-search_algorithm
Gil (1989). "The number of faces of centrally-symmetric polytopes". Graphs and Combinatorics. 5 (1): 389–391. doi:10.1007/BF01788696. MR 1554357. S2CID 8917264
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Tessellation of convex uniform polyhedron cells
generated. If a hole has two branches, a Vinberg polytope is generated, although only Vinberg polytope with mirror symmetry are related to the simplex
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
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
Study of geometric properties of sets through measure theory
measures. Each is the Gaussian curvature measure of a polytope, and the sequence of polytopes converge to K {\displaystyle K} . If μ K {\displaystyle
Geometric_measure_theory
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
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
Segmentochora Dr. Richard Klitzing, Symmetry: Culture and Science, Vol. 11, Nos. 1-4, 139-181, 2000 (4.23 tetrahedron || cuboctahedron) Segmentochora: tetaco
Tetrahedral_cupola
Varga 1996, p. 42. Henahan 1971. Simon 2003. Russcol 1972, p. 154. Frankenstein 1970, p. 116. Henahan 1970. Rockwell 1975. "Polytope of Persepolis".
Bohor_(Xenakis)
Graph formed by subdivision of triangles
graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing
Apollonian_network
Triangulation method
29 October 2018. Seidel, Raimund (1995). "The upper bound theorem for polytopes: an easy proof of its asymptotic version". Computational Geometry. 5 (2):
Delaunay_triangulation
Volume space bounded by a sphere
distance is a hypercube, and a ball under the taxicab distance is a cross-polytope. A closed ball also need not be compact. For example, a closed ball in
Ball_(mathematics)
Visual depiction of a partially ordered set
3-dimensional cubes, and that a tetrahedron (abstract 3-polytope) likewise merges two triangles (abstract 2-polytopes). The third diagram shows some of the internal
Hasse_diagram
Formula for the "volume" of an n-simplex
n} -dimensional polytope and the convex hull of n + 1 {\displaystyle n+1} points which do not lie in any ( n − 1 ) {\displaystyle (n-1)} -dimensional plane
Cayley–Menger_determinant
Polyhedron that tiles space by translation
mathematics Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram? More
Parallelohedron
Method of drawing geometric objects
in the sequence 7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97
Straightedge and compass construction
Straightedge_and_compass_construction
Convex polyhedron with regular faces
by gyration, diminishment, or dissection. Near-miss Johnson solid Blind polytope Araki, Yoshiaki; Horiyama, Takashi; Uehara, Ryuhei (2015). "Common Unfolding
Johnson_solid
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