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N-dimensional polytope with vertices adjacent to N facets
d-dimensional simple polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges (also d facets). The vertex figure of a simple d-polytope
Simple_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
Combinitorics of Polyhedra
convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for
Polyhedral_combinatorics
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
is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly
Unique_sink_orientation
Polytope whose facets are all simplices
simplicial polytopes In geometry, a simplicial polytope is a polytope whose facets are all simplices. It is topologically dual to simple polytopes. Polytopes that
Simplicial_polytope
Isogonal polytope with regular facets
definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled
Semiregular_polytope
Geometric operation
omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each
Omnitruncation
Point where two or more curves, lines, or edges meet
is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In
Vertex_(geometry)
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
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
Polytope in 8-dimensional geometry
root vectors of the simple Lie group E8, this polytope is sometimes referred to as the E8 root polytope. The vertices of this polytope can also be obtained
4_21_polytope
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
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
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)
polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes.
Dehn–Sommerville_equations
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
Polynomial sequence
Eulerian numbers also form the h {\displaystyle h} -vector of the simple polytope which is dual to the n {\displaystyle n} -dimensional permutohedron
Eulerian_number
Classification of symplectic toric manifolds
simple polytope, i.e. for each vertex x {\displaystyle x} , exactly n {\displaystyle n} edges meet at v {\displaystyle v} ; it is a rational polytope
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
5-dimensional hypercube
deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the
5-cube
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
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
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
6-dimensional hypercube
being a 6-dimensional polytope constructed from 12 regular facets. Acronym: ax It is a part of an infinite family of polytopes, called hypercubes. The
6-cube
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
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 polytope constructed recursively
geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after
Hanner_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
Type of convex polytope
etc. Every simple 0/1-polytope is a Cartesian product of 0/1 simplexes. Ziegler, Günter M. (2000). "Lectures on 0/1-polytopes". Polytopes—combinatorics
0/1-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
On lengths of shortest paths in convex polytopes
graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected
Hirsch_conjecture
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
Polytope combining two smaller polytopes
geometry of convex polytopes, the Blaschke sum of two polytopes is a polytope that has a facet parallel to each facet of the two given polytopes, with the same
Blaschke_sum
Uniform Polytope
6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E7. This polytope is the vertex figure for a uniform tessellation of 7-dimensional
2_31_polytope
Isogonal polytope with uniform facets
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means
Uniform_polytope
combinatorial generation problems: Vertices of simple convex polytopes If a d {\displaystyle d} -dimensional convex polytope is defined as an intersection of half-spaces
Reverse-search_algorithm
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 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
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 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
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
Graph-theoretic description of polyhedra
polytope isomorphisms", Aequationes Mathematicae, 34 (2–3): 287–297, doi:10.1007/BF01830678, MR 0921106, S2CID 120222616 Kalai, Gil (1988), "A simple
Steinitz's_theorem
Polytope
mathematics, the permutoassociahedron is an n {\displaystyle n} -dimensional polytope whose vertices correspond to the bracketings of the permutations of n +
Permutoassociahedron
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
In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3{3}3{3}3{3}3, and Coxeter diagram . It has 240 vertices
Witting_polytope
five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three unique
Rectified_5-simplexes
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
Natural number between 89 and 91
The root vectors of simple Lie group E8 are represented by the vertex arrangement of the 4 21 {\displaystyle 4_{21}} polytope, which shares 240 vertices
90_(number)
locally standard with the orbit space a simple convex polytope. The aim is to do combinatorics on the quotient polytope and obtain information on the manifold
Toric_manifold
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
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
upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of
Upper_bound_theorem
Geometric space with six dimensions
interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the
Six-dimensional_space
3D shape made of polyhedra sharing a common center
regular polytopes. Coxeter lists a few of these in his book Regular Polytopes. McMullen added six in his paper New Regular Compounds of 4-Polytopes. Self-duals:
Polytope_compound
solid. The coordinates of uniform 4-polytopes with pentachoric symmetry can be generated as permutations of simple integers in 5-space, all in hyperplanes
A4_polytope
221 polytope, , in 6-dimensional space, sharing the same 27 vertices. The 216 edges in 221 can be seen as the 72 3{} edges represented as 3 simple edges
Hessian_polyhedron
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
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)
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
Regular 5-polytope
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and
5-simplex
248-dimensional exceptional simple Lie group
are the vertices of a semi-regular polytope discovered by Thorold Gosset in 1900, sometimes known as the 421 polytope. In the so-called even coordinate
E8_(mathematics)
Infinitely detailed mathematical structure
for instance, could only be visualized through a few iterations as very simple drawings). That changed, however, in the 1960s, when Benoit Mandelbrot started
Fractal
the 50 root vectors of the B5 and C5 simple Lie groups. E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr51 as a first rectification
Rectified_5-orthoplexes
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
Generalization of a rectangle for higher dimensions
database theory or ranges of integers, rather than real numbers. The dual polytope of an n-orthotope has been variously called a rectangular n-orthoplex,
Hyperrectangle
Type of 7-polytope
seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex
Hexicated_7-simplexes
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)
Topological space of dimension zero
Concepts Features Dimension Straightedge and compass constructions Angle Polytope Centroid Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence
Zero-dimensional_space
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
German mathematician and politician
she and Mani-Levitska proved that the combinatorial structure of simple polytopes is completely determined by their graphs. This result has been called
Roswitha_Blind
British mathematician
1017/s0305004100051665, MR 0394436, S2CID 63778391. —— (1993), "On simple polytopes", Inventiones Mathematicae, 113 (2): 419–444, Bibcode:1993InMat.113
Peter_McMullen
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)
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
Regular non-convex polygon
difference when retrograde polygons are incorporated in higher-dimensional polytopes. For example, an antiprism formed from a prograde pentagram {5/2} results
Star_polygon
Topics referred to by the same term
structure or topological triangulation E8 polytope, alternate name for the 421 semiregular (uniform) polytope Elementary abelian group of order 8 E8 Theory
E8
Closed volume that completely contains the union of a set of objects
the union of a finite set of points, its convex hull is a polytope. A discrete oriented polytope (DOP) generalizes the bounding box. A k-DOP is the Boolean
Bounding_volume
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
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
Pictorial representation of symmetry
hexagonal lattice. An associated polytope – for example Gosset 421 polytope may be referred to as "the E8 polytope", as its vertices are derived from
Dynkin_diagram
In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry
Vertex_enumeration_problem
Straight figure with zero width and depth
previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates (r, θ) of the points of a
Line_(geometry)
Straight line segment that passes through the centre of a circle
Concepts Features Dimension Straightedge and compass constructions Angle Polytope Centroid Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence
Diameter
Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen and
H-vector
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
Field of mathematics which studies incidence structures
polar spaces are near polygons. Many near polygons are related to finite simple groups like the Mathieu groups and the Janko group J2. Moreover, the generalized
Incidence_geometry
Mathematical space with two coordinates
Concepts Features Dimension Straightedge and compass constructions Angle Polytope Centroid Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence
Two-dimensional_space
Theorem about projections of coadjoint orbits of a connected compact Lie group
described above for the copies of SU(2) corresponding to simple roots, so the whole convex polytope lies in P(Ad(K)⋅X). Heckman (1982) gave another proof
Kostant's_convexity_theorem
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
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
Finite simple group type not classified as Lie, cyclic or alternating
20007. Hartley, Michael I.; Hulpke, Alexander (2010), "Polytopes Derived from Sporadic Simple Groups", Contributions to Discrete Mathematics, 5 (2), Alberta
Sporadic_group
Group that admits a formal description in terms of reflections
Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include
Coxeter_group
Euclidean geometry without distance and angles
Euclidean geometry but also in Minkowski's geometry of time and space (in the simple case of 1 + 1 dimensions, whereas the special theory of relativity needs
Affine_geometry
Partition of a simple polygon into triangles
create a triangulation based on a set of points. The associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon. Polygon
Polygon_triangulation
d-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot be obtained
Simplicial_sphere
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
Concept in geometry
used to draw diagrams of higher-dimensional polytopes and root systems – the vertices and edges of the polytope, or roots (and some edges connecting these)
Coxeter_element
Shape with five sides
polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting regular
Pentagon
Natural number
positive root vectors in the seven-dimensional space. There are 63 uniform polytopes in the sixth dimension that are generated from the abstract hypercubic
63_(number)
five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex. There
Stericated_5-simplexes
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