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Regular polytope dual to the hypercube in any number of dimensions
A 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
Cross-polytope
Geometric object with flat sides
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any
Polytope
Four-dimensional analog of the octahedron
]. It is the 4-dimensional member of an infinite family of polytopes called cross-polytopes, orthoplexes, or hyperoctahedrons which are analogous to the
16-cell
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
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 regular 5-polytope in geometry
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron
5-orthoplex
10-polytope 10-cube 10-demicube 10-orthoplex 10-simplex Regular polytope and List of regular polytopes Simplex Hypercube Cross-polytope Uniform polytope
List_of_mathematical_shapes
A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
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
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
Regular_octahedron
Geometric space with five dimensions
cube), and 10 hypercells (each a tesseract). The 5-orthoplex of the cross polytope family, {3,3,3,4}, with 10 vertices, 40 edges, 80 faces (each a triangle)
Five-dimensional_space
Regular 6 dimensional polytope
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell
6-orthoplex
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
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
Shape with four equal sides and angles
family that includes the regular octahedron in three dimensions and the cross-polytopes in higher dimensions. The cube and hypercubes can be given vertex coordinates
Square
Multi-dimensional generalization of triangle
is the first of three regular polytope families, labeled by Donald Coxeter as αn, the other two being the cross-polytope family, labeled as βn, and the
Simplex
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
Geometric space with seven dimensions
were seven-dimensional. A polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes, of which there are only three
Seven-dimensional_space
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
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 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
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)
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
Balanced complete multipartite graph
Roberts graph. This graph is also the 1-skeleton of an n-dimensional cross-polytope; for instance, the graph T(6,3) = K2,2,2 is the octahedral graph, the
Turán_graph
Geometric space with eight dimensions
geometric constructions. A polytope in eight dimensions is called an 8-polytope. The most studied are the regular polytopes, of which there are only three
Eight-dimensional_space
Manifold or algebraic variety of dimension n in a space of dimension n+1
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Hypersurface
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
Maximum number of equidistant points
it is exactly 2 d {\displaystyle 2d} , achieved by the vertices of a cross polytope. The equilateral dimension has been particularly studied for Lebesgue
Equilateral_dimension
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
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
Non-empty convex set in Euclidean space
examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope. Write K n {\displaystyle {\mathcal {K}}^{n}} for the set of convex
Convex_body
Geometric space with six dimensions
Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and
Six-dimensional_space
Star polygon
compound of 5-cube and 5-orthoplex; that is, the compound of a n-cube and cross-polytope in their respective dual positions. An octagonal star can be seen as
Octagram
Topological space of dimension zero
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Zero-dimensional_space
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
Number of independent parameters of a system
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Degrees_of_freedom
Number of paths between grid corners, allowing diagonal steps
{\displaystyle n} , the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in cellular automata
Delannoy_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
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
Natural number
is also a figurate number based on the 5-orthoplex or 5-dimensional cross polytope. In the gematria of Eleazar of Worms, the Hebrew words "temunah" (image)
501_(number)
Measure of a mathematical object studied in the field of algebraic geometry
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
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
Any of the five regular polyhedra
are only three convex regular polytopes: the simplex as {3,3,...,3}, the hypercube as {4,3,...,3}, and the cross-polytope as {3,3,...,4}. In three dimensions
Platonic_solid
Notation for polytopes and tessellations
Schläfli symbol is a notation of the form {p,q,r, ...} that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century
Schläfli_symbol
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
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
N-sphere
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
Group that admits a formal description in terms of reflections
Weyl group; this corresponds to the hypercube and cross-polytope being different regular polytopes but having the same symmetry group. Some properties
Coxeter_group
Mathematical space with two coordinates
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Two-dimensional_space
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
Generalization of a quadrant to any dimension
in many constrained optimization problems. Cross polytope (or orthoplex) – a family of regular polytopes in n-dimensions which can be constructed with
Orthant
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
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
Subspace of n-space whose dimension is (n-1)
and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces. In non-Euclidean geometry, the ambient
Hyperplane
Type of metric geometry
rotationally 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
Method of determining fractal dimension
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Minkowski–Bouligand_dimension
6-dimensional hypercube
can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed of various 5-cubes, at perpendicular angles on the
6-cube
Number of vectors in any basis of the vector space
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Dimension_(vector_space)
Family of regular honeycombs in geometry
is constructed from 4 n-hypercubes per ridge. The vertex figure is a cross-polytope {3...3,4}. The hypercubic honeycombs are self-dual. Coxeter named this
Hypercubic_honeycomb
Length in a vector space
of 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
Norm_(mathematics)
Method for producing composition algebras
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Cayley–Dickson_construction
Geometrical figure in a Euclidean space
and minus the set of standard basis vectors (i.e., the vertices of a cross-polytope) from a higher-dimensional space onto a subspace. Such stars were called
Eutactic_star
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
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
Four-dimensional number system
geometry Quaternionic matrix – Concept in linear algebra Quaternionic polytope – Concept in geometry Quaternionic projective space – Concept in mathematics
Quaternion
Polytope constructed from alternation of a hypercube
(also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled
Demihypercube
Invariant measure of fractal dimension
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Hausdorff_dimension
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
In mathematics, dimension of a ring
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Krull_dimension
Topologically invariant definition of the dimension of a space
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Lebesgue_covering_dimension
Invariant of topological spaces
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Inductive_dimension
1994 book by Michio Kaku
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Hyperspace_(book)
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
In mathematics, a module that has a basis
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Free_module
Number associated with symmetric convex bodies
polyhedron or polytope is its dual polyhedron or dual polytope. In particular, the polar body of a cube or hypercube is an octahedron or cross polytope. Its Mahler
Mahler_volume
Faster-than-light travel in science fiction
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Hyperspace
Fundamental object of geometry
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Point_(geometry)
Completion of the usual space with "points at infinity"
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Projective_space
Theorem about admissible crystal symmetries
is the aforementioned eightfold symmetry of the hypercube (and the cross-polytope): A = [ 0 0 0 − 1 1 0 0 0 0 − 1 0 0 0 0 − 1 0 ] . {\displaystyle
Crystallographic restriction theorem
Crystallographic_restriction_theorem
Space with one dimension
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
One-dimensional_space
Statistical method
a square rotated so that its corners lie on the axes (in general a cross-polytope), while the region defined by the ℓ 2 {\displaystyle \ell ^{2}} norm
Lasso_(statistics)
Mathematical transformation in physics
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Time-translation_symmetry
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)
Quasiregular space-filling tesselation
alternation of a hypercubic honeycomb and being composed of demihypercube and cross-polytope facets. It is also part of another infinite family of uniform honeycombs
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
Smallest convex set containing a given set
of vertices of a square, regular octahedron, or higher-dimensional cross-polytope provide examples where exactly 2 d {\displaystyle 2d} points are needed
Convex_hull
Maths conjecture
symmetric polytope have at least 3 d {\displaystyle 3^{d}} nonempty faces? More unsolved problems in mathematics In geometry, more specifically in polytope theory
Kalai's_3^d_conjecture
it in 1912 as a semiregular polytope, identifying it as Cr51 as a first rectification of a 5-dimensional cross polytope. Rectified pentacross Rectified
Rectified_5-orthoplexes
3-dimensional geometric figure
ISSN 0138-4821, MR 1447981 Gaifullin, Alexander A. (2014), "Flexible cross-polytopes in spaces of constant curvature", Proceedings of the Steklov Institute
Flexible_polyhedron
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
Ball_(mathematics)
Topics referred to by the same term
Concentration ratio, a measure of market concentration in economics Cross-polytope of n-dimensions, in geometry US Climate Reference Network, a network
CRN
Matrix with one nonzero entry in each row and column
2^{n}n!} . It is the symmetry group of the hypercube and (dually) of the cross-polytope. Its index 2 subgroup of matrices with determinant equal to their underlying
Generalized permutation matrix
Generalized_permutation_matrix
Double cover Lie group of the special orthogonal group
group (symmetries of the hypercube, or equivalently of its dual, the cross-polytope). For point groups that reverse orientation, the situation is more complicated
Spin_group
Real-valued number of spatial dimensions
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Fractal_dimension
Measure of complexity of real-valued functions
Rademacher complexity 1 / 2 {\displaystyle 1/2} . Similarly, the unit cross-polytope { x ∈ R m : ‖ x ‖ 1 ≤ 1 } {\displaystyle \{x\in \mathbb {R} ^{m}:\|x\|_{1}\leq
Rademacher_complexity
Topics referred to by the same term
4-cross may refer to: Four-cross bike racing 4-orthoplex polytope This disambiguation page lists articles associated with the title 4-cross. If an internal
4-cross
Theorem in functional analysis
{\displaystyle \mathbf {R} ^{n}} has a linear image which contains the unit cross-polytope (the unit ball for the ℓ 1 n {\displaystyle \ell _{1}^{n}} norm) and
Auerbach's_lemma
Hypercomplex number system
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Sedenion
N-dimensional generalisation of a pyramid
construction gets generalised to n dimensions. The base becomes a (n – 1)-polytope in a (n – 1)-dimensional hyperplane. A point called apex is located outside
Hyperpyramid
Hypercomplex number system
Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Trigintaduonion
Element of a unital algebra over the field of real numbers
algebras have been styled the "generalized complex numbers". The idea of cross-ratio of four complex numbers can be extended to the 2-dimensional real
Hypercomplex_number
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