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Cross-polytope

A cross-polytope, also known as an orthoplex, hyperoctahedron, or cocube, is a regular convex polytope in n-dimensional Euclidean space defined as the convex hull of the 2n points formed by all permutations of the coordinates (±1, 0, ..., 0).^[1] It generalizes the octahedron to higher dimensions and is one of the three regular convex polytopes that exist in every dimension n ≥ 3 (the others being the n-simplex and the n-hypercube).^[1] The cross-polytope has 2n vertices, 2n facets (each an (n-1)-simplex), and its skeleton is the complete multipartite graph K2,2,...,2 (n parts), which is isomorphic to the cocktail party graph.^[1] Its Schläfli symbol is {3,3,...,3,4} with (n-2) threes, reflecting its regular structure where all edges are equal and faces are equilateral triangles up to the facets.^[2] Notably, the cross-polytope is the dual polytope of the n-dimensional hypercube, meaning their vertices correspond to the hypercube's facets and vice versa, and this duality extends to their symmetry groups, both belonging to the hyperoctahedral group of order 2n n!.^[1] In low dimensions, the cross-polytope takes familiar forms: in 1D, it is a line segment; in 2D, a square with vertices at (±1,0) and (0,±1); in 3D, the regular octahedron with 6 vertices and 8 triangular faces; and in 4D, the 16-cell (or hexadecachoron) with 8 vertices, 32 triangular faces, and 16 tetrahedral cells.^[1] The volume of an n-dimensional cross-polytope with edge length s is given by (2n / n!) * (s / √2)n, derived by decomposing it into n! hyperpyramids from the center.^[2] These polytopes appear in convex geometry, optimization (as the unit ball in the ℓ1 norm), and symmetry studies, with applications in linear programming and combinatorial designs.^[3]

Definition and Construction

Geometric Definition

The n-dimensional cross-polytope, also known as the orthoplex or hyperoctahedron, is defined as the convex hull of the $2npoints consisting of the [standard basis](/page/Standard_basis) vectorse_iand their negatives-e_ifori=1,\dots,nin\mathbb{R}^n. This construction places all vertices at unit distance from the origin along the coordinate axes, forming a centrally symmetric figure that is the L_1$-ball of radius 1.^[4]^[5] As a regular polytope, the cross-polytope exhibits the highest degree of symmetry among convex polytopes in n dimensions, with all facets being regular simplices and vertex figures being regular (n-1)-dimensional cross-polytopes. Its Schläfli symbol is \{3,3,\dots,3,4\}, consisting of n-2 entries of 3 followed by a 4 (for n \geq 2). This symbol encodes the recursive structure where each ridge is surrounded by four facets, distinguishing it from other regular polytopes like the simplex or hypercube.^[6]^[7] The name "cross-polytope" derives from the star-like or cross-shaped appearance in low dimensions, where the vertices align along perpendicular axes intersecting at the center, evoking a generalized cross. It plays a foundational role as the simplest uniform polytope dual to the n-dimensional hypercube, meaning their faces and vertices correspond in a polarity that preserves the regular symmetry.^[1]^[1]

Vertex Coordinates

The vertices of the n-dimensional cross-polytope are given by the $2npoints\pm \mathbf{e}_i \in \mathbb{R}^nfori = 1, \dots, n, where \mathbf{e}_idenotes thei

-th [standard basis](/page/Standard_basis) vector with a &#36;1

in the i-th coordinate and $0elsewhere.[1] These points have coordinates featuring exactly one\pm 1and zeros in all other positions, and the cross-polytope is the [convex hull](/page/Convex_hull) of this vertex set.[1] In this standard embedding, each vertex lies at an\ell_2

-distance of &#36;1

from the origin, yielding a circumradius of $1$.^[1] An equivalent representation arises as the unit ball under the \ell_1-norm:

\{ x \in \mathbb{R}^n : \sum_{i=1}^n |x_i| \leq 1 \}.

This formulation confirms the same $2nvertices\pm \mathbf{e}_i$, as these points saturate the norm bound while all other points in the set satisfy it strictly.^[8] The facets of the cross-polytope, which are (n-1)-dimensional simplices, are defined by the supporting half-spaces

\sum_{i=1}^n \epsilon_i x_i \leq 1,

where each \epsilon = (\epsilon_1, \dots, \epsilon_n) is a choice of signs with \epsilon_i \in \{ \pm 1 \}, producing $2^n such inequalities in total.^[9] These hyperplanes touch the polytope at its facets, with the normal vectors \epsilon corresponding to the vertices of the dual hypercube (up to scaling).^[9] Alternative scalings of the cross-polytope are common for specific applications, such as normalizing to unit inradius (the distance from the origin to a facet) or unit edge length; for instance, scaling the standard vertex coordinates by $1/\sqrt{n} adjusts the circumradius while preserving combinatorial structure.

Low-Dimensional Examples

One- and Two-Dimensional Cases

In one dimension, the cross-polytope is the line segment spanning from -1 to 1 on the real line, with vertices at the points -1 and 1.^[1] This simplest case consists of 2 vertices connected by a single edge, representing the convex hull of these two points equidistant from the origin.^[1] Embedded in \mathbb{R}^1, it serves as the foundational example of a cross-polytope, illustrating the structure of opposite vertices along a single axis.^[1] In two dimensions, the cross-polytope takes the form of a square rotated by 45 degrees relative to the coordinate axes, with vertices at (-1, 0), (1, 0), (0, -1), and (0, 1).^[1] This figure, denoted by the Schläfli symbol \{4\}, has 4 vertices and 4 edges, forming the convex hull of these points, each lying along the perpendicular axes at unit distance from the origin.^[1] Visually, it appears as a diamond shape inscribed in a circle of radius 1, providing an intuitive planar embedding that highlights the cross-like arrangement of its vertices.^[1] As the dual of the 2-dimensional hypercube (square), it exemplifies the regular polytope structure in the plane beyond the simplex.^[1] This 2D case progresses naturally to the 3D octahedron in higher dimensions.^[1]

Three-Dimensional Octahedron

The three-dimensional cross-polytope is the regular octahedron, a Platonic solid characterized by its high degree of symmetry and uniform structure. It consists of 6 vertices located at the coordinates (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1), which form the convex hull in Euclidean 3-space. These vertices connect via 12 edges, each of length √2, and bound 8 equilateral triangular faces, making it the only Platonic solid with triangular faces meeting four at each vertex. This configuration allows for straightforward visualizations, often depicted as two square pyramids glued at their bases or as the outline of a diamond shape rotated in 3D. As the dual polyhedron to the cube, the regular octahedron exhibits a reciprocal relationship where its 6 vertices correspond to the 6 faces of the cube, and its 8 faces correspond to the 8 vertices of the cube, with face centers of one forming the vertices of the other.^[10] This duality highlights the octahedron's role in geometric complementarity, where the triangular faces align with the cube's vertices to create a self-intersecting star-like projection in certain views. The dihedral angle between adjacent faces, measuring the angle between their planes, is precisely \arccos\left(-\frac{1}{3}\right) \approx 109.47^\circ, which governs the solid's angular rigidity and distinguishes it from other polyhedra. Beyond pure geometry, the octahedral structure finds prominent application in coordination chemistry, where it models the arrangement of six ligands around a central transition metal ion, as in octahedral complexes like [Co(NH₃)₆]³⁺. This geometry is prevalent for coordination number 6 due to the stability provided by the symmetric ligand placement along the coordinate axes, influencing electronic properties and reactivity in inorganic compounds.

Four-Dimensional 16-Cell

The four-dimensional cross-polytope, known as the 16-cell or hexadecachoron, is a regular convex 4-polytope that serves as the direct analog of the octahedron in higher dimensions. It consists of 16 regular tetrahedral cells, 32 equilateral triangular faces, 24 edges, and 8 vertices, with four tetrahedra meeting at each vertex in an octahedral arrangement. This structure embodies the cross-polytope's defining property of being formed by the convex hull of points at the centers of the facets of a dual hypercube (tesseract), resulting in a self-dual polytope.^[11] The 16-cell is denoted by the Schläfli symbol {3,3,4}, indicating that its cells are regular tetrahedra {3,3} and its vertex figures are regular octahedra {3,4}. This makes it one of the six regular 4-polytopes, alongside the 5-cell, tesseract, 24-cell, 120-cell, and 600-cell. The polytope's combinatorial uniformity ensures that all elements—cells, faces, edges, and vertices—are symmetrically equivalent under its symmetry group.^[11]^[12] First described by Swiss mathematician Ludwig Schläfli in his 1852 enumeration of regular polytopes in dimensions greater than three, the 16-cell highlighted the richness of higher-dimensional geometry beyond Euclidean 3-space. Schläfli's work systematically classified these figures, proving the existence of exactly six convex regular 4-polytopes.^[12]^[13] Visualizing the 16-cell requires projections into lower dimensions, such as Schlegel diagrams, which represent the polytope as a central tetrahedral cell surrounded by the remaining 15 tetrahedra projected onto its boundary, akin to a 3D perspective drawing. Net projections, unfolding the cells into 3D space, further aid comprehension by displaying the polytope's surface as an arrangement of tetrahedra connected edge-to-edge. These techniques reveal the 16-cell's intricate connectivity, where 24 edges form six disjoint squares in hyperplanes parallel to coordinate planes.^[14]^[11] In the broader context of cross-polytopes, the 16-cell exemplifies the combinatorial pattern where the n-dimensional case has 2n vertices and 2n(n-1) edges, scaling symmetrically from lower dimensions.^[1]

General Properties

Combinatorial Structure

The combinatorial structure of the n-dimensional cross-polytope is captured by its f-vector, which enumerates the number of faces of each dimension: f_k = 2^{k+1} \binom{n}{k+1} for k = 0, 1, \dots, n-1.^[15] This formula arises from the polytope's construction as the convex hull of the standard basis vectors and their negatives in \mathbb{R}^n. In particular, the cross-polytope has f_0 = 2n vertices and f_{n-1} = 2^n facets, with each facet being an (n-1)-simplex.^[15] As the polar dual of the n-dimensional hypercube, the f-vector of the cross-polytope is the reverse of the hypercube's f-vector, reflecting the bijection between k-faces of one and (n-1-k)-faces of the other.^[15] For verification in low dimensions, the 3-dimensional cross-polytope (regular octahedron) has 8 triangular facets.^[15] In the standard realization with vertices at \pm \mathbf{e}_i (the signed standard basis vectors in \mathbb{R}^n), all edges of the cross-polytope have equal length \sqrt{2}. The polytope is convex by construction as the convex hull of its vertices, ensuring no self-intersections and that it is star-free.

Volumes and Hypervolumes

The n-dimensional cross-polytope, constructed as the convex hull of the 2n points ±e_i (where e_i denotes the i-th standard basis vector in \mathbb{R}^n), possesses a hypervolume given by the formula

V_n = \frac{2^n}{n!}.

This expression holds for the standard embedding where the vertices lie on the coordinate axes at distance 1 from the origin.^[16] The hypervolume can be computed by decomposing it into 2^n full-dimensional simplices (one for each choice of signs, corresponding to the orthants), each with volume 1/n!, or equivalently via the integral over the dual l_1 unit ball scaled appropriately.^[17] The boundary of the cross-polytope consists of 2^n facets, each a regular (n-1)-dimensional simplex with edge length \sqrt{2}. The (n-1)-dimensional content of each such facet is \sqrt{n} / (n-1)!, yielding a total hypersurface content (surface area in the generalized sense) of

S_n = \frac{2^n \sqrt{n}}{(n-1)!}.

This measure quantifies the "surface" complexity and scales factorially with dimension, reflecting the increasing number and size of facets.^[17] The dihedral angle \theta_n between two adjacent facets satisfies

\theta_n = \arccos\left( \frac{2-n}{n} \right).

For example, in three dimensions (the regular octahedron), this yields \theta_3 \approx 109.47^\circ, while in four dimensions (the 16-cell), \theta_4 = 120^\circ. This angle governs the internal geometry and increases toward 180^\circ as n grows, indicating that higher-dimensional cross-polytopes become "flatter" in their facet arrangements.^[18] For large n, the hypervolume exhibits rapid decay: using Stirling's approximation n! \sim \sqrt{2\pi n} (n/e)^n, one obtains the asymptotic

V_n \sim \frac{(2e/n)^n}{\sqrt{2\pi n}},

demonstrating exponential shrinkage relative to the embedding space, consistent with the concentration of measure in high dimensions. The hypersurface content S_n similarly follows S_n \sim 2^n \sqrt{n} / (n-1)! \approx c \sqrt{n} (2e/n)^{n-1} for some constant c, peaking before eventual decay. These behaviors highlight the cross-polytope's diminishing volumetric presence in higher dimensions compared to other regular polytopes like the hypercube.^[17]

Symmetries

Rotation Group

The rotation group of the n-dimensional cross-polytope consists of the orientation-preserving isometries that map the polytope to itself, forming an index-2 subgroup of the full symmetry group. This group has order $2^{n-1} n!, half that of the hyperoctahedral group, which is the full symmetry group of order $2^n n!.^[19] The rotation group is isomorphic to the index-2 orientation-preserving subgroup of the Weyl group of type B_n/C_n, the hyperoctahedral group acting as signed permutations on the coordinates.^[20] It can be generated by permutations of the coordinates combined with an even number of sign changes on the coordinates, ensuring the overall transformation has determinant 1. Equivalently, it is generated by n-1 basic rotations R_{0j} for j=1,\dots,n-1, where R_{0j} rotates by 90 degrees in the plane spanned by the first and j-th standard basis vectors, defined by R_{0j}(x_0, \dots, x_j, \dots) = (-x_j, \dots, x_0, \dots) in augmented coordinates (treating the hypercube vertices as \pm1 in each coordinate).^[19]^[21] The group acts transitively on the n pairs of opposite vertices \{\pm e_i\} (i=1,\dots,n), where e_i are the standard basis vectors, as permutations of coordinates map any pair to any other while even sign adjustments preserve orientation. The stabilizer of a fixed pair is isomorphic to the rotation group of the (n-1)-dimensional cross-polytope.^[19] The natural action of the rotation group on \mathbb{R}^n provides an irreducible representation within the special orthogonal group \mathrm{SO}(n), as the only invariant subspaces under the full hyperoctahedral action are trivial, and the subgroup inherits this irreducibility for the defining representation.^[20] This representation underscores the cross-polytope's role in realizing finite subgroups of \mathrm{SO}(n) with high symmetry.

Full Symmetry Group

The full symmetry group of the n-dimensional cross-polytope is the hyperoctahedral group B_n, which consists of all signed permutations of the n coordinates and has order $2^n n!. This group includes both orientation-preserving and orientation-reversing isometries, encompassing reflections and improper rotations that map the polytope to itself. As a finite Coxeter group of type B_n, it admits a presentation generated by reflections across the hyperplanes x_i = 0 (for i=1,\dots,n) and x_i = x_j (for $1 \leq i < j \leq n). These reflections correspond to sign flips on individual coordinates and permutations of coordinates, respectively, preserving the \ell_1-norm defining the polytope. The hyperoctahedral group acts transitively on the $2nvertices of the cross-polytope, which are the standard basis vectors\pm e_ifori=1,\dots,n. By the orbit-stabilizer theorem, the stabilizer of any vertex has order 2^n n! / 2n = 2^{n-1} (n-1)!and is isomorphic to the hyperoctahedral groupB_{n-1}, reflecting the symmetries fixing one axis while acting on the orthogonal complement. Similarly, the group acts transitively on the 2^nfacets, each an(n-1)-simplex; the stabilizer of a facet has order 2^n n! / 2^n = n!and is isomorphic to the symmetric groupS_n$, which permutes the coordinates defining the facet. Due to the duality between the cross-polytope and the n-hypercube, the two polytopes share the identical full symmetry group B_n. The rotational subgroup, of index 2 in B_n, consists solely of the orientation-preserving symmetries.

Generalizations

Generalized Orthoplex

The generalized orthoplex, also known as the β_p^n polytope, extends the standard cross-polytope to regular polytopes in complex Euclidean space of dimension n, where p ≥ 2 is an integer specifying the order of rotational symmetry. These polytopes are constructed as the convex hull of vertices formed by taking the p-th roots of unity and associating them with the standard basis vectors in ℂ^n through appropriate group actions of the unitary group, yielding p^n vertices in total. For p = 2, the construction reduces to the familiar real cross-polytope embedded in ℝ^n, with vertices at (±1, 0, ..., 0) and permutations thereof.^[22]^[23] The facets of the generalized n-orthoplex consist of p^n copies of the (n-1)-dimensional regular simplex (which are real).^[24] The vertex figure is the generalized (n-1)-hypercube γ_p^{n-1}, and its dual is the generalized n-hypercube γ_p^n, obtained by interchanging the roles of vertices and facets in the Schläfli symbol {3^p, 4}. The full symmetry group has order n! p^n, reflecting the wreath product structure involving the symmetric group S_n and cyclic groups of order p. These properties hold in the finite-dimensional complex Hilbert space ℂ^n, where the polytope is invariant under unitary transformations preserving the Hermitian inner product.^[22]^[23]^[24] This generalization was first systematically explored by G. C. Shephard in 1952, who classified all regular complex polytopes as realizations of finite complex reflection groups, with the generalized orthoplexes corresponding to specific irreducible representations. H. S. M. Coxeter further developed the theory in 1974, providing explicit constructions and enumerations, building on earlier work in the 1920s by D. M. Y. Sommerville on real regular polytopes, including the orthoplex, though Sommerville's contributions focused on Euclidean cases without the complex extension. In real finite-dimensional spaces, an analogous construction relates to the dual of the unit ball in the L_p norm, where for p=1 the dual is the standard cross-polytope and for p=∞ it approaches the hypercube, but these are not polytopal for 1 < p < ∞.^[22]^[23]

Extensions to Other Spaces

Cross-polytopes extend naturally to complex vector spaces, where they are defined using the Hermitian inner product. In \mathbb{C}^n, the Hermitian cross-polytope is the convex hull of the $2n points consisting of the standard basis vectors e_i and their negatives -e_i, for i = 1, \dots, n. This construction preserves the combinatorial structure of the real case while adapting to the complex geometry, with symmetries realized by unitary transformations. Such objects appear in the study of regular complex polytopes, where the vertices form highly symmetric tight frames in the complex space.^[25] In infinite-dimensional settings, cross-polytopes are generalized to separable Hilbert spaces \mathcal{H}, as the convex hull of the points \pm e_i, where \{e_i\}_{i=1}^\infty is an orthonormal basis. This infinite-dimensional analog serves as the unit ball for the \ell_1-norm induced by the basis, despite \mathcal{H} being equipped with the \ell_2-norm. It plays a role in functional analysis, particularly in problems involving approximation, operator norms, and geometric properties of convex bodies in Banach spaces. For instance, projections and sections of this object inform bounds on distortion in embedding theorems and Kashin-type decompositions. Cross-polytopes also manifest in non-Euclidean geometries, notably on spheres, where their vertices form spherical polytopes. The spherical cross-polytope in S^{n-1} is obtained by intersecting the Euclidean cross-polytope with the unit sphere, yielding a regular spherical polytope with $2n vertices that constitutes a spherical 2-design. This configuration ensures uniform integration against quadratic polynomials on the sphere, with applications in numerical analysis and coding theory. In hyperbolic geometry, analogous finite convex polytopes can be realized within hyperbolic space \mathbb{H}^n, maintaining regularity under the hyperbolic metric, though they are less commonly emphasized compared to their spherical counterparts. In modern research, cross-polytopes and their higher analogs find applications in quantum information theory, particularly in the construction of morphophoric positive operator-valued measures (POVMs), which generalize symmetric informationally complete POVMs (SIC-POVMs) via polytopal structures. Distributing equal probability mass over the vertices of a cross-polytope—corresponding to an orthonormal basis and its antipodes—yields a 2-design POVM in finite dimensions, useful for quantum state tomography and frame theory. These connections extend to morphophoric POVMs, with implications for quantum measurements beyond traditional bases. Finite p-generalizations of cross-polytopes, such as the unit balls of \ell_p-norms for $1 < p < \infty, provide further analogs in normed spaces, bridging to broader convex optimization contexts.

Relations to Other Polytopes

Duality with Hypercube

The cross-polytope and the hypercube form a dual pair in the theory of convex polytopes, where the polar dual of the hypercube is the cross-polytope, and vice versa. In this duality, the vertices of the cross-polytope correspond to the outward unit normals of the facets of the hypercube. For instance, in three dimensions, the regular octahedron (cross-polytope) is the dual of the cube (hypercube).^[26]^[27] A standard realization places the n-dimensional hypercube as the set [-1,1]^n, whose facets are defined by the equations x_i = \pm 1 for i=1,\dots,n, with outward normals given by the vectors \pm \mathbf{e}_i, where \mathbf{e}_i are the standard basis vectors in \mathbb{R}^n. The polar dual of this hypercube is then the cross-polytope, defined as the convex hull \operatorname{conv}\{\pm \mathbf{e}_i \mid i=1,\dots,n\}, whose vertices are precisely these facet normals of the hypercube.^[28]^[27] This polar duality induces a lattice anti-isomorphism between the face lattices of the two polytopes, establishing an inclusion-reversing bijection between their faces. Specifically, a k-dimensional face of the cross-polytope is dual to an (n-k-1)-dimensional face of the hypercube.^[26]^[27] Both polytopes share the same full symmetry group, known as the hyperoctahedral group B_n \cong (\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n, which consists of all signed permutations of the coordinates and acts transitively on their vertices and facets.^[29]

Comparisons with Simplex and Others

The n-dimensional cross-polytope possesses 2n vertices, whereas the n-simplex has only n+1 vertices.^[1]^[30] Its facets consist of (n-1)-simplices, numbering 2^n in total, in sharp contrast to the n+1 facets of the n-simplex.^[31]^[30] This structural difference highlights the cross-polytope's greater combinatorial complexity, as its facets form a more expansive arrangement of simplices despite sharing the same facet type. Among regular convex polytopes, the cross-polytope stands alongside the simplex and hypercube as one of only three families that exist in dimensions n > 4.^[13]^[32] No other convex regular polytopes appear in these higher dimensions, underscoring the cross-polytope's unique role in the classification of such objects beyond the exceptional cases in lower dimensions.^[13] Uniform variants of the cross-polytope include the rectified form, which is a vertex-transitive polytope constructed by truncating vertices to edge midpoints, resulting in facets consisting of lower-dimensional cross-polytopes and rectified simplices.^[33] In applications, the cross-polytope frequently arises as the unit ball in the L1 (Manhattan) norm, facilitating analyses in optimization and sparse signal processing where promoting sparsity is key.^[3] In contrast, the simplex underpins barycentric coordinate systems, enabling affine combinations and interpolation within convex sets, as commonly used in computational geometry and finite element methods.^[34] These distinct roles emphasize the simplex's utility in coordinate representations versus the cross-polytope's prominence in norm-based geometric constraints.

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Cross-polytope - EPFL Graph Search
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space.
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[PDF] REGULAR POLYTOPES IN Zn Contents 1. Introduction 1 2. Some ...
Aug 26, 2011 · Definition 2.3. A regular cross polytope of dimension n is the dual polytope of a hypercube of dimension n. For dimension 5 and up, there are ...
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[PDF] Barycentric Coordinates for Convex Sets - Applied Geometry Lab
Aug 10, 2005 · Abstract. In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates ...Missing: L1 | Show results with:L1