cross-polytope

In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. 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 facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

The vertices of a cross-polytope can be chosen as the unit vectors pointing along each co-ordinate axis – i.e. all the permutations of . The cross-polytope is the convex hull of its vertices.
The ''n''-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ₁-norm on R^''n'', those points satisfying

:$, x_1, + , x_2, + \cdots + , x_n, \le 1.$

An ''n''-orthoplex can be constructed as a bipyramid with an (''n''−1)-orthoplex base.

The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of an ''n''-dimensional cross-polytope is the Turán graph ''T''(2''n'', ''n'') (also known as a ''cocktail party graph'' ).

Low-dimensional examples

In 1 dimension the cross-polytope is simply the line segment minus;1, +1

In 2 dimensions the cross-polytope is a square with vertices , sometimes called a ''diamond''.

In 3 dimensions it is a regular octahedron—one of the five convex regular polyhedra known as the Platonic solids.

The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of the six convex regular 4-polytopes. These 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. The vertices of the 4-dimensional hypercube, or tesseract, can be divided into two sets of eight, the convex hull of each set forming a cross-polytope. Moreover, the polytope known as the 24-cell can be constructed by symmetrically arranging three cross-polytopes.

''n'' dimensions

The cross-polytope family is one of three regular polytope families, labeled by Coxeter as ''β_n'', the other two being the hypercube family, labeled as ''γ_n'', and the simplex family, labeled as ''α_n''. A fourth family, the infinite tessellations of hypercubes, he labeled as ''δ_n''.

The ''n''-dimensional cross-polytope has 2''n'' vertices, and 2^''n'' facets ((''n'' − 1)-dimensional components) all of which are (''n'' − 1)-simplices. The vertex figures are all (''n'' − 1)-cross-polytopes. The Schläfli symbol of the cross-polytope is .

The dihedral angle of the ''n''-dimensional cross-polytope is $\delta_n = \arccos\left(\frac\right)$. This gives: δ₂ = arccos(0/2) = 90°, δ₃ = arccos(−1/3) = 109.47°, δ₄ = arccos(−2/4) = 120°, δ₅ = arccos(−3/5) = 126.87°, ... δ_∞ = arccos(−1) = 180°.

The hypervolume of the ''n''-dimensional cross-polytope is
:$\frac.$

For each pair of non-opposite vertices, there is an edge joining them. More generally, each set of ''k'' + 1 orthogonal vertices corresponds to a distinct ''k''-dimensional component which contains them. The number of ''k''-dimensional components (vertices, edges, faces, ..., facets) in an ''n''-dimensional cross-polytope is thus given by (see binomial coefficient):
:$2^$

The extended f-vector for an ''n''-orthoplex can be computed by (1,2)^''n'', like the coefficients of polynomial products. For example a 16-cell is (1,2)⁴ = (1,4,4)² = (1,8,24,32,16).

There are many possible orthographic projections that can show the cross-polytopes as 2-dimensional graphs. Petrie polygon projections map the points into a regular 2''n''-gon or lower order regular polygons. A second projection takes the 2(''n''−1)-gon petrie polygon of the lower dimension, seen as a bipyramid, projected down the axis, with 2 vertices mapped into the center.

The vertices of an axis-aligned cross polytope are all at equal distance from each other in the Manhattan distance ( L¹ norm). Kusner's conjecture states that this set of 2''d'' points is the largest possible equidistant set for this distance.

Generalized orthoplex

Regular complex polytopes can be defined in complex Hilbert space called ''generalized orthoplexes'' (or cross polytopes), β = ₂₂..._2''p'', or ... Real solutions exist with ''p'' = 2, i.e. β = β_''n'' = ₂₂...₂₂ = . For ''p'' > 2, they exist in $\mathbb^n$. A ''p''-generalized ''n''-orthoplex has ''pn'' vertices. ''Generalized orthoplexes'' have regular simplexes (real) as facets.Coxeter, Regular Complex Polytopes, p. 108 Generalized orthoplexes make complete multipartite graphs, β make K_''p'',''p'' for complete bipartite graph, β make K_{''p'',''p'',''p''} for complete tripartite graphs. β creates K_''p''^''n''. An orthogonal projection can be defined that maps all the vertices equally-spaced on a circle, with all pairs of vertices connected, except multiples of ''n''. The regular polygon perimeter in these orthogonal projections is called a petrie polygon.

Related polytope families

Cross-polytopes can be combined with their dual cubes to form compound polytopes:

*In two dimensions, we obtain the octagrammic star figure ,
*In three dimensions we obtain the compound of cube and octahedron,
*In four dimensions we obtain the compound of tesseract and 16-cell.

See also

* List of regular polytopes
* Hyperoctahedral group, the symmetry group of the cross-polytope

Citations

References

*
** pp. 121–122, §7.21, illustration Fig 7.2B
** p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)

External links

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{{DEFAULTSORT:Cross-Polytope
Regular polytopes
Multi-dimensional geometry