In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a
regular,
convex polytope
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-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...
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
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es of the previous dimension, while the cross-polytope's
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
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
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
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'':
:
In 1 dimension the cross-polytope is simply the
line segment minus;1, +1 in 2 dimensions it is a
square (or diamond) with vertices . In 3 dimensions it is an
octahedron—one of the five convex regular
polyhedra known as the
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s. This can be generalised to higher dimensions with an ''n''-orthoplex being constructed as a
bipyramid
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices.
The "-gonal" in the name of a bipyramid does ...
with an (''n''−1)-orthoplex base.
The cross-polytope is the
dual polytope of the
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
. The 1-
skeleton
A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
of a ''n''-dimensional cross-polytope is a
Turán graph ''T''(2''n'', ''n'').
4 dimensions
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.
Higher dimensions
The cross polytope family is one of three
regular polytope families, labeled by
Coxeter as ''β
n'', the other two being the
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
family, labeled as ''γ
n'', and the
simplices, 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 figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
s are all (''n'' − 1)-cross-polytopes. The
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
of the cross-polytope is .
The
dihedral angle of the ''n''-dimensional cross-polytope is
. This gives: δ
2 = arccos(0/2) = 90°, δ
3 = arccos(−1/3) = 109.47°, δ
4 = arccos(−2/4) = 120°, δ
5 = arccos(−3/5) = 126.87°, ... δ
∞ = arccos(−1) = 180°.
The hypervolume of the ''n''-dimensional cross-polytope is
:
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
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
):
:
There are many possible
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogona ...
s 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
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices.
The "-gonal" in the name of a bipyramid does ...
, 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 In mathematics, an equidistant set (also called a midset, or a bisector) is a set each of whose elements has the same distance (measured using some appropriate distance function) from two or more sets. The equidistant set of two singleton sets in ...
for this distance.
Generalized orthoplex
Regular
complex polytopes can be defined in
complex Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
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
. A ''p''-generalized ''n''-orthoplex has ''pn'' vertices. ''Generalized orthoplexes'' have regular
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es (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. see illustration Fig 7.2
B
** p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
External links
*
{{DEFAULTSORT:Cross-Polytope
Polytopes
Multi-dimensional geometry