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 ...
, 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
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, 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 l ...
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 spac ...
of its vertices.
The ''n''-dimensional cross-polytope can also be defined as the closed
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
(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
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
minus;1, +1 in 2 dimensions it is a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ad ...
(or diamond) with vertices . In 3 dimensions it is an
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
—one of the five convex regular
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
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 edge ...
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 n ...
with an (''n''−1)-orthoplex base.
The cross-polytope is the
dual polytope
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...
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, perp ...
. 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
The Turán graph, denoted by T(n,r), is a complete multipartite graph; it is formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only if they belong to dif ...
''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-polytope
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, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
s were first described by the Swiss mathematician
Ludwig Schläfli
Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spac ...
in the mid-19th century.
Higher dimensions
The cross polytope family is one of three
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
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, perp ...
family, labeled as ''γ
n'', and the
simplices
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. ...
, 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
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. ...
. 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 l ...
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 mor ...
of the cross-polytope is .
The
dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
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 te ...
):
:
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 orthogonal ...
s that can show the cross-polytopes as 2-dimensional graphs.
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a re ...
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 n ...
, 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
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian co ...
(
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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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 graph
In graph theory, a part of mathematics, a -partite graph is a graph whose vertices are (or can be) partitioned into different independent sets. Equivalently, it is a graph that can be colored with colors, so that no two endpoints of an edge h ...
s, β make K
''p'',''p'' for
complete bipartite graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17.
Graph theory i ...
, β make K
''p'',''p'',''p'' for complete tripartite graphs. β creates K
''p''''n''. An
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
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
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
perimeter in these orthogonal projections is called a
petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a re ...
.
Related polytope families
Cross-polytopes can be combined with their dual cubes to form compound polytopes:
*In two dimensions, we obtain the
octagram
In geometry, an octagram is an eight-angled star polygon.
The name ''octagram'' combine a Greek numeral prefix, '' octa-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line".
Deta ...
mic star figure ,
*In three dimensions we obtain the
compound of cube and octahedron
The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. Construction
The 14 Cartesian coordinates of the vertices of the compound are.
: 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2)
: ...
,
*In four dimensions we obtain the
compound of tesseract and 16-cell.
See also
*
List of regular polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ' ...
*
Hyperoctahedral group
In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a paramete ...
, 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