Cross-polytope

Encyclopedia

In geometry

, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular

, convex polytope

that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull

of its vertices. Its facets are simplex

es of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

The n-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ

In 1 dimension the cross-polytope is simply the line segment

[−1, +1], in 2 dimensions it is a square

(or diamond) with vertices {(±1, 0), (0, ±1)}. In 3 dimensions it is an octahedron

—one of the five convex regular polyhedra

known as the Platonic solid

s. Higher-dimensional cross-polytopes are generalizations of these.

The cross-polytope is the dual polytope of the hypercube

. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph

T(2n,n).

. It is one of six convex regular 4-polytope

s. These polychora

were first described by the Swiss mathematician Ludwig Schläfli

in the mid-19th century.

families, labeled by Coxeter as β

family, labeled as γ

, labeled as α

The n-dimensional cross-polytope has 2n vertices, and 2

. The vertex figure

s are all n − 1 cross-polytopes. The Schläfli symbol of the cross-polytope is {3,3,…,3,4}.

The number of k-dimensional components (vertices, edges, faces, …, facets) in an n-dimensional cross-polytope is given by (see binomial coefficient

):

The volume of the n-dimensional cross-polytope is

There are many possible orthographic projection

s that can show the cross-polytopes as 2-dimensional graphs. Petrie polygon

projections map the points into a regular 2n-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

(L

). Kusner's conjecture states that this set of 2d points is the largest possible equidistant set for this distance.

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

, convex polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull

Convex hull

In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....

of its vertices. Its facets are simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

es of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

The n-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ

_{1}-norm on R^{n}:In 1 dimension the cross-polytope is simply the line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

[−1, +1], in 2 dimensions it is a square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

(or diamond) with vertices {(±1, 0), (0, ±1)}. In 3 dimensions it is an octahedron

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

—one of the five convex regular polyhedra

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

known as the Platonic solid

Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s. Higher-dimensional cross-polytopes are generalizations of these.

2 dimensions square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... |
3 dimensions octahedron Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... |
4 dimensions 16-cell 16-cell In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.... |

The cross-polytope is the dual polytope of the hypercube

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, perpendicular to each other and of the same length.An...

. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph

Turán graph

The Turán graph T is a graph formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices by an edge whenever they belong to different subsets. The graph will have subsets of size \lceil n/r\rceil, and r- subsets of size \lfloor n/r\rfloor...

T(2n,n).

## 4 dimensions

The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell16-cell

In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....

. It is one of six convex regular 4-polytope

Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....

s. These polychora

Polychoron

In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...

were first described by the Swiss mathematician Ludwig Schläfli

Ludwig Schläfli

Ludwig Schläfli was a Swiss geometer and complex analyst who was one of the key figures in developing the notion of higher dimensional spaces. The concept of multidimensionality has since come to play a pivotal role in physics, and is a common element in science fiction...

in the mid-19th century.

## Higher dimensions

The cross polytope family is one of three regular polytopeRegular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

families, labeled by Coxeter as β

_{n}, the other two being the hypercubeHypercube

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, perpendicular to each other and of the same length.An...

family, labeled as γ

_{n}, and the simplicesSimplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

, labeled as α

_{n}. A fourth family, the infinite tessellations of hypercubes, he labeled as δ_{n}.The n-dimensional cross-polytope has 2n vertices, and 2

^{n}facets (n−1 dimensional components) all of which are n−1 simplicesSimplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

. The vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

s are all n − 1 cross-polytopes. The Schläfli symbol of the cross-polytope is {3,3,…,3,4}.

The number of k-dimensional components (vertices, edges, faces, …, facets) in an n-dimensional cross-polytope is given by (see binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

):

The volume of the n-dimensional cross-polytope is

There are many possible orthographic projection

Orthographic projection

Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

s that can show the cross-polytopes as 2-dimensional graphs. Petrie polygon

Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...

projections map the points into a regular 2n-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

Bipyramid

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...

, projected down the axis, with 2 vertices mapped into the center.

n Polytope In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions... |
β_{n}k _{11} |
Name(s) Graph Graph (mathematics) In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges... |
Graph 2n-gon |
Graph 2(n-1)-gon |
Schläfli | Coxeter-Dynkin diagrams Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
Vertices Vertex (geometry) In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:... |
Edges Edge (geometry) In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects.... |
Faces Face (geometry) In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube... |
Cells Cell (geometry) In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made... |
4-faces | 5-faces | 6-faces | 7-faces | 8-faces | 9-faces | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | β_{1} |
Line segment Edge (geometry) In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects.... 1-orthoplex |
{} | 2 | |||||||||||||

2 | β_{2}−1 _{11} |
Bicross square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... 2-orthoplex |
{4} {} x {} |
4 | 4 | ||||||||||||

3 | β_{3}0 _{11} |
Tricross octahedron Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... 3-orthoplex |
{3,4} {3 ^{0,1,1}} |
6 | 12 | 8 | |||||||||||

4 | β_{4}1 _{11} |
Tetracross 16-cell 16-cell In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.... 4-orthoplex |
{3,3,4} {3 ^{1,1,1}} |
8 | 24 | 32 | 16 | ||||||||||

5 5-polytope In five-dimensional geometry, a 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two polychoron facets. A proposed name for 5-polytopes is polyteron.-Definition:... |
β_{5}2 _{11} |
Pentacross Pentacross 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 cells, 32 5-cell hypercells.... 5-orthoplex |
{3^{3},4}{3 ^{2,1,1}} |
10 | 40 | 80 | 80 | 32 | |||||||||

6 6-polytope In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera.... |
β_{6}3 _{11} |
Hexacross Hexacross 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 4-faces, and 64 5-faces.... 6-orthoplex |
{3^{4},4}{3 ^{3,1,1}} |
12 | 60 | 160 | 240 | 192 | 64 | ||||||||

7 7-polytope In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.... |
β_{7}4 _{11} |
Heptacross Heptacross In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.... 7-orthoplex |
{3^{5},4}{3 ^{4,1,1}} |
14 | 84 | 280 | 560 | 672 | 448 | 128 | |||||||

8 8-polytope In eight-dimensional geometry, a polyzetton is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.... |
β_{8}5 _{11} |
Octacross Octacross In geometry, an 8-orthoplex, or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.... 8-orthoplex |
{3^{6},4}{3 ^{5,1,1}} |
16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 | ||||||

9 9-polytope In nine-dimensional geometry, a polyyotton is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.... |
β_{9}6 _{11} |
Enneacross 9-orthoplex |
{3^{7},4}{3 ^{6,1,1}} |
18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | |||||

10 10-polytope In ten-dimensional geometry, a 10-polytope is a 10 dimensional polytope contained by 9-polytope facets. Each 8-polytope ridge being shared by exactly two 9-polytope facets.... |
β_{10}7 _{11} |
Decacross 10-orthoplex |
{3^{8},4}{3 ^{7,1,1}} |
20 | 180 | 960 | 3360 | 8064 | 13440 | 15360 | 11520 | 5120 | 1024 | ||||

... | |||||||||||||||||

n | β_{n}k _{11} |
n-cross n-orthoplex |
{3^{n − 2},4}{3 ^{n − 3,1,1}} |
... ... |
2n 0-faces, ... k-faces ..., 2^{n} (n-1)-faces |

The vertices of an axis-aligned cross polytope are all at equal distance from each other in the Manhattan distance

Taxicab geometry

Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the 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 coordinates...

(L

^{1}normLp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

). Kusner's conjecture states that this set of 2d points is the largest possible equidistant set for this distance.

## See also

- List of regular polytopes
- Hyperoctahedral groupHyperoctahedral groupIn 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. Groups of this type are identified by a parameter n, the dimension of the hypercube....

, the symmetry group of the cross-polytope

## External links

- Polytope Viewer (Click
to select cross polytope.)