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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 (±1, 0, 0, ..., 0). 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 Rn, those points x = (x1, x2..., xn) satisfying
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 vertex-edge graph of an n-dimensional cross-polytope is the Turán graph T(2n, n) (also known as a cocktail party graph ).
In 1 dimension the cross-polytope is a line segment, which can be chosen as the interval [−1, +1].
In 2 dimensions the cross-polytope is a square. If the vertices are chosen as {(±1, 0), (0, ±1)}, the square's sides are at right angles to the axes; in this orientation a square is often called a diamond.
In 3 dimensions the cross-polytope 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.
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Cross-polytope AI simulator
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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 (±1, 0, 0, ..., 0). 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 Rn, those points x = (x1, x2..., xn) satisfying
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 vertex-edge graph of an n-dimensional cross-polytope is the Turán graph T(2n, n) (also known as a cocktail party graph ).
In 1 dimension the cross-polytope is a line segment, which can be chosen as the interval [−1, +1].
In 2 dimensions the cross-polytope is a square. If the vertices are chosen as {(±1, 0), (0, ±1)}, the square's sides are at right angles to the axes; in this orientation a square is often called a diamond.
In 3 dimensions the cross-polytope 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.