Cross-polytope
As a cross-polytope (ie, a polytope ) is called in geometry, an n -dimensional, bounded polyhedron, which is combinatorially equivalent to the unit cross-polytope. Cross-polytopes are increasingly well used in both (especially linear ) optimization as well as in crystallography.
The unit cross-polytope
The unit cross-polytope can be represented as follows as a convex hull of its vertices:
It denotes the - th unit vector of the vector space. The unit cross-polytope is convex, closed and connected ( with respect to the Euclidean metric ) and has the following special features:
- It is point-symmetrical with respect to, i.e..
- It is symmetrical with respect to mirroring at the coordinate planes, i.e.
- It has corners, just the ( positive and negative ) unit vectors.
- It has edges, because every corner is out with the opposite corner connected with each other via an edge.
- Is equivalent to the above representation as solution set of the inequality
- This inequality can be rewritten in a system of linear inequalities. It can be seen by the fact that exactly facets ( ie -dimensional bounding hyperplanes ) has.
- Is the (n- dimensional ) volume of the unit Kreuzpolytops. It is arbitrarily small for high dimensions.
- The standard cross-polytope is the unit sphere of the norm with respect to the sum, that is. Herein lies the importance of Kreuzpolytops in crystallography: At the molecular level, some substances have the tendency with respect to the induced by the 1- norm distance possible term " tight" to arrange; the results are crystals in the form of Kreuzpolytopen.
- The two-dimensional cross-polytope is a ( excepted to the tip) square.
- The three-dimensional cross-polytope is also called octahedron and is one of the Platonic solids.
- The coordinate planes divide the unit cross-polytope in the Einheitssimplices.
- The facets of Kreuzpolytops are simplices of.
- The cross-polytope is the prototype of a polyhedron ( in relation to the dimension) has very few corners, but very many facets. This property is particularly important in linear programming, since the simplex algorithm, the standard method for solving linear optimization problems, specifically corners checked for optimality. The counterpart to this is the cube whose edge number exponentially, the number of facets but only linearly increases.
General cross-polytopes
Alternatively, each polyhedron is also called cross-polytope which is combinatorially equivalent to the standard cross-polytope. Precisely formulated, this means:
Thus, a general cross-polytope has the same number of vertices, edges and facets as the standard cross-polytope, but the symmetries are lost.
A more rigorous ( and more geometrically motivated ) Definition of Kreuzpolytops is as follows:
Each cross-polytope according to the second definition meets the first, but not vice versa. Cross-polytopes according to the second definition, see "optical" as the standard of cross-polytope, and the symmetry properties ( the center of symmetry, and mirror layers are correspondingly transformed as ) and the volume of retained formula (except for the extra factor).
The Department of Mathematics at the Technical University of Munich is a three-dimensional cross-polytope in their logo.