A Wisdom Archive on Polytope

Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is therefore an n-dimensional manifold with boundary. In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of ...

Read more here: » Simplex: Encyclopedia II - Simplex - Topology

The volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is where each column of the n × n determinant is the difference between two vertices. Any determinant which involves taking the difference between pairs of vertices, where the pairs connect the vertices as a simply connected graph will also give the (same) volume. Without the 1/n! it is the formula for the volume of an n-parallelepiped. One wa ...

Read more here: » Simplex: Encyclopedia II - Simplex - Geometric properties

The standard n-simplex is the subset of Rn+1 given by Removing the restriction ti ≥ 0 in the above gives an n-dimensional affine subspace of Rn+1 containing the standard n-simplex. The vertices of the standard n-simplex are the points e0 = (1, 0, 0, …, 0), e1 = (0, 1, 0, …, 0), e

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Fourth dimension - Vector spaces. In treating space to be akin to a vector space, that is, a set of vectors which we can think of as arrows, fixed from some single place in space which we call the origin (geometric vectors), that point to other places in space, we can look at the following intuitive concepts to build up a definition of dimension. A point is a zero-dimensional object. It has no extension in space, and no properties. If we were to think of this point as a geometric vector, like an arrow, it would have no length. This vector is called the zero vector, ...

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For any (topologically) discrete set S of points in Euclidean space and for almost any point x, there is one point of S to which x is closer than x is to any other point of S. The word "almost" is occasioned by the fact that a point x may be equally close to two or more points of S. If S contains only two points, a and b, then the set of all points equidistant from a and b is a hyperplane — an affine subspace of codimension 1. That hyperplane ...

Read more here: » Voronoi diagram: Encyclopedia II - Voronoi diagram - Definition

The universe that we inhabit seems to be three-dimensional. We can move ourselves and other objects in three dimensions — up/down, left/right and forwards/backwards. We have great difficulty in imagining the existence of a fourth dimension — if you have three axes at right angles to each other, how would you add a fourth at right angles to the existing three? From our previous discussion, visualizing such a four-dimensional space may be difficult, but analyzing mathematically a four-dimensional space is straightforward. We just th ...

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All symmetry groups of regular polytopes are finite Coxeter groups. The dihedral groups, which are the symmetry groups of regular polygons, form the series I2(p). The symmetry group of a regular n-simplex is the symmetric group Sn+1, also known as the Coxeter group of type An. The symmetry group of the n-cube is the same as that of the n-cross-polytope, namely BCn. The symmetry group of the regular dodecahedron and the regular icosahedr ...

Read more here: » Coxeter group: Encyclopedia II - Coxeter group - Symmetry groups of regular polytopes

The affine Weyl groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal abelian subgroup such that the corresponding quotient group is finite. In each case, the quotient group is itself a Weyl group, and the Coxeter graph is obtained from the Coxeter graph of the Weyl group by adding an additional vertex and one or two additional edges. For example, for n ≥ 2, the graph consisting of n+1 vertices in a circle is obtained from An in this wa ...

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Every Weyl group can be realized as a Coxeter group. The Coxeter graph can be obtained from the Dynkin diagram by replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. The example given above corresponds to the Weyl group of the root system of type An. The Weyl groups include most of the finite Coxeter groups, but there are additional examples as well. The following list gives all connect ...

Read more here: » Coxeter group: Encyclopedia II - Coxeter group - Finite Coxeter groups