In combinatorics a polyhedron is the solution set of a finite system of linear inequalities. The solution set is in for integer . Hence, it is a convex set. Each extreme point of such a polyhedron is also called a vertex (or corner point) of the polyhedron. A solution set could be empty. If the solution set is bounded (that is, is contained in some sphere) the polyhedron is said to be bounded.
Careful Treatments of Geometry
In treatments of geometry that are carefully done a definition due to Lennes is sometimes used . The intent is to rule out certain objects that one does not want to consider and to simplify the theory of dissection. A polyhedron is a set of points consisting of a finite set of triangles , not all coplanar, and their interiors such that
every side of a triangle is common to an even number of triangles of the set, and
there is no subset of such that (i) is true of a proper subset of .
Notice that condition (ii) excludes, for example, two cubes that are disjoint. But two tetrahedra having a common edge are allowed. The faces of the polyhedron are the insides of the triangles. Note that the condition that the faces be triangles is not important, since a polygon an be dissected into triangles. Also note since a triangle meets an even number of other triangles, it is possible to meet 4,6 or any other even number of triangles. So for example, a configuration of 6 tetrahedra all sharing a common edge is allowed.
By dissections of the triangles one can create a set of triangles in which no face intersects another face, edge or vertex. If this done the polyhedron is said to be .
A convex polyhedron is one such that all its inside points lie on one side of each of the planes of its faces.
An Euler polyhedron is a set of points consisting of a finite set of polygons, not all coplanar, and their insides such that
each edge is common to just two polygons,
there is a way using edges of from a given vertex to each vertex, and
any simple polygon made up of edges of , divides the polygons of into two sets and such that any way, whose points are on from any point inside a polygon of to a point inside a polygon of , meets .
It is a theorem, proved here (http://planetmath.org/ClassificationOfPlatonicSolids), that for a regular polyhedron, the number of polygons with the same vertex is the same for each vertex and that there are 5 types of regular polyhedra.
Notice that a cone, and a cylinder are not polyhedra since they have “faces” that are not polygons.
In algebraic topology another definition is used:
If is a simplicial complex in , then denotes the union of the elements of , with the subspace topology induced by the topology of . is called a polyhedron. If is a finite complex, then is called a finite polyhedron.
|Date of creation||2013-03-22 12:14:43|
|Last modified on||2013-03-22 12:14:43|
|Last modified by||Mathprof (13753)|