polyhedron

At least four definitions of a polyhedron are used.

Combinatorics

In combinatorics a polyhedron is the solution set of a finite systemof linear inequalities. The solution set is in ${ℝ}^n$ for integer$n$. 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 solutionset could be empty. If the solution set is bounded (that is, is contained insome sphere) the polyhedron is said to be bounded.

Elementary Geometry

In elementary geometry a polyhedron is a solid bounded by a finite number of plane faces,each of which is a polygon. This of course is not a precise definition as itrelies on the undefined term “solid”. Also, this definition allows a polyhedronto be non-convex.

Careful Treatments of Geometry

In treatments of geometry that are carefully done a definition due to Lennes issometimes used [2]. The intent is to rule out certain objects that one does not wantto consider and to simplify the theory of dissection.A polyhedron is a set of points consisting of a finite set oftriangles $T$, not all coplanar, and their interiors such that

• every side of a triangle is common to an even number of triangles of theset, and

• there is no subset $T^{\prime }$ of $T$ such that (i) is true of a proper subsetof $T^{\prime }$.

Notice that condition (ii) excludes, for example, two cubes that are disjoint. But twotetrahedra having a common edge are allowed. The faces of the polyhedron are the insidesof the triangles. Note that the condition that the faces be trianglesis 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 whichno face intersects another face, edge or vertex. If this done thepolyhedron is said to be .

A convex polyhedron is one such that all its inside points lie on one side ofeach of the planes of its faces.

An Euler polyhedron $P$ is a set of points consisting of a finite setof polygons, not all coplanar, and their insides such that

• each edge is common to just two polygons,

• there is a way using edges of $P$ from a given vertex to each vertex, and

• any simple polygon $p$ made up of edges of $P$, divides the polygonsof $P$ into two sets $A$ and $B$ such that any way, whose points are on $P$from any point inside a polygon of $A$ to a point inside a polygon of $B$,meets $p$.

A regular polyhedron is a convex Euler polyhedron whose faces are congruentregular polygons and whose dihedral angles are congruent.

It is a theorem, proved here (http://planetmath.org/ClassificationOfPlatonicSolids), that for a regular polyhedron, the number of polygons with the samevertex 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.

A simple polyhedron is one that is homeomorphic to a sphere. For such a polyhedronone has $V-E+F=2$, where $V$ is the number of vertices, $E$ is the number of edgesand $F$ is the number of faces. This is called Euler’s formula.

Algebraic Topology

In algebraic topology another definition is used:

If $K$ is a simplicial complex in ${ℝ}^n$, then $|K|$ denotes the union of the elements of$K$, with the subspace topology induced by the topology of ${ℝ}^n$.$|K|$ is called a polyhedron. If $K$ is a finite complex, then$|K|$ is called a finite polyhedron.

It should be noted that we allow the complex to have an infinite number ofsimplexes. As a result, spaces such as $ℝ$ and${ℝ}^n$ are polyhedra.

Some authors require the simplicial complex to be locally finite.That is, given $x\in \sigma \in K$ there is a neighborhood of $x$ that meets only finitely many $\tau \in K$.