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# polyhedron

At least four definitions of a polyhedron are used.

# Combinatorics

In combinatorics a *polyhedron* is the solution set of a finite system
of linear inequalities. The solution set is in ${\mathbbmss{R}}^{n}$ 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 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*.

# Elementary Geometry

# Careful Treatments of Geometry

In treatments of geometry that are carefully done a definition due to Lennes is sometimes used [2]. 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 $T$, not all coplanar, and their interiors such that

- (i)
every side of a triangle is common to an even number of triangles of the set, and

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

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 *normal*.

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* $P$ is a set of points consisting of a finite set
of *polygons*, not all coplanar, and their insides such that

- (i)
each edge is common to just

*two*polygons, - (ii)
there is a way using edges of $P$ from a given vertex to each vertex, and

- (iii)
any simple polygon $p$ made up of edges of $P$, divides the polygons of $P$ 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 congruent
regular polygons and whose dihedral angles are congruent.

It is a theorem, proved here, 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.

A *simple polyhedron* is one that is homeomorphic to a sphere. For such a polyhedron
one has $V-E+F=2$, where $V$ is the number of vertices, $E$ is the number of edges
and $F$ 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 ${\mathbbmss{R}}^{n}$, then $|K|$ denotes the union of the elements of
$K$, with the subspace topology induced by the topology of ${\mathbbmss{R}}^{n}$.
$|K|$ 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 of simplexes. As a result, spaces such as $\mathbbmss{R}$ and ${\mathbbmss{R}}^{n}$ 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 $\tau\in K$.

# References

- 1
Henry George Forder,
*The Foundations of Euclidean Geometry*, Dover Publications, New York , 1958. - 2
N.J. Lennes,
*On the simple finite polygon and polyhedron*, Amer. J. Math. 33, (1911), p. 37

## Mathematics Subject Classification

51M20*no label found*57Q05

*no label found*

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## Corrections

edges by pahio ✘

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subsubsection by CWoo ✓