polyhedron

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

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

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 it
relies on the undefined term “solid”. Also, this definition allows a polyhedron
to be non-convex.

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

′

of

T

such that (i) is true of a

proper subset

of

T

′

.

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

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 (

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.

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

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 of

simplexes

. 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

∈

σ

∈

K

there is a

neighborhood

of

x

that meets only finitely many

τ

∈

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

Title

polyhedron

Canonical

name

Polyhedron

Date of creation

2013-03-22 12:14:43

Last modified on

2013-03-22 12:14:43

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

24

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51M20

Classification

msc 57Q05

Related topic

RegularPolygon

Related topic

Polytope

Related topic

Diagonal

Related topic

CubicallyThinHomotopy

Defines

vertex

Defines

corner point

Defines

finite polyhedron

Defines

locally finite

Defines

polyhedra

Defines

bounded polyhedron

Defines

normal polyhedron

Defines

regular polyhedron

Defines

Euler polyhedron

Defines

convex polyhedron

Defines

simple polyhedron

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