polygon

polygon

1

Definitions

We follow Forder

[

2

]

for most of this entry.
The term

polygon

can be defined if one has a

definition

of an

interval

. For this
entry we use betweenness

geometry

. A betweenness geometry
is just one for which there is a set of points and a

betweenness relation

B

defined.
Rather than write

(

a

,

b

,

c

)

∈

B

we write

a

*

b

*

c

.

1.

If

a

and

b

are distinct points, the

line

a

⁢

b

is the set of
all points

p

such that

p

*

a

*

b

or

a

*

p

*

b

or

a

*

b

*

p

. It can be shown
that the line

a

⁢

b

and the line

b

⁢

a

are the same set of points.

2.

If

o

and

a

are distinct points, a

ray

[

o

a

is the set of all points

p

such that

p

=

o

or

o

*

p

*

a

or

o

*

a

*

p

.

3.

If

a

and

b

are distinct points, the

open interval

is the set of points

p

such that

a

*

p

*

b

. It is denoted by

(

a

,

b

)

.

4.

If

a

and

b

are distinct points, the

closed interval

is

(

a

,

b

)

∪

{

a

}

∪

{

b

}

, and denoted by

[

a

,

b

]

.

5.

The

way

a

1

⁢

a

2

⁢

…

⁢

a

n

is the

finite set

of points

{

a

1

,

…

,

a

n

}

along with the open intervals

(

a

1

,

a

2

)

,

(

a

2

,

a

3

)

,

…

,

(

a

n

-

1

,

a

n

)

.
The points

a

1

,

…

,

a

n

are called the

vertices

of the way, and the
open intervals are called the

sides

of the way.
A way is also called a

broken line

.
The closed intervals

[

a

1

,

a

2

]

,

…

,

[

a

n

-

1

,

a

n

]

are called the

side-intervals

of
the way. The lines

a

1

⁢

a

2

,

…

,

a

n

-

1

⁢

a

n

are called the

side-lines

of the way.
The way

a

1

⁢

a

2

⁢

…

⁢

a

n

is said to

join

a

1

to

a

n

.
It is assumed that

a

i

-

1

,

a

i

,

a

i

+

1

are not

collinear

.

6.

A way is said to be

simple

if it does not meet itself. To be precise,
(i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervals
meet in any point.

7.

A

polygon

is a way

a

1

⁢

a

2

⁢

…

⁢

a

n

for which

a

1

=

a

n

. Notice that there is
no

assumption

that the points are

coplanar

.

8.

A

simple polygon

is polygon for which the way is simple.

9.

A

region

is a set of points not all collinear, any two of which can be joined by points of a way using
only points of the region.

10.

A region

R

is

convex

if for each pair of points

a

,

b

∈

R

the open interval

(

a

,

b

)

is

contained

in

R

.

11.

Let

X

and

Y

be two sets of points. If there is a set of points

S

such that every way
joining a point of

X

to a point of

Y

meets

S

then

S

is said to

separate

X

from

Y

.

12.

If

a

1

⁢

a

2

⁢

…

⁢

a

n

is a polygon, then the

angles of the polygon

are

∠

⁢

a

n

⁢

a

1

⁢

a

2

,

∠

⁢

a

1

⁢

a

2

⁢

a

3

, and so on.

Now assume that all points of the geometry are in one plane. Let

P

be a polygon. (

P

is called
a

plane polygon

.)

1.

A ray or line which does not go through a vertex of

P

will be called

suitable

.

2.

An

inside point

a

of

P

is one for which a suitable ray from

a

meets

P

an

odd number

of times. Points that are not on or inside

P

are said to be

outside

P

.

3.

Let

{

P

i

}

be a set of polygons. We say that

{

P

i

}

dissect

P

if the following
three conditions are satisfied: (i)

P

i

and

P

j

do not have a common inside point for

i

≠

j

,
(ii) each inside point of

P

is inside or on some

P

i

and (iii) each inside point of

P

i

is
inside

P

.

4.

A

convex polygon

is one whose inside points are all on the same side of any side-line
of the polygon.

2

Theorems

Assume that all points are in one plane. Let

P

be a polygon.

1.

It can be shown that

P

separates the other points of the plane into at least two regions and that
if

P

is simple there are exactly two regions. Moise proves this directly in

[

3

]

, pp. 16-18.

2.

It can be shown that

P

can be dissected into

triangles

{

T

i

}

such that
every vertex of a

T

i

is a vertex of

P

.

3.

The following

theorem

of Euler can be shown: Suppose

P

is dissected into

f

>

1

polygons
and that the total number of vertices of these polygons is

v

, and the number of open intervals
which are sides is

e

. Then

v

-

e

+

f

=

1

.

A plane simple polygon with

n

sides is called an

n

-gon, although for small

n

there are more traditional names:

Number of sides

Name of the polygon

3

triangle

4

quadrilateral

5

pentagon

6

hexagon

7

heptagon

8

octagon

10

decagon

A plane simple polygon is also called a

Jordan polygon

.

References

1

K. Borsuk and W. Szmielew,

Foundations of Geometry

,
North-Holland Publishing Company, 1960.

2

H.G. Forder,

The

Foundations

of

Euclidean Geometry

,
Dover Publications, 1958.

3

E.E. Moise,

Geometric Topology in

Dimensions

2 and 3

,
Springer-Verlag, 1977.

Title

polygon

Canonical

name

Polygon

Date of creation

2013-03-22 12:10:15

Last modified on

2013-03-22 12:10:15

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

43

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51-00

Classification

msc 51G05

Related topic

RegularPolygon

Related topic

Semiperimeter

Related topic

EquilateralPolygon

Related topic

EquiangularPolygon

Related topic

Pentagon

Related topic

BasicPolygon

Related topic

Hexagon

Related topic

GeneralizedPythagoreanTheorem

Defines

side

Defines

vertex

Defines

vertices

Defines

simple polygon

Defines

side-lines

Defines

ray

Defines

simple way

Defines

way

Defines

region

Defines

convex region

Defines

Jordan polygon

Defines

angles of a polygon

Defines

plane polygon

Defines

broken line

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