neutral geometry

neutral geometry

Dedekind Cuts

.

Let

ℓ

be a line in a

linear ordered
geometry

S

and let

A

,

B

be two subsets on

ℓ

. A point

p

is
said to be between

A

and

B

if for any pair of points

q

∈

A

and

r

∈

B

,

p

is between

q

and

r

. Note that

p

necessarily
lies on

ℓ

.

For example, given a ray

ρ

on a line

ℓ

. If

p

is the
source of

ρ

, then

p

is a point between

ρ

and its
opposite ray

-

ρ

, regardless whether the ray is defined to be
open or closed. It is easy to see that

p

is the unique point
between

ρ

and

-

ρ

.

Given a line

ℓ

, a Dedekind cut on

ℓ

is a pair of subsets

A

,

B

⊆

ℓ

such that

A

∪

B

=

ℓ

and there is a unique
point

p

between

A

and

B

. A ray

ρ

on a line

ℓ

and
its compliment

ρ

¯

constitute a Dedekind cut on

ℓ

.

If

A

,

B

form a Dedekind cut on

ℓ

, then

A

and

B

have two additional properties:

1.

no point on

A

is

strictly between

two points on

B

, and

2.

no point on

B

is strictly between two points on

A

.

Conversely

, if

A

,

B

satisfy

the above two conditions, can we say that

A

and

B

constitute a Dedekind cut? In a

neutral geometry

, the answer is yes.

Neutral Geometry.

A neutral geometry is a linear ordered
geometry satisfying

1.

the

congruence axioms

, and

2.

the

continuity axiom

:
given any line

ℓ

with

ℓ

=

A

∪

B

such that

(a)

no point on

A

is (strictly) between two points on

B

, and

(b)

no point on

B

is (strictly) between two points on

A

.

then

A

and

B

constitute a Dedekind cut on

ℓ

. In other
words, there is a unique point

o

between

A

and

B

.

Clearly,

A

∩

B

contains at most one point. The continuity axiom is also known as Dedekind’s Axiom.

Properties.

1.

Let

ℓ

=

A

∪

B

be a line, satisfying (a) and (b)
above and let

p

∈

A

. Suppose

ρ

lying on

ℓ

is a ray
emanating from

p

. Then either

ρ

⊆

A

or

B

⊆

ρ

.

2.

Let

ℓ

=

A

∪

B

be a line, satisfying (a) and (b)
above and let

o

be the unique point as mentioned above. Then a

closed ray

emanating from

o

is either

A

or

B

.
This implies that every Dedekind cut on a line

ℓ

consists of at least one ray.

3.

We can similarly propose a continuity axiom on a ray as follows: given any ray

ρ

with

ρ

=

A

∪

B

such that

–

no point on

A

is strictly between two points on

B

, and

–

no point on

B

is strictly between two points on

A

.

then there is a unique point

o

on

ρ

between

A

and

B

.
It turns out that the two continuity axioms are

equivalent

.

4.

Archimedean Property

Given two

line segments

p

⁢

q

¯

and

r

⁢

s

¯

, then there is a unique

natural number

n

and a unique point

t

, such that

(a)

t

lies on the line segment

n

⋅

r

⁢

s

¯

⊆

r

⁢

s

→

,

(b)

t

does not lie on the line segment

(

n

-

1

)

⋅

r

⁢

s

¯

, and

(c)

p

⁢

q

¯

≅

r

⁢

t

¯

.

This property usually appears in the study of

ordered fields

.

5.

For any given line

ℓ

and any point

p

, there exists a line

m

passing through

p

that is

perpendicular

to

ℓ

.

6.

Consequently, for any given line

ℓ

and any point

p

not lying on

ℓ

, there exists at leaast one line passing through

p

that is

parallel

to

ℓ

. If there is more than one line passing through

p

parallel to

ℓ

, then there are infinitely many of these lines.

Examples

.

•

A

Euclidean geometry

is a neutral geometry satisfying the Euclid’s
parallel axiom: for any given line and any given point not lying on
the line, there is a unique line passing through the point and
parallel to the given line.

•

A

hyperbolic geometry

(or Bolyai-Lobachevsky geometry) is a neutral
geometry satisfying the hyperbolic axiom: for any given line and any
given point not lying on the line, there are at least two distinct (hence infinitely many)
lines passing through the point and parallel to the given line.

•

In fact, one can replace the

indefinite article

“a” in the
first letter of each of the above examples by the

definite
article

“the”. It can be shown that any two Euclidean geometries
are geometrically

isomorphic

(preserving incidence, order,

congruence

, and continuity). Similarly, any two hyperbolic
geometries are isomorphic. Such

geometries

are said to be

categorical

.

•

An elliptic geometry is not a neutral geometry, because pairwise distinct

parallel lines

do not exist.

Title

neutral geometry

Canonical name

NeutralGeometry

Date of creation

2013-03-22 15:33:49

Last modified on

2013-03-22 15:33:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51F10

Classification

msc 51F05

Synonym

absolute geometry

Synonym

Dedekind axiom

Defines

hyperbolic axiom

Defines

Bolyai-Lobachevsky geometry

Defines

continuity axiom

Defines

categorical

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