# neutral geometry

Let $\ell$ be a line in a linear ordered geometry $S$ and let $A,B$ be two subsets on $\ell$. A point $p$ is said to be between $A$ and $B$ if for any pair of points $q\in A$ and $r\in B$, $p$ is between $q$ and $r$. Note that $p$ necessarily lies on $\ell$.

For example, given a ray $\rho$ on a line $\ell$. If $p$ is the source of $\rho$, then $p$ is a point between $\rho$ and its opposite ray $-\rho$, regardless whether the ray is defined to be open or closed. It is easy to see that $p$ is the unique point between $\rho$ and $-\rho$.

Given a line $\ell$, a Dedekind cut on $\ell$ is a pair of subsets $A,B\subseteq\ell$ such that $A\cup B=\ell$ and there is a unique point $p$ between $A$ and $B$. A ray $\rho$ on a line $\ell$ and its compliment $\overline{\rho}$ constitute a Dedekind cut on $\ell$.

If $A,B$ form a Dedekind cut on $\ell$, then $A$ and $B$ have two additional properties:

1. 1.

no point on $A$ is strictly between two points on $B$, and

2. 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. 1.
2. 2.

the continuity axiom: given any line $\ell$ with $\ell=A\cup B$ such that

1. (a)

no point on $A$ is (strictly) between two points on $B$, and

2. (b)

no point on $B$ is (strictly) between two points on $A$.

then $A$ and $B$ constitute a Dedekind cut on $\ell$. In other words, there is a unique point $o$ between $A$ and $B$.

Clearly, $A\cap B$ contains at most one point. The continuity axiom is also known as Dedekind’s Axiom.

Properties.

1. 1.

Let $\ell=A\cup B$ be a line, satisfying (a) and (b) above and let $p\in A$. Suppose $\rho$ lying on $\ell$ is a ray emanating from $p$. Then either $\rho\subseteq A$ or $B\subseteq\rho$.

2. 2.

Let $\ell=A\cup B$ 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 $\ell$ consists of at least one ray.

3. 3.

We can similarly propose a continuity axiom on a ray as follows: given any ray $\rho$ with $\rho=A\cup B$ 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 $\rho$ between $A$ and $B$. It turns out that the two continuity axioms are equivalent      .

4. 4.

Archimedean Property Given two line segments  $\overline{pq}$ and $\overline{rs}$, then there is a unique natural number  $n$ and a unique point $t$, such that

1. (a)

$t$ lies on the line segment $n\cdot\overline{rs}\subseteq\overrightarrow{rs}$,

2. (b)

$t$ does not lie on the line segment $(n-1)\cdot\overline{rs}$, and

3. (c)

$\overline{pq}\cong\overline{rt}$.

This property usually appears in the study of ordered fields.

5. 5.

For any given line $\ell$ and any point $p$, there exists a line $m$ passing through $p$ that is perpendicular  to $\ell$.

6. 6.

Consequently, for any given line $\ell$ and any point $p$ not lying on $\ell$, there exists at leaast one line passing through $p$ that is parallel   to $\ell$. If there is more than one line passing through $p$ parallel to $\ell$, then there are infinitely many of these lines.

Examples.

 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