neutral geometry

Dedekind Cuts. Let $\mathrm{\ell }$ be a line in a linear orderedgeometry $S$ and let $A,B$ be two subsets on $\mathrm{\ell }$. A point $p$ issaid 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 $p$ necessarilylies on $\mathrm{\ell }$.

For example, given a ray $\rho$ on a line $\mathrm{\ell }$. If $p$ is thesource of $\rho$, then $p$ is a point between $\rho$ and itsopposite ray $-\rho$, regardless whether the ray is defined to beopen or closed. It is easy to see that $p$ is the unique pointbetween $\rho$ and $-\rho$.

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

If $A,B$ form a Dedekind cut on $\mathrm{\ell }$, 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 orderedgeometry satisfying

1. the congruence axioms, and

2. the continuity axiom:given any line $\mathrm{\ell }$ with $\mathrm{\ell }=A\cup B$ such that

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

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

   then $A$ and $B$ constitute a Dedekind cut on $\mathrm{\ell }$. In otherwords, 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. Let $\mathrm{\ell }=A\cup B$ be a line, satisfying (a) and (b)above and let $p\in A$. Suppose $\rho$ lying on $\mathrm{\ell }$ is a rayemanating from $p$. Then either $\rho \subseteq A$ or $B\subseteq \rho$.

2. Let $\mathrm{\ell }=A\cup B$ be a line, satisfying (a) and (b)above and let $o$ be the unique point as mentioned above. Then aclosed ray emanating from $o$ is either $A$ or $B$.This implies that every Dedekind cut on a line$\mathrm{\ell }$ consists of at least one ray.

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

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

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

   This property usually appears in the study of ordered fields.

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

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

Examples.

• A Euclidean geometry is a neutral geometry satisfying the Euclid’sparallel axiom: for any given line and any given point not lying onthe line, there is a unique line passing through the point andparallel to the given line.

• A hyperbolic geometry (or Bolyai-Lobachevsky geometry) is a neutralgeometry satisfying the hyperbolic axiom: for any given line and anygiven 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 thefirst letter of each of the above examples by the definitearticle “the”. It can be shown that any two Euclidean geometriesare geometrically isomorphic (preserving incidence, order,congruence, and continuity). Similarly, any two hyperbolicgeometries are isomorphic. Such geometries are said to becategorical.

• 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