neutral geometry


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

For example, given a ray ρ on a line . 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 , a Dedekind cut on is a pair of subsets A,B such that AB= and there is a unique point p between A and B. A ray ρ on a line and its compliment ρ¯ constitute a Dedekind cut on .

If A,B form a Dedekind cut on , 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 with =AB 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 . In other words, there is a unique point o between A and B.

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

Properties.

  1. 1.

    Let =AB be a line, satisfying (a) and (b) above and let pA. Suppose ρ lying on is a ray emanating from p. Then either ρA or Bρ.

  2. 2.

    Let =AB 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 consists of at least one ray.

  3. 3.

    We can similarly propose a continuity axiom on a ray as follows: given any ray ρ with ρ=AB 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

  4. 4.

    Archimedean Property Given two line segmentsMathworldPlanetmath pq¯ and rs¯, then there is a unique natural numberMathworldPlanetmath n and a unique point t, such that

    1. (a)

      t lies on the line segment nrs¯rs,

    2. (b)

      t does not lie on the line segment (n-1)rs¯, and

    3. (c)

      pq¯rt¯.

    This property usually appears in the study of ordered fields.

  5. 5.

    For any given line and any point p, there exists a line m passing through p that is perpendicularPlanetmathPlanetmath to .

  6. 6.

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

Examples.

  • A Euclidean geometryMathworldPlanetmath 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, congruencePlanetmathPlanetmath, 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