Ordered geometry

Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry).

History

Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of ''The Elements'': "a straight line is a line which lies evenly with the points on itself".

Primitive concepts

The only primitive notions in ordered geometry are points ''A'', ''B'', ''C'', ... and the ternary relation of intermediacy 'ABC''which can be read as "''B'' is between ''A'' and ''C''".

Definitions

The ''segment'' ''AB'' is the set of points ''P'' such that 'APB''

The ''interval'' ''AB'' is the segment ''AB'' and its end points ''A'' and ''B''.

The ''ray'' ''A''/''B'' (read as "the ray from ''A'' away from ''B''") is the set of points ''P'' such that 'PAB''

The ''line'' ''AB'' is the interval ''AB'' and the two rays ''A''/''B'' and ''B''/''A''. Points on the line ''AB'' are said to be ''collinear''.

An ''angle'' consists of a point ''O'' (the ''vertex'') and two non-collinear rays out from ''O'' (the ''sides'').

A ''triangle'' is given by three non-collinear points (called ''vertices'') and their three ''segments'' ''AB'', ''BC'', and ''CA''.

If three points ''A'', ''B'', and ''C'' are non-collinear, then a ''plane'' ''ABC'' is the set of all points collinear with pairs of points on one or two of the sides of triangle ''ABC''.

If four points ''A'', ''B'', ''C'', and ''D'' are non-coplanar, then a ''space'' ( 3-space) ''ABCD'' is the set of all points collinear with pairs of points selected from any of the four ''faces'' (planar regions) of the tetrahedron ''ABCD''.

Axioms of ordered geometry

# There exist at least two points.
# If ''A'' and ''B'' are distinct points, there exists a ''C'' such that BC
# If 'ABC'' then ''A'' and ''C'' are distinct (''A'' ≠ ''C'').
# If 'ABC'' then 'CBA''but not 'CAB''
# If ''C'' and ''D'' are distinct points on the line ''AB'', then ''A'' is on the line ''CD''.
# If ''AB'' is a line, there is a point ''C'' not on the line ''AB''.
# ( Axiom of Pasch) If ''ABC'' is a triangle and 'BCD''and 'CEA'' then there exists a point ''F'' on the line ''DE'' for which 'AFB''
# Axiom of dimensionality:
## For planar ordered geometry, all points are in one plane. Or
## If ''ABC'' is a plane, then there exists a point ''D'' not in the plane ''ABC''.
# All points are in the same plane, space, etc. (depending on the dimension one chooses to work within).
# (Dedekind's Axiom) For every partition of all the points on a line into two nonempty sets such that no point of either lies between two points of the other, there is a point of one set which lies between every other point of that set and every point of the other set.

These axioms are closely related to Hilbert's axioms of order. For a comprehensive survey of axiomatizations of ordered geometry see.

Results

Sylvester's problem of collinear points

The Sylvester–Gallai theorem can be proven within ordered geometry.

Parallelism

Gauss, Bolyai, and Lobachevsky developed a notion of parallelism which can be expressed in ordered geometry.

Theorem (existence of parallelism): Given a point ''A'' and a line ''r'', not through ''A'', there exist exactly two limiting rays from ''A'' in the plane ''Ar'' which do not meet ''r''. So there is a ''parallel'' line through ''A'' which does not meet ''r''.

Theorem (transmissibility of parallelism): The parallelism of a ray and a line is preserved by adding or subtracting a segment from the beginning of a ray.

The transitivity of parallelism cannot be proven in ordered geometry. Therefore, the "ordered" concept of parallelism does not form an equivalence relation on lines.

See also

* Incidence geometry
* Euclidean geometry
** Hilbert's axioms
** Tarski's axioms
* Affine geometry
* Absolute geometry
* Non-Euclidean geometry
* Erlangen program
* Cyclic order
* Separation relation

References

{{Reflist

Fields of geometry
Order theory