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).
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".
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 [ABC].
- 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.
Sylvester's problem of collinear points
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.
- Incidence geometry
- Euclidean geometry
- Affine geometry
- Absolute geometry
- Non-Euclidean geometry
- Erlangen program
- Coxeter (1969) p.176
- Heath, Thomas (1956) . The Thirteen Books of Euclid's Elements (Vol 1). New York: Dover Publications. p. 165. ISBN 0-486-60088-2.
- Pambuccian, Victor (2011). "The axiomatics of ordered geometry: I. Ordered incidence spaces". Expositiones Mathematicae 29: 24–66. doi:10.1016/j.exmath.2010.09.004.
- Coxeter (1969) pp.181–182
- Pambuccian, Victor (2009). "A Reverse Analysis of the Sylvester–Gallai Theorem". Notre Dame Journal of Formal Logic 50: 245–260. doi:10.1215/00294527-2009-010.
- Coxeter (1969) pp.189–190
- Bussemann, Herbert (1955). Geometry of Geodesics. New York: Academic Press. p. 139. ISBN 0-12-148350-9.
- Coxeter, H.S.M. (1969). Introduction to geometry (2nd ed.). John Wiley and Sons. ISBN 0-471-18283-4. Zbl 0181.48101.