Ordered geometry is a form of

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

featuring the concept of intermediacy (or "betweenness") but, like

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

, omitting the basic notion of

measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to ...

. Ordered geometry is a fundamental geometry forming a common framework for

affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...

, Euclidean,

absolute Absolute may refer to: Companies * Absolute Entertainment, a video game publisher * Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK * Absolute Software Corporation, specializes in security and data risk ma ...

, and

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

(but not for projective geometry).

History

Moritz Pasch Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish ancestry specializing in the foundations of geometry. He completed his Ph.D. at the Uni ...

first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889),

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad ...

(1899), and Veblen (1904).

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

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 notion In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to Intuition (knowledge), intuition or taken ...

s in ordered geometry are

points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...

''A'', ''B'', ''C'', ... and the

ternary relation In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relatio ...

of intermediacy 'ABC''which can be read as "''B'' is between ''A'' and ''C''".

Definitions

The ''segment'' ''AB'' is the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

''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 Pambuccian (2011).

Results

Sylvester's problem of collinear points

The

Sylvester–Gallai theorem The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, ...

can be proven within ordered geometry.

Parallelism

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

, 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

on lines.

References

{{Reflist Fields of geometry Order theory