Ordered geometry is a form of

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

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, ...

, omitting the basic notion of measurement. 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 relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...

, Euclidean, absolute, and

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

(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 Univer ...

first defined a geometry without reference to measurement in 1882. His axioms were improved upon by

Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...

(1889),

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

(1899), and Veblen (1904).

Euclid Euclid (; grc-gre, Εὐκλείδης; 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 ...

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 and everyday experience. In an ...

s 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 Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...

) ''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 (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

''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 In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...

: ## 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 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 (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

, Bolyai, and

Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...

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.

References

{{Reflist Fields of geometry Order theory