Ordered geometry
Encyclopedia
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).
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".
s in ordered geometry are points
A, B, C, ... and the relation
of intermediacy [ABC] which can be read as "B is between A and C".
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.
These axioms are closely related to Hilbert's axioms of order. For a comprehensive survey of axiomatizations of ordered geometry see.
, 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 symmetry of parallelism cannot be proven in ordered geometry. Therefore, the "ordered" concept of parallelism does not form an equivalence relation
on lines.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
featuring the concept of intermediacy (or "betweenness") but, like projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine
Affine geometry
In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
, Euclidean
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, absolute
Absolute geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832...
, and hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
(but not for projective geometry).
History
Moritz PaschMoritz Pasch
Moritz Pasch was a German mathematician specializing in the foundations of geometry. He completed his Ph.D. at the University of Breslau at only 22 years of age...
first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano
Giuseppe Peano
Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. 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 standard axiomatization of the natural numbers is named the Peano axioms in...
(1889), Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
(1899), and Veblen
Oswald Veblen
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...
(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 notionPrimitive notion
In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or...
s in ordered geometry are points
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
A, B, C, ... and the relation
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
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
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
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.
These axioms are closely related to Hilbert's axioms of order. For a comprehensive survey of axiomatizations of ordered geometry see.
Sylvester's problem of collinear points
The Sylvester-Gallai theorem can be proven within ordered geometry.,Parallelism
GaussCarl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...
, and Lobachevsky developed a notion of parallelism
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
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 symmetry of parallelism cannot be proven in ordered geometry. Therefore, the "ordered" concept of parallelism does not form an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
on lines.
See also
- Incidence geometry
- Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
- Hilbert's axiomsHilbert's axiomsHilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...
- Tarski's axiomsTarski's axiomsTarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory . Other modern axiomizations of Euclidean geometry are those by Hilbert and George...
- Hilbert's axioms
- Affine geometryAffine geometryIn mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
- Absolute geometryAbsolute geometryAbsolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832...
- Non-Euclidean geometryNon-Euclidean geometryNon-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
- Erlangen programErlangen programAn influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...