Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. Ordered geometry is a fundamental geometry forming a common framework for affine geometry, absolute geometry, and Euclidean geometry. Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths Absolute geometry is a Geometry based on an Axiom system that does not assume the Parallel postulate or any of its alternatives Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.
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History
Moritz Pasch first defined a geometry without reference to measurement in 1882. Moritz Pasch (8 November 1843 Breslau, Germany (now Wrocław, Poland) --20 September 1930 Bad Homburg, Germany) was a His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904) [1]. Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Oswald Veblen ( 24 June 1880 in Decorah Iowa &ndash 10 August, 1960) was an American Mathematician, 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" [2].
Primitive concepts
The only primitive notions in ordered geometry are points A, B, C, . In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume . . and the relation of intermediacy [ABC] which can be read as "B is between A and C". This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations
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. Three-dimensional space is a geometric model of the physical Universe in which we live A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex.
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]. In Geometry, Pasch's axiom, is a result of Plane geometry used by Euclid, but yet which cannot be derived from Euclid's postulates.
- Axiom of dimensionality:
- For planar ordered geometry, all points are in one plane. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it 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. Hilbert's axioms are a set of 20 assumptions (originally 21 David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry.
Results
Sylvester's problem of collinear points
The Sylvester-Gallai theorem can be proven within ordered geometry[3].
Parallelism
Gauss, Bolyai, and Lobachevsky developed a notion of parallelism which can be expressed in ordered geometry[4]. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive
Theorem (existence of parallelism): Given a point A and a line r, not through A, there exist exactly two 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[5]. Therefore, the "ordered" concept of parallelism does not form an equivalence relation on lines. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent"
See also
- Incidence geometry
- Euclidean geometry
- Affine geometry
- Absolute geometry
- Non-Euclidean geometry
- Erlangen program
References
- ^ Coxeter, H. An incidence geometry is a Mathematical structure composed of objects of various types and an Incidence relation between them Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Hilbert's axioms are a set of 20 assumptions (originally 21 David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary" that is formulable in Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths Absolute geometry is a Geometry based on an Axiom system that does not assume the Parallel postulate or any of its alternatives In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen S. M. (1969). Introduction to Geometry. New York: John Wiley & Sons, p. 176. ISBN 0471504580.
- ^ Heath, Thomas (1956). The Thirteen Books of Euclid's Elements (Vol 1). New York: Dover Publications, p. 165. ISBN 0486600882.
- ^ Coxeter, H. S. M. (1969). Introduction to Geometry. New York: John Wiley & Sons, p. 181-182. ISBN 0471504580.
- ^ Coxeter, H. S. M. (1969). Introduction to Geometry. New York: John Wiley & Sons, p. 189-190. ISBN 0471504580.
- ^ Bussemann, Herbert (1955). Geometry of Geodesics. New York: Academic Press, p. 139. ISBN 0121483509.
External links
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