User:Boris Tsirelson/Sandbox1: Difference between revisions

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{{Image|Latgen.png|right|480px|Two finite lists of vectors generate (by summation) the same infinite set of vectors. Similarly, different finite lists of axioms can be used for generating the same infinite set of theorems.}}
In [[Euclidean geometry]], a '''line''' (sometimes called, more explicitly, a '''straight line''') is an abstract concept that models the common notion of a (uniform) curve that does not bend, has no thickness and extends infinitely in both directions.  


It is closely related to other basic concepts of geometry, especially, distance: it provides the shortest path between any two of its points. Moreover, in space it can also be described as the intersection of two planes.


In [[Euclidean geometry]], a plane is defined as a flat surface that for any two of its points entirely contains the [[line (geometry)|straight line]] joining them. Until well into the nineteenth century it was thought that the only geometry possible was Euclidean and consequently this definition of   "plane" was considered satisfactory. However, with the birth of non-Euclidean geometry and attention to  the logical foundations of mathematics in the second half of the nineteenth century, doubts arose about the exactness and the limitations of the Euclidean definition of a plane.
It is, however, difficult to give a self-contained definition of straight lines. Assuming an (intuitive or physical) idea of the geometry of a plane, "line" can be defined in terms of distances, orthogonality, coordinates etc. (as we shall do below).  


In 1899 [[David Hilbert]] published his seminal book ''Grundlagen der Geometrie'' [Foundations of Geometry]<ref name="Hilbert">
In a more abstract approach (vector spaces) lines are defined as one-dimensional affine subspaces.
D. Hilbert, ''Grundlagen der Geometrie'', B. G. Teubner, Leipzig (1899) [http://www.archive.org/stream/grunddergeovon00hilbrich#page/n9/mode/2up 2nd German edition]</ref> in which he re-investigated and rephrased Euclid's  two-millennia-old axioms and propositions.  Hilbert begins with listing  undefined concepts, among which are "point", "line", and "plane". In terms of these undefined concepts Hilbert formulates sets of axioms. The first axiom regarding the plane is axiom '''I4''': Three points ''A'', ''B'', ''C'' that are not on one and the same line determine always a plane &alpha;. He adds that this is expressed as "''A'', ''B'', and ''C'' lie in &alpha;", or "''A'', ''B'', and ''C'' are points of &alpha;". His axiom '''I5''' is a subtle extension of I4: Any three points in plane &alpha; that are not on one line determine plane &alpha;.


This article discusses some possible geometrical definitions of a plane as a locus of points and mentions its modern definition as two-dimensional subspace of an [[affine space]].
In an axiomatic approach, "line", together with "point", is a basic concept of elementary geometry. It is an undefined primitive.
{{Image|StreetArtTA.jpg|right|250px|The surface of this metallic body consists of [[rectangle]]s situated in various planes.}}
 
 
 
In Euclidean geometry, a line (sometimes called a straight line) is a straight curve having no thickness and extending infinitely in both directions. Line, together with point, is a basic concept of elementary geometry. It is closely related to other basic concepts, especially, distance: it provides the shortest way between any two of its points. Line can be defined in terms of distances, orthogonality, coordinates etc. In the axiomatic approach it is an undefined primitive. In a more abstract approach a line is defined as a one-dimensional affine subspace.

Revision as of 13:41, 26 May 2010

In Euclidean geometry, a line (sometimes called, more explicitly, a straight line) is an abstract concept that models the common notion of a (uniform) curve that does not bend, has no thickness and extends infinitely in both directions.

It is closely related to other basic concepts of geometry, especially, distance: it provides the shortest path between any two of its points. Moreover, in space it can also be described as the intersection of two planes.

It is, however, difficult to give a self-contained definition of straight lines. Assuming an (intuitive or physical) idea of the geometry of a plane, "line" can be defined in terms of distances, orthogonality, coordinates etc. (as we shall do below).

In a more abstract approach (vector spaces) lines are defined as one-dimensional affine subspaces.

In an axiomatic approach, "line", together with "point", is a basic concept of elementary geometry. It is an undefined primitive.

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