User:Boris Tsirelson/Sandbox1: Difference between revisions

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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.
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The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.


In 1899 [[David Hilbert]] published his seminal book ''Grundlagen der Geometrie'' [Foundations of Geometry]<ref name="Hilbert">
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
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]].
{{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.

Latest revision as of 03:25, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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