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In [[Euclidean geometry]] a plane is defined as a flat surface in which a straight line joining any of its two points lies entirely on that surface. Until the middle of 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 limitations of the Euclidean definition of plane. This article will discuss some of the reasons for these doubts and give an introduction to the modern definition.
{{AccountNotLive}}
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.


==Non-axiomatic approach==
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).
 
By lines we mean straight lines.
 
Lines are treated both in plane geometry and in solid geometry.
Plane geometry (called also "planar geometry") is a part of solid geometry that restricts itself to a single plane ("the plane") treated as a geometric universe.
In other words, plane geometry is the theory of the two-dimensional Euclidean space, while solid geometry is the theory of the three-dimensional Euclidean space.
 
===Definitions===
 
====A remark====
 
To define a line is more complicated than it may seem.
 
It is tempting to define a line as a curve of zero curvature, where a curvature is defined as a geometric object having length but no breadth or depth. However, this is not a good idea; such definitions are useless in mathematics, since they cannot be used when proving theorems. Straight lines are treated by elementary geometry, but the notions of curvatures and curvature are not elementary, they need more advanced mathematics and more sophisticated definitions. Fortunately, it is possible to define a line via more elementary notions, and this way is preferred in mathematics. Still, the definitions given below are tentative. They are criticized afterwards, see [[#Axiomatic approach|axiomatic approach]].
 
Four equivalent definitions of plane are given below. Any other definition is equally acceptable provided that it is equivalent to these. Note that a part of a line is not a plane. In particular, a line segment is not a line.
 
The first definition (via betweenness) works both in plane geometry and in solid geometry. The other three definitions apply in plane geometry only.
 
====Definition via betweenness====
 
First we define betweenness via distances. A point ''B'' is said to lie between points ''A'' and ''C'' if <math>|AB|+|BC|=|AC|</math>.
 
Now we define a line as a set of points that satisfies the two conditions:
*If three points belong to the given set then at least one of them lies between the others.
*If one of three points lies between the two others, and at least two of the three points belong to the given set, then the third point also belongs to the given set.
 
====Another remark====
 
''Below, all points and lines are situated in the plane (assumed to be a two-dimensional [[Euclidean space]]).
 
The definition of line given below may be compared with the definition of [[Circle (mathematics)|circle]] as consisting of those points in a plane that are a given distance (the radius) away from a given point (the center). A circle is a set of points chosen according to their relation to some given parameters (center and radius). Similarly, a line is a set of points chosen according to their relation to some given objects (points, lines etc). However, a circle determines its center and radius uniquely; for a line, the situation is different.
 
====Definition via distances====
 
Let two different points ''A'' and ''B'' be given. The set of all points ''C'' that are equally far from ''A'' and ''B'' (that is, <math>|AC|=|BC|</math>) is a line.
 
This is the line orthogonal to the line ''AB'' through the middle point of the line segment ''AB''.
 
====Definition via Cartesian coordinates====
 
In terms of [[Cartesian coordinates]] ''x'', ''y'' ascribed to every point of the plane, a line is the set of points whose coordinates satisfy the [[linear equation]] <math>ax+by=c</math>. Here real numbers ''a'', ''b'' and ''c'' are parameters such that at least one of ''a'', ''b'' does not vanish.
 
===Some properties of lines===
 
====Most basic properties====
 
For every two different points there exists one and only one line that contains these two points.
 
Every line contains at least two points.
 
There exist three points not lying on a line.
 
For every line and every point outside the line there exists one and only one line through the given point which does not intersect the given line.
 
====Further properties====
 
Two lines in a plane either do not intersect (are parallel), or intersect in a single point, or coincide.
 
Two lines perpendicular to the same line are parallel to each other (or coincide).
 
==Axiomatic approach==
 
===What is wrong with the definitions given above?===
 
The definitions given above assume implicitly that the Euclidean plane (or alternatively the 3-dimensional Euclidean space) is already defined, together with such notions as distances and/or Cartesian coordinates, while lines are not defined yet. However, this situation never appears in mathematics.
 
In the axiomatic approach points, lines are ''undefined primitives''.
 
The [[#Modern approach|modern approach]] (below) defines lines in a completely different way.
 
===How does it work===
 
Axiomatic approach is similar to [[chess]] in the following aspect.
 
A chess piece, say a [[Rook (chess)|rook]], cannot be defined before the whole chess game is defined, since such a phrase as "the rook moves horizontally or vertically, forward or back, through any number of unoccupied squares" makes no sense unless it is already known that "chess is played on a square board of eight rows and eight columns" etc. And conversely, the whole chess game cannot be defined before each piece is defined; the properties of the rook are an indispensable part of the rules of the game. No chess without rooks, no rooks outside chess! One must introduce the game, its pieces and their properties in a singe combined definition.
 
Likewise, Euclidean space, its points, lines, planes and their properties are introduced simultaneously in a set of 20 assumptions known as Hilbert's axioms of Euclidean geometry. The "most basic properties of planes" listed above are roughly the plane-related assumptions (Hilbert's axioms), while "further properties" are first plane-related consequences (theorems).
 
==Modern approach==
 
The modern approach defines the three-dimensional Euclidean space more algebraically, via [[Vector space|linear spaces]] and quadratic forms, namely, as an [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[space (mathematics)]].
 
In this approach a plane in an ''n''-dimensional affine space (''n'' &ge; 2)  is defined as a (proper or improper) ''two-dimensional affine subspace''.
 
A less formal version of this approach uses points, vectors and [[scalar product]] (called also [[dot product]] or [[inner product]]) of vectors without mentioning linear and affine spaces. Optionally, Cartesian coordinates of points and vectors are used. See [[#Algebraic equations|algebraic equations]] below. There, in particular, equivalence between the definition via right angles (orthogonality) and the definition via Cartesian coordinates is explained.
 
==Plane geometry==
 
 
A plane figure is a combination of points and/or lines that fall on the same plane. In plane geometry every figure is plane, in contrast to solid geometry.
 
A rectilinear figure is a plane figure consisting of points, straight lines and straight line segments only. Rectilinear figures include [[triangle]]s and [[polygon]]s.
 
==Beyond mathematics==
 
In industry, a [http://en.wikipedia.org/wiki/Surface_plate surface plate] is a piece of cast iron or other appropriate material whose surface (or rather a part of it) is made as close as possible to a geometric plane. An old method of their manufacturing is the ''three-plate method'': three roughly flat surfaces become more and more flat when rubbing against each other: first and second; second and third; third and first; first and second again, and so on. It is possible to achieve a surface close to a plane up to 10<sup>–5</sup> of its size.

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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