View source for Point (geometry)

{{Short description|Fundamental object of geometry}}
{{More citations needed|date=March 2022}}
[[Image:ACP 3.svg|thumb|A finite set of points (in red) in the [[Euclidean plane]].]]
{{General geometry |0d/1d}}

In [[geometry]], a '''point''' is an abstract idealization of an exact [[position (geometry)|position]], without size, in [[physical space]],{{sfnp|Ohmer|1969|p=34&ndash;37}} or its generalization to other kinds of mathematical [[space (mathematics)|space]]s. As [[zero-dimensional]] objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which [[one-dimensional]] [[curve]]s, [[two-dimensional]] [[surface (mathematics)|surface]]s, and [[higher-dimensional]] objects consist. 

In classical [[Euclidean geometry]], a point is a [[primitive notion]], defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called [[axiom]]s, that they must satisfy; for example, ''"there is exactly one [[straight line]] that passes through two distinct points"''. As physical diagrams, [[geometric figure]]s are made with tools such as a [[compass (drawing tool)|compass]], [[scriber]], or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve.

A point can also be determined by the [[intersection (geometry)|intersection]] of two curves or three surfaces, called a ''[[vertex (geometry)|vertex]]'' or ''corner''.
Since the advent of [[analytic geometry]], points are often defined or represented in terms of numerical [[coordinate system|coordinates]]. In modern mathematics, a space of points is typically treated as a [[set (mathematics)|set]], a '''point set'''.

An ''[[isolated point]]'' is an element of some subset of points which has some [[neighbourhood (mathematics)|neighborhood]] containing no other points of the subset.

==Points in Euclidean geometry{{anchor|In Euclidean geometry}}==

Points, considered within the framework of [[Euclidean geometry]], are one of the most fundamental objects. [[Euclid]] originally defined the point as "that which has no part".{{sfnp|Heath|1956|p=153}} In the two-dimensional [[Euclidean plane]], a point is represented by an [[ordered pair]] ({{mvar|x}}, {{mvar|y}}) of numbers, where the first number [[Convention (norm)|conventionally]] represents the [[Horizontal plane|horizontal]] and is often denoted by {{mvar|x}}, and the second number conventionally represents the [[Vertical direction|vertical]] and is often denoted by {{mvar|y}}. This idea is easily generalized to three-dimensional [[Euclidean space]], where a point is represented by an ordered triplet ({{mvar|x}}, {{mvar|y}}, {{mvar|z}}) with the additional third number representing depth and often denoted by {{mvar|z}}. Further generalizations are represented by an ordered [[tuple]]t of {{mvar|n}} terms, {{math|(''a''<sub>1</sub>, ''a''<sub>2</sub>, … , ''a''<sub>''n''</sub>)}} where {{mvar|n}} is the [[dimension (mathematics)|dimension]] of the space in which the point is located.{{sfnp|Silverman|1969|p=7}}

Many constructs within Euclidean geometry consist of an [[infinity|infinite]] collection of points that conform to certain axioms. This is usually represented by a [[Set (mathematics)|set]] of points; As an example, a [[line (mathematics)|line]] is an infinite set of points of the form
<math display="block"> L = \lbrace (a_1,a_2,...a_n) \mid a_1c_1 + a_2c_2 + ... a_nc_n = d \rbrace,</math>where {{math|''c''<sub>1</sub>}} through {{math|''c<sub>n</sub>''}} and {{mvar|d}} are constants and {{mvar|n}} is the dimension of the space. Similar constructions exist that define the [[plane (geometry)|plane]], [[line segment]], and other related concepts.{{sfnp|de Laguna|1922}} A line segment consisting of only a single point is called a [[degeneracy (mathematics)|degenerate]] line segment.{{citation needed|date=February 2023}}

In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, that any two points can be connected by a straight line.{{sfnp|Heath|1956|p=154}} This is easily confirmed under modern extensions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts known at the time. However, Euclid's postulation of points was neither complete nor definitive, and he occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions.<ref>{{Citation |title=Hilbert's axioms |date=2024-09-24 |work=Wikipedia |url=https://en.wikipedia.org/wiki/Hilbert's_axioms |access-date=2024-09-29 |language=en}}</ref>

==Dimension of a point==
{{Unreferenced section|date=March 2022}}
There are several inequivalent definitions of [[dimension (mathematics and physics)|dimension]] in mathematics. In all of the common definitions, a point is 0-dimensional.

=== Vector space dimension ===
{{Main|Dimension (vector space)}}

The dimension of a vector space is the maximum size of a [[linearly independent]] subset. In a vector space consisting of a single point (which must be the zero vector '''0'''), there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non-trivial linear combination making it zero: <math>1 \cdot \mathbf{0}=\mathbf{0}</math>.

===Topological dimension===
{{Main|Lebesgue covering dimension}}

The topological dimension of a topological space <math>X</math> is defined to be the minimum value of ''n'', such that every finite [[open cover]] <math>\mathcal{A}</math> of <math>X</math> admits a finite open cover <math>\mathcal{B}</math> of <math>X</math> which [[refinement (topology)|refines]] <math>\mathcal{A}</math> in which no point is included in more than ''n''+1 elements. If no such minimal ''n'' exists, the space is said to be of infinite covering dimension.

A point is [[zero-dimensional space|zero-dimensional]] with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set.

=== Hausdorff dimension ===
Let ''X'' be a [[metric space]]. If {{math|''S'' ⊂ ''X''}} and {{math|''d'' ∈ [0, ∞)}}, the ''d''-dimensional '''Hausdorff content''' of ''S'' is the [[infimum]] of the set of numbers {{math|''δ'' ≥ 0}} such that there is some (indexed) collection of [[metric space|balls]] <math>\{B(x_i,r_i):i\in I\}</math> covering ''S'' with {{math|''r<sub>i</sub>'' > 0}} for each {{math|''i'' ∈ ''I''}} that satisfies <math display="block">\sum_{i\in I} r_i^d<\delta. </math>

The '''Hausdorff dimension''' of ''X'' is defined by
<math display="block">\operatorname{dim}_{\operatorname{H}}(X):=\inf\{d\ge 0: C_H^d(X)=0\}.</math>

A point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius.

==Geometry without points==
Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e.g. [[noncommutative geometry]] and [[pointless topology]]. A "pointless" or "pointfree" space is defined not as a [[set (mathematics)|set]], but via some structure ([[C*-algebra|algebraic]] or [[complete Heyting algebra|logical]] respectively) which looks like a well-known function space on the set: an algebra of [[continuous function]]s or an [[algebra of sets]] respectively. More precisely, such structures generalize well-known spaces of [[Function (mathematics)|functions]] in a way that the operation "take a value at this point" may not be defined.{{sfnp|Gerla|1995}} A further tradition starts from some books of [[A. N. Whitehead]] in which the notion of [[region (mathematics)|region]] is assumed as a primitive together with the one of ''inclusion'' or ''connection''.<ref>{{harvs|last=Whitehead|year=1919|year2=1920|year3=1929|txt}}.</ref>

==Point masses and the Dirac delta function ==
{{Main|Dirac delta function}}

Often in physics and mathematics, it is useful to think of a point as having non-zero mass or charge (this is especially common in [[classical electromagnetism]], where electrons are idealized as points with non-zero charge). The '''Dirac delta function''', or '''{{mvar|δ}} function''', is (informally) a [[generalized function]] on the real number line that is zero everywhere except at zero, with an [[integral]] of one over the entire real line.{{sfnmp|1a1 = Dirac|1y = 1958|1p = 58|1loc = More specifically, see §15. The δ function|2a1 = Gelfand|2a2 = Shilov|2y = 1964|2pp = 1&ndash;5|2loc=See §§1.1, 1.3|3a1 = Schwartz|3y = 1950|3p = 3}} The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized [[point mass]] or [[point charge]].{{sfnp|Arfken|Weber|2005|p=84}} It was introduced by theoretical physicist [[Paul Dirac]]. In the context of [[signal processing]] it is often referred to as the '''unit impulse symbol''' (or function).{{sfnp|Bracewell|1986|loc=Chapter 5}} Its discrete analog is the [[Kronecker delta]] function which is usually defined on a finite domain and takes values 0 and 1.

==See also==
{{Div col|colwidth=25em}}
*[[Accumulation point]]
*[[Affine space]]
*[[Boundary point]]
*[[Critical point (mathematics)|Critical point]]
*[[Cusp (singularity)|Cusp]]
*[[Event (relativity)]]
*[[Foundations of geometry]]
*[[Position (geometry)]]
*[[Point at infinity]]
*[[Point cloud]]
*[[Point process]]
*[[Point set registration]]
*[[Pointwise]]
*[[Singular point of a curve]]
*[[Whitehead point-free geometry]]
{{Div col end}}

==Notes==
{{reflist|colwidth=30em}}

==References==
{{refbegin|30em}}
*{{cite book
 |last1            = Arfken | first1 = George B. |author-link1 = George B. Arfken
 |last2            = Weber | first2 = Hans J. |author-link2 = Hans Weber (mathematician)
 |year             = 2005
 |title            = Mathematical Methods For Physicists International Student Edition
 |url              = https://books.google.com/books?id=tNtijk2iBSMC&pg=PA83
 |edition          = 6th
 |publisher        = Academic Press| isbn = 978-0-08-047069-6 }}
*{{cite book
  |last              = Bracewell |first = Ronald N. | author-link = Ronald Bracewell
  |title             = The Fourier transform and its applications
  |edition           = 3rd
  |publisher         = McGraw-Hill Series
  |publication-place = New York
  |year              = 1986
  |isbn              = 0-07-007015-6}} 
*{{cite journal
  |last      = Clarke |first = Bowman
  |year      = 1985
  |url       = http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1093870761
  |title     = Individuals and Points
  |journal   = Notre Dame Journal of Formal Logic
  |volume    = 26
  |issue     = 1
  |pages     = 61–75}}
*{{cite journal
 |last       = de Laguna |first = T. |author-link = Theodore de Laguna
 |year       = 1922
 |title      = Point, line and surface as sets of solids
 |journal    = The Journal of Philosophy
 |volume     = 19
 |issue      = 17
 |pages      = 449–461
 |jstor      = 2939504
 |doi        = 10.2307/2939504}}
*{{cite book
  |last      = Dirac | first = Paul |author-link = Paul Dirac
  |title     = The Principles of Quantum Mechanics
  |year      = 1958
  |url       = https://books.google.com/books?id=XehUpGiM6FIC&pg=PA58
  |edition   = 4th
  |publisher = Oxford University Press
| isbn = 978-0-19-852011-5 }}
*{{cite book
 |last1 = Gelfand | first1 = Israel | author-link1 = Israel Gelfand
 |last2 = Shilov | first2 = Georgiy | author-link2 = Georgiy Shilov
 |title = Generalized Functions: Properties and Operations
 |year = 1964
 |volume = 1
 |url = https://archive.org/details/gelfand-shilov-generalized-functions-vol-1-properties-and-operations/page/5/mode/2up
 |publisher = Academic Press
 |isbn = 0-12-279501-6}}
*{{cite book
 |last             = Gerla |first = G
 |year             = 1995
 |contribution-url = https://web.archive.org/web/20110717210751/http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf
 |contribution     = Pointless Geometries
 |editor1-last     = Buekenhout |editor1-first = F.
 |editor2-last     = Kantor |editor2-first = W
 |title            = Handbook of Incidence Geometry: Buildings and Foundations
 |publisher        = North-Holland
 |pages             = 1015–1031}}
*{{cite book
 |last        = Heath |first = Thomas L.
 |author-link = Thomas Little Heath
 |title       = The Thirteen Books of Euclid's Elements
 |volume      = 1
 |edition     = 2nd
 |year        = 1956
 |publisher   = Dover Publications
 |location    = New York
 |isbn        = 0-486-60088-2
 |url         = https://archive.org/details/thirteenbooksofe00eucl
}}
*{{cite book
 |last        = Ohmer |first = Merlin M.
 |title       = Elementary Geometry for Teachers
 |url         = https://archive.org/details/elementarygeomet00ohme/page/34
 |url-access  = registration
 |location    = Reading
 |publisher   = Addison-Wesley
 |year        = 1969
 |oclc        = 00218666}}
*{{cite book
  |last = Schwartz |first = Laurent | author-link = Laurent Schwartz
  |title = Théorie des distributions
  |volume = 1
  |url = https://books.google.com/books?id=Io_ruAEACAAJ
  |language = French
  |year = 1950}}
*{{cite book
 |last = Silverman |first = Richard A.
 |title = Modern Calculus and Analytic Geometry
 |url = https://books.google.com/books?id=DcWHAwAAQBAJ&pg=PA7
 |year = 1969
 |publisher = Macmillan|isbn = 978-0-486-79398-6
 }}
*{{cite book
 |last       = Whitehead |first = A. N. |author-link = Alfred North Whitehead
 |year       = 1919
 |title      = An Enquiry Concerning the Principles of Natural Knowledge
 |publisher  = Cambridge: University Press
 |url        = https://archive.org/details/enquiryconcernpr00whitrich/mode/2up}}
*{{cite book
  |last = Whitehead |first = A. N.
  |year = 1920
  |url  = http://www.gutenberg.org/files/18835/18835-h/18835-h.htm
  |title = The Concept of Nature
  |publisher = Cambridge: University Press}}. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at [[Trinity College, Cambridge|Trinity College]].
*{{cite book
 |last       = Whitehead |first = A. N
 |year       = 1929
 |title      = Process and Reality: An Essay in Cosmology
 |title-link = Process and Reality
 |publisher = Free Press}}
{{refend}}

==External links==
{{Commons category|Points (mathematics)}}
*{{PlanetMath reference|urlname=Point|title=Point}}
*{{MathWorld |title=Point |id=Point}}
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[[Category:Point (geometry)| ]]