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In classical
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, a point is a primitive notion that models an exact location in the
space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gre ...
, and has no length, width, or thickness. In modern
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a point refers more generally to an
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
of some set called a
space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gre ...
. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s, that it must satisfy; for example, ''"there is exactly one
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

that passes through two different points"''.

# Points in Euclidean geometry

Points, considered within the framework of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, are one of the most fundamental objects.
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

originally defined the point as "that which has no part". In two-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, a point is represented by an
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

(, ) of numbers, where the first number conventionally represents the horizontal and is often denoted by , and the second number conventionally represents the
vertical Vertical may refer to: * Vertical direction In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects ...
and is often denoted by . This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (, , ) with the additional third number representing depth and often denoted by . Further generalizations are represented by an ordered
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
t of terms, where is the
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
of the space in which the point is located. Many constructs within Euclidean geometry consist of an
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...
is an infinite set of points of the form $\scriptstyle$, where through and are constants and is the dimension of the space. Similar constructions exist that define the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
,
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

and other related concepts. A line segment consisting of only a single point is called a
degenerate Degeneracy may refer to: Science * Codon degeneracy * Degeneracy (biology), the ability of elements that are structurally different to perform the same function or yield the same output * Degeneration (medical) ** Degenerative disease, a diseas ...
line segment. 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. 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.

# Dimension of a point

There are several inequivalent definitions of
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
in mathematics. In all of the common definitions, a point is 0-dimensional.

## Vector space dimension

The dimension of a vector space is the maximum size of a
linearly independent In the theory of vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
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: $1 \cdot \mathbf=\mathbf$.

## Topological dimension

The topological dimension of a topological space $X$ is defined to be the minimum value of ''n'', such that every finite
open cover In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
$\mathcal$ of $X$ admits a finite open cover $\mathcal$ of $X$ which refines $\mathcal$ 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 In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical il ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. If ''S'' ⊂ ''X'' and ''d'' ∈ ,_∞),_the_''d''-dimensional_Hausdorff_content_of_''S''_is_the_infimum_of_the_set_of_numbers_δ_≥_0_such_that_there_is_some_(indexed)_collection_of_metric_space.html" "title="infimum.html" ;"title=", ∞), the ''d''-dimensional Hausdorff content of ''S'' is the infimum">, ∞), the ''d''-dimensional Hausdorff content of ''S'' is the infimum of the set of numbers δ ≥ 0 such that there is some (indexed) collection of metric space">balls A ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( p ...
$\$ covering ''S'' with ''ri'' > 0 for each ''i'' ∈ ''I'' that satisfies $\sum_ r_i^d<\delta$. The Hausdorff dimension of ''X'' is defined by :$\operatorname_\left(X\right):=\inf\.$ 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s or an
algebra of sets In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
respectively. More precisely, such structures generalize well-known spaces of
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
in a way that the operation "take a value at this point" may not be defined. A further tradition starts from some books of A. N. Whitehead in which the notion of
region In geography Geography (from Ancient Greek, Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and Solar System, planets. The ...
is assumed as a primitive together with the one of ''inclusion'' or ''connection''.

# Point masses and the 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 Classical electromagnetism or classical electrodynamics is a branch of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and ...
, where electrons are idealized as points with non-zero charge). The Dirac delta function, or function, is (informally) a
generalized function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
on the real number line that is zero everywhere except at zero, with an
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of one over the entire real line., p. 58 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 A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, l ...

or
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, l ...
. It was introduced by theoretical physicist
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. Dirac made fundamental contributions to the early develop ...

. In the context of
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

it is often referred to as the unit impulse symbol (or function). Its discrete analog is the
Kronecker delta In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
function which is usually defined on a finite domain and takes values 0 and 1.

*
Accumulation point In mathematics, a limit point (or cluster point or accumulation point) of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood (mathematics), neighbourhood ...
*
Affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
*
Boundary point In topology and mathematics in general, the boundary of a subset ''S'' of a topological space ''X'' is the set of points which can be approached both from ''S'' and from the outside of ''S''. More precisely, it is the set of points in the closure ...
* Critical point *
Cusp Cusp may refer to: Mathematics *Cusp (singularity) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
*
Foundations of geometryFoundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometry, non-Euclidean geometries. These are fundamental to the study and of histor ...
*
Position (geometry) In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a Point (geometry), point ''P'' in space#Classical mechanics, space in relation to an arbitrary refer ...
*
Point cloud A point cloud image of a torus A point cloud is a set of data points in space. The points represent a 3D shape or object. Each point has its set of X, Y and Z coordinates. Point clouds are generally produced by 3D scanners or by photogrammetry so ...
*
Point process In statistics and probability theory, a point process or point field is a collection of Point (mathematics), mathematical points randomly located on a mathematical space such as the real line or Euclidean space.Olav Kallenberg, Kallenberg, O. (1986) ...
*
Point set registration In computer vision, pattern recognition, and robotics, point set registration, also known as point cloud registration or scan matching, is the process of finding a spatial mathematical transformation, transformation (''e.g.,'' Scaling (geometry), sc ...
*
PointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...
*
Singular point of a curveIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

# References

* Clarke, Bowman, 1985,
Individuals and Points
" ''Notre Dame Journal of Formal Logic 26'': 61–75. * De Laguna, T., 1922, "Point, line and surface as sets of solids," ''The Journal of Philosophy 19'': 449–61. * Gerla, G., 1995,
Pointless Geometries
in Buekenhout, F., Kantor, W. eds., ''Handbook of incidence geometry: buildings and foundations''. North-Holland: 1015–31. * Whitehead, A. N., 1919. ''An Enquiry Concerning the Principles of Natural Knowledge''. Cambridge Univ. Press. 2nd ed., 1925. * Whitehead, A. N., 1920.
The Concept of Nature
'. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at
Trinity CollegeTrinity College may refer to: Australia * Trinity Anglican College, an Anglican Church of Australia, Anglican coeducational primary and secondary school in , New South Wales * Trinity Catholic College, Auburn, a coeducational school in the inner-we ...
. * Whitehead, A. N., 1979 (1929). ''
Process and Reality ''Process and Reality'' is a book by Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical scho ...

''. Free Press.