In classical Euclidean geometry
, a point is a primitive notion
that models an exact location in the space
, and has no length, width, or thickness. In modern mathematics
, a point refers more generally to an element
of some set
called a space
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
s, that it must satisfy; for example, ''"there is exactly one line
that passes through two different points"''.
Points in Euclidean geometry
A finite set of points in two-dimensional Euclidean space
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". In two-dimensional Euclidean space
, a point is represented by an ordered pair
(, ) of numbers, where the first number conventionally
represents the horizontal
and is often denoted by , and the second number conventionally represents the vertical
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
t of terms, where is the dimension
of the space in which the point is located.
Many constructs within Euclidean geometry consist of an infinite
collection of points that conform to certain axioms. This is usually represented by a set
of points; As an example, a line
is an infinite set of points of the form
, where through and are constants and is the dimension of the space. Similar constructions exist that define the plane
, line segment
and other related concepts. A line segment consisting of only a single point is called a degenerate
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 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
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:
The topological dimension of a topological space ''X'' is defined to be the minimum value of ''n'', such that every finite open cover
of ''X'' admits a finite open cover
of ''X'' which refines
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
with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set.
Let ''X'' be a metric space
. 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" style="text-decoration: none;"class="mw-redirect" title="infimum.html" style="text-decoration: none;"class="mw-redirect" 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
covering ''S'' with ''ri
'' > 0 for each ''i'' ∈ ''I'' that satisfies
The Hausdorff dimension of ''X'' is defined by
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 functions
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 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
, where electrons are idealized as points with non-zero charge). The Dirac delta function, or 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.
[, 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
or point charge
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).
Its discrete analog is the Kronecker delta
function which is usually defined on a finite domain and takes values 0 and 1.
*Foundations of geometry
*Singular point of a curve
*Whitehead point-free geometry
* 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 College
* Whitehead, A. N., 1979 (1929). ''Process and Reality
''. Free Press.