# Point (geometry)

Geometry :
Projecting a sphere to a plane :
Branches
Euclidean
Non-Euclidean
Elliptic
Spherical
Hyperbolic
Non-Archimedean geometry
Projective
Affine
Synthetic
Analytic
Algebraic
Arithmetic
Diophantine
Differential
Riemannian
Symplectic
Discrete differential
Complex
Finite
Discrete/Combinatorial
Digital
Convex
Computational
Fractal
Incidence
Noncommutative geometry
Noncommutative algebraic geometry :
ConceptsFeaturesDimension
Straightedge and compass constructions
Angle
Curve
Diagonal
Orthogonality (Perpendicular)
Parallel
Vertex
Congruence
Similarity
Symmetry :
Zero-dimensional
Point :
One-dimensional
Line
segment
ray
Length :
Two-dimensional
Plane
Area
Polygon
Triangle
Altitude
Hypotenuse
Pythagorean theorem
Parallelogram
Square
Rectangle
Rhombus
Rhomboid
Quadrilateral
Trapezoid
Kite
Circle
Diameter
Circumference
Area :
Three-dimensional
Volume
Cube
cuboid
Cylinder
Dodecahedron
Icosahedron
Octahedron
Pyramid
Platonic Solid
Sphere
Tetrahedron :
Four-/other-dimensional
Tesseract
Hypersphere :
Geometers :
by name
Aida
Aryabhata
Ahmes
Alhazen
Apollonius
Archimedes
Atiyah
Baudhayana
Bolyai
Brahmagupta
Cartan
Chern
Coxeter
Descartes
Euclid
Euler
Gauss
Gromov
Hilbert
Huygens
Jyeṣṭhadeva
Kātyāyana
Khayyám
Klein
Lobachevsky
Manava
Minkowski
Minggatu
Pascal
Pythagoras
Parameshvara
Poincaré
Riemann
Sakabe
Sijzi
al-Tusi
Veblen
Virasena
Yang Hui
al-Yasamin
Zhang
List of geometers :
by period
BCE
Ahmes
Baudhayana
Manava
Pythagoras
Euclid
Archimedes
Apollonius
1–1400s
Zhang
Kātyāyana
Aryabhata
Brahmagupta
Virasena
Alhazen
Sijzi
Khayyám
al-Yasamin
al-Tusi
Yang Hui
Parameshvara
1400s–1700s
Jyeṣṭhadeva
Descartes
Pascal
Huygens
Minggatu
Euler
Sakabe
Aida
1700s–1900s
Gauss
Lobachevsky
Bolyai
Riemann
Klein
Poincaré
Hilbert
Minkowski
Cartan
Veblen
Coxeter
Chern
Present day
Atiyah
Gromov :
vte :
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, 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 axioms, that they must satisfy; for example, "there is exactly one straight line that passes through two distinct points". As physical diagrams, geometric figures are made with tools such as a 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 of two curves or three surfaces, called a vertex or corner. Since the advent of analytic geometry, points are often defined or represented in terms of numerical coordinates. In modern mathematics, a space of points is typically treated as a set, a point set.
An isolated point is an element of some subset of points which has some neighborhood containing no other points of the subset.

## Points 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". In the two-dimensional Euclidean plane, a point is represented by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a$_1$, a$_2$, … , a$_n$) where n 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 $L=\lbrace (a_{1},a_{2},...a_{n})\mid a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d\rbrace ,$where c$_1$ through c$_n$ and d are constants and n 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 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 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: $1\cdot \mathbf {0} =\mathbf {0}$.

### 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 ${\mathcal {A}}$ of $X$ admits a finite open cover ${\mathcal {B}}$ of $X$ which refines ${\mathcal {A}}$ 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.

### Hausdorff dimension

Let X be a metric space. If S ⊂ X and d ∈ [0, ∞), 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 balls $\{B(x_{i},r_{i}):i\in I\}$ covering S with r$_i$ > 0 for each i ∈ I that satisfies $\sum _{i\in I}r_{i}^{d}<\delta .$
The Hausdorff dimension of X is defined by $\operatorname {dim} _{\operatorname {H} }(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.$
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, but via some structure (algebraic or logical respectively) which looks like a well-known function space on the set: an algebra of continuous functions 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 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. 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.