Point (geometry)

Encyclopedia

In geometry

, topology

and related branches of mathematics a

upon which other concepts may be defined. In geometry, points are zero-dimensional

; i.e., they do not have volume

, area

, length

, or any other higher-dimension

al analogue. In branches of mathematics dealing with set theory

, an element is often referred to as a

, where they are one of the fundamental objects. Euclid

originally defined the point vaguely, 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 tuplet of n terms, 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 , where through and are constants and n is the dimension of the space. Similar constructions exist that define the plane

, line segment

and other related concepts.

In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line. This is easily confirmed under modern expansions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's postulation of points was neither complete nor definitive, as he occasionally assumed facts about points that didn't 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.

.

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 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 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 point is also a location and it has no size.

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

and related branches of mathematics a

**spatial point**is a primitive notionPrimitive notion

In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or...

upon which other concepts may be defined. In geometry, points are zero-dimensional

Zero-dimensional space

In mathematics, a zero-dimensional topological 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...

; i.e., they do not have volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

, area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

, length

Length

In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

, or any other higher-dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

al analogue. In branches of mathematics dealing with set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, an element is often referred to as a

**point**. A point could also be defined as a sphere which has a diameter of zero.## Points in Euclidean geometry

Points are most often considered within the framework of Euclidean geometryEuclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, where they are one of the fundamental objects. Euclid

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

originally defined the point vaguely, as "that which has no part". In two-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, a point is represented by an ordered pair

Ordered pair

In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...

, , of numbers, where the first number conventionally

Convention (norm)

A convention is a set of agreed, stipulated or generally accepted standards, norms, social norms or criteria, often taking the form of a custom....

represents the horizontal

Horizontal plane

In geometry, physics, astronomy, geography, and related sciences, a plane is said to be horizontal at a given point if it is perpendicular to the gradient of the gravity field at that point— in other words, if apparent gravity makes a plumb bob hang perpendicular to the plane at that point.In...

and is often denoted by , and the second number conventionally represents the vertical

Vertical direction

In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i.e., with the direction of the gravitational force at that point...

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 tuplet of n terms, where n is the dimension of the space in which the point is located.

Many constructs within Euclidean geometry consist of an infinite

Infinity

Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line

Line (mathematics)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

is an infinite set of points of the form , where through and are constants and n is the dimension of the space. Similar constructions exist that define the plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

, line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

and other related concepts.

In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line. This is easily confirmed under modern expansions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's postulation of points was neither complete nor definitive, as he occasionally assumed facts about points that didn't 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.

## Points in branches of mathematics

A point in point-set topology is defined as a member of the underlying set of a topological spaceTopological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

.

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

Noncommutative geometry

Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

and pointless topology

Pointless topology

In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...

. A “pointless space” is defined not as a set, but via some structure (algebraic or logical

Complete Heyting algebra

In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...

respectively) which looks like a well-known function space on the set: an algebra of continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

s or an algebra of sets

Algebra of sets

The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion...

respectively. More precisely, such structures generalize well-known spaces of functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

in a way that the operation “take a value at this point” may not be defined. A point is also a location and it has no size.

## See also

- Accumulation point
- Affine spaceAffine spaceIn mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
- Boundary point
- Critical pointCritical point (mathematics)In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
- CuspCusp (singularity)In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
- Singular point

## External links

- Definition of Point with interactive applet
- Points definition pages, with interactive animations that are also useful in a classroom setting. Math Open Reference