Geometry |
---|

Geometers |

This article needs additional citations for verification .(March 2022) |

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.^{ [1] } In modern mathematics, a **point** refers more generally to an element of some set called a space.

- Points in Euclidean geometry
- Dimension of a point
- Vector space dimension
- Topological dimension
- Hausdorff dimension
- Geometry without points
- Point masses and the Dirac delta function
- See also
- References
- External links

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 axioms, that it must satisfy; for example, *"there is exactly one line that passes through two different points"*.

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 (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 , 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.

There are several inequivalent definitions of dimension in mathematics. In all of the common definitions, a point is 0-dimensional.

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 is defined to be the minimum value of *n*, such that every finite open cover of admits a finite open cover of 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* ∈ [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 covering *S* with *r _{i}* > 0 for each

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.

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*.

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.^{ [2] }^{ [3] }^{ [4] } 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.^{ [5] } 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).^{ [6] } Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1.

**Euclidean space** is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's *Elements*, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the *Euclidean plane*. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

In mathematics, **Hausdorff dimension** is a measure of *roughness*, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the *Hausdorff–Besicovitch dimension.*

In mathematics, a **metric space** is a non-empty set together with a metric on the set. The metric is a function that defines a concept of *distance* between any two members of the set, which are usually called points. The metric satisfies the following properties:

In mathematics, the **Dirac delta distribution**, also known as the **unit impulse** symbol, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.

**Distance** is a numerical measurement of how far apart objects or points are. In Physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria. The distance from a point A to a point B is sometimes denoted as . In most cases, "distance from A to B" is interchangeable with "distance from B to A". In mathematics, a distance function or metric is a generalization of the concept of physical distance; it is a way of describing what it means for elements of some space to be "close to", or "far away from" each other. In psychology and social sciences, distance is a non-numerical measurement; Psychological distance is defined as "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality.

In mathematics, the **real line**, or **real number line** is the line whose points are the real numbers. That is, the real line is the set **R** of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

In mathematics, a **curve** is an object similar to a line, but that does not have to be straight.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, **homogeneous coordinates** or **projective coordinates**, introduced by August Ferdinand Möbius in his 1827 work *Der barycentrische Calcul*, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.

In mathematics, an **affine space** is a geometric 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 only the properties related to parallelism and ratio of lengths for parallel line segments.

In mathematics, **Hausdorff measure** is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space.

In mathematics, **Sard's theorem**, also known as **Sard's lemma** or the **Morse–Sard theorem**, is a result in mathematical analysis that asserts that the set of critical values of a smooth function *f* from one Euclidean space or manifold to another is a null set, i.e., it has Lebesgue measure 0. This makes the set of critical values "small" in the sense of a generic property. The theorem is named for Anthony Morse and Arthur Sard.

In mathematics, a **Dirac measure** assigns a size to a set based solely on whether it contains a fixed element *x* or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.

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 *P* in space in relation to an arbitrary reference origin *O*. Usually denoted **x**, **r**, or **s**, it corresponds to the straight line segment from *O* to *P*. In other words, it is the displacement or translation that maps the origin to *P*:

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.

In mathematics, a **space** is a set with some added structure.

In mathematics, **tightness** is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity".

In mathematics, the **support** of a measure *μ* on a measurable topological space is a precise notion of where in the space *X* the measure "lives". It is defined to be the largest (closed) subset of *X* for which every open neighbourhood of every point of the set has positive measure.

In mathematics, the notion of an (**exact**) **dimension function** is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple "diameter to the dimension" power law used in the construction of *s*-dimensional Hausdorff measure.

**Two-dimensional Euclidean space** or simply **two-dimensional space** is a geometric setting in which two values are required to determine the position of an element on the plane. The set of pairs of real numbers with appropriate structure often serves as the canonical example of a **Euclidean plane**, the two-dimensional Euclidean space; for a generalization of the concept, see dimension. Two-dimensional space can be seen as a projection of the physical universe onto a plane. Usually, it is thought of as a Euclidean space and the two dimensions are called length and width.

- ↑ Ohmer, Merlin M. (1969).
*Elementary Geometry for Teachers*. Reading: Addison-Wesley. p. 34–37. OCLC 00218666. - ↑ Dirac 1958 , §15 The δ function , p. 58
- ↑ Gel'fand & Shilov 1968 , Volume I, §§1.1, 1.3
- ↑ Schwartz 1950 , p. 3
- ↑ Arfken & Weber 2000 , p. 84
- ↑ Bracewell 1986 , Chapter 5

- 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.

Wikimedia Commons has media related to Points (mathematics) . |

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.