A point is a 0-dimensional mathematical object which can be specified in n-dimensional space using an n-tuple (, , ..., ) consisting of coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors. Although the notion of a point is intuitively rather clear, the mathematical machinery used to deal with points and point-like objects can be surprisingly slippery. This difficulty was encountered by none other than Euclid himself who, in his Elements, gave the vague definition of a point as "that which has no part" (Euclid 1908, Book I, Definition 1).

The basic geometric structures of higher dimensional geometry--the line, plane, space, and hyperspace--are all built up of infinite numbers of points arranged in particular ways.

These facts lead to the mathematical pun, "without geometry, life is pointless."

The decimal point in a decimal expansion is voiced as "point" in the United States, e.g., 3.1415 is voiced "three point one four one five," whereas a comma is used for this purpose in continental Europe.

See also

2-Dimensional Point-Line Distance, 2-Dimensional Point-Point Distance, 3-Dimensional Point-Line Distance, 3-Dimensional Point-Point Distance, Accumulation Point, Boundary Point, Branch Point, Cartesian Coordinates, Comma, Concur, Concurrent, Critical Point, Double Point, Endpoint, Euclidean Space, Fixed Point, Incidence Structure, Isolated Point, Limit Point, Line Line Picking, Midpoint, n-Tuple, n-Vector, Ordinary Point, Point Set, Singular Point Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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References

Casey, J. "The Point." Ch. 1 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin, Ireland: Hodges, Figgis, & Co., pp. 1-29, 1893.Euclid. The Thirteen Books of Euclid's Elements, Vol. 1: Introduction and Books I, II. Translated from the text of Heiberg, with introduction and commentary by T. L. Heath. Cambridge, England: Cambridge University Press, 1908. https://archive.org/details/thirteenbookseu02heibgoog.Lachlan, R. "Special Points Connected with a Triangle." §112-117 in An Elementary Treatise on Modern Pure Geometry. London, England: Macmillan, pp. 62-66, 1893.

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Point

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Point." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Point.html

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