Point (geometry)

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Geometers
by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan 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
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v t e

In

or corner.

In classical

, 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.

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

Many constructs within Euclidean geometry consist of an

line

is an infinite set of points of the form

$${\displaystyle L=\lbrace (a_{1},a_{2},...a_{n})\mid a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d\rbrace ,}$$

where c₁ through c_n and d are constants and n is the dimension of the space. Similar constructions exist that define the ]

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.[5] 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.^{[citation needed]}

Dimension of a point

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There are several inequivalent definitions of

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

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: ${\displaystyle 1\cdot \mathbf {0} =\mathbf {0} }$.

Topological dimension

The topological dimension of a topological space ${\displaystyle X}$ is defined to be the minimum value of n, such that every finite

${\displaystyle {\mathcal {A}}}$ of ${\displaystyle X}$ admits a finite open cover ${\displaystyle {\mathcal {B}}}$ of ${\displaystyle X}$ which ${\displaystyle {\mathcal {A}}}$ 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

${\displaystyle \{B(x_{i},r_{i}):i\in I\}}$ covering S with r_i > 0 for each i ∈ I that satisfies

$${\displaystyle \sum _{i\in I}r_{i}^{d}<\delta .}$$

The Hausdorff dimension of X is defined by

$${\displaystyle \operatorname {dim} _{\operatorname {H} }(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.}$$

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.

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

function which is usually defined on a finite domain and takes values 0 and 1.

See also

Notes

References

External links

• "Point". PlanetMath.
• Weisstein, Eric W. "Point". MathWorld.