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In
geometry Geometry (from the grc, γεωμετρία; ''geo-'' "earth", ''-metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and ...
, a disk (also spelled disc). is the region in a plane bounded by a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. T ...
. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not.


Formulas

In
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the s ...
, the ''open disk'' of center (a, b) and radius ''R'' is given by the formula :D=\ while the ''closed disk'' of the same center and radius is given by :\overline=\. The
area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of ...
of a closed or open disk of radius ''R'' is π''R''2 (see
area of a disk In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.1416. One method of deriving this formula, which originate ...
).


Properties

The disk has
circular symmetry In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the spec ...
. The open disk and the closed disk are not topologically equivalent (that is, they are not
homeomorphic and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function. In the mathematics, mathematical field ...
), as they have different topological properties from each other. For instance, every closed disk is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British N ...
whereas every open disk is not compact. However from the viewpoint of
algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that cla ...
they share many properties: both of them are
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
and so are
homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός ''homós'' "same, similar" and τόπος ''tópos'' "place") if one can be "continuously deformed" i ...
to a single point. This implies that their
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
s are trivial, and all
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. ...
s are trivial except the 0th one, which is isomorphic to Z. The
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
of a point (and therefore also that of a closed or open disk) is 1. Every
continuous map In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to suff ...
from the closed disk to itself has at least one fixed point (we don't require the map to be
bijective In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element ...
or even
surjective In mathematics, a function ''f'' from a set ''X'' to a set ''Y'' is surjective (also known as onto, or a surjection), if for every element ''y'' in the codomain ''Y'' of ''f'', there is at least one element ''x'' in the domain ''X'' of ''f'' such ...
); this is the case ''n''=2 of the
Brouwer fixed point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest f ...
. The statement is false for the open disk:, Ex. 1, p. 135. Consider for example the function f(x,y)=\left(\frac,y\right) which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the closed unit disk it fixes every point on the half circle x^2 + y^2 = 1 , x >0 .


See also

*
Unit disk An open Euclidean unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is t ...
, a disk with radius one *
Annulus (mathematics) 150px, An annulus In mathematics, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''an ...
, the region between two concentric circles *
Ball (mathematics) In mathematics, a ball is the volume space bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not ...
, the usual term for the 3-dimensional analogue of a disk *
Disk algebraIn functional and complex analysis, the disk algebra ''A''(D) (also spelled disc algebra) is the set of holomorphic functions :''f'' : D → C, where D is the open unit disk in the complex plane C, ''f'' extends to a continuous function on the c ...
, a space of functions on a disk *
Orthocentroidal disk In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and its centroid at opposite ends of a diameter. This diameter also contains the triangle's nine-point center and is a subset ...

Orthocentroidal disk
, containing certain centers of a triangle


References

{{DEFAULTSORT:Disk (Mathematics) Euclidean geometry