 #jsDisabledContent { display:none; } My Account | Register | Help Flag as Inappropriate This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate?          Excessive Violence          Sexual Content          Political / Social Email this Article Email Address:

# Disk (mathematics)

Article Id: WHEBN0000329542
Reproduction Date:

 Title: Disk (mathematics) Author: World Heritage Encyclopedia Language: English Subject: Collection: Euclidean Geometry Publisher: World Heritage Encyclopedia Publication Date:

### Disk (mathematics)

In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle. A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary.

• Formulas 1
• Properties 2
• References 4

## Formulas

In Cartesian coordinates, the open disk of center (a, b) and radius R is given by the formula

D=\{(x, y)\in {\mathbb R^2}: (x-a)^2+(y-b)^2 < R^2\}

while the closed disk of the same center and radius is given by

\overline{ D }=\{(x, y)\in {\mathbb R^2}: (x-a)^2+(y-b)^2 \le R^2\}.

The area of a closed or open disk of radius R is πR2 (see area of a disk).

## Properties

The Euclidean disk has circular symmetry.

The open disk and the closed disk are not topologically equivalent (that is, they are not homeomorphic), as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact. However from the viewpoint of algebraic topology they share many properties: both of them are contractible and so are homotopy equivalent to a single point. This implies that their fundamental groups are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to Z. The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1.

Every continuous map from the closed disk to itself has at least one fixed point (we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed point theorem. The statement is false for the open disk: consider for example the function

f(x,y)=\left(\frac{x+\sqrt{1-y^2}}{2},y\right)

which maps every point of the open unit disk to another point of the open unit disk slightly to the right of the given one.