Disk (mathematics)
Disk (mathematics)
Formulas
while the closed disk of the same center and radius is given by
The area of a closed or open disk of radius R is πR2 (see area of a disk).[3]
Properties
The disk has circular symmetry.[4]
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.[5] However from the viewpoint of algebraic topology they share many properties: both of them are contractible[6] 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.[7]
See also
Unit disk, a disk with radius one
Annulus (mathematics), the region between two concentric circles
Ball (mathematics), the usual term for the 3-dimensional analogue of a disk
Disk algebra, a space of functions on a disk
Orthocentroidal disk, containing certain centers of a triangle