3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
N-sphere a central point, where the radius may be any positive real number. Thus, the -sphere would be defined by: In particular: An "n"-sphere embedded in an -dimensional Euclidean space is called a hypersphere. The -sphere of unit radius is called the unit -sphere, denoted , often referred to as "the" -sphere. When embedded as described, an -sphere is the surface or boundary of an -dimensional ball. For , the -spheres are the simply connected -dimensional manifolds of constant, positive curvature. The -spheres admit several other topological descriptions: for example, they can be constructed by gluing two -dimensional Euclidean spaces together,