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
Sphere -sphere of unit radius centered at the origin is denoted and is often referred to as "the" -sphere. Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface (which is embedded in 3-dimensional space). The surface area of the unit ()-sphere is where is Euler's gamma function. Another expression for the surface area is and the volume is the surface area times or General recursive formulas also exist for the volume of an -ball. More generally, in a metric space , the sphere of center and radius is the set of points such that . If