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 Two spheres intersect at the same angle at all points of their circle of intersection. They intersect at right angles (are orthogonal) if and only if the squares of the distance between their centers is equal to the sum of the squares of their radii. If and are the equations of two distinct spheres then is also the equation of a sphere for arbitrary values of the parameters and . The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a