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 The shortest distance along the surface between two distinct non-antipodal points on the sphere is the length of the smaller of the two arcs on the unique great circle that includes the two points. Equipped with this "great-circle distance", a great circle becomes the Riemannian circle. If a particular point on a sphere is (arbitrarily) designated as its "north pole", then the corresponding antipodal point is called the "south pole", and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the