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 and is called the equation of an imaginary sphere. If formula_5 the only solution of formula_6 is the point formula_7 and the equation is said to be the equation of a point sphere. Finally, in the case formula_8, formula_6 is an equation of a sphere whose center is formula_10 and whose radius is formula_11. If in the above equation is zero then is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity. The points on the sphere with radius formula_12 and center formula_13 can