There are 16 different ways in which four different spheres may be made tangent to a fifth sphere. When the six planes of a tetrahedron are extended, three dimensional space is partitioned into 15 separate regions, including that space within the tetrahedron. Equation 2 is of degree higher than first, and solution of it will produce multiple roots. This is true since two intersecting spheres produce a circle, a third intersecting sphere produces two points, and the fourth intersecting sphere contains one of these points. A Professor of Engineering and a Professor of Mathematics, friends at the same prestigious University, encounter each other on campus one day.
Several days pass, and they meet again.
A few more days pass, and they meet again.