Descartes' theorem in space... and beyond ---------------------------------------------------
In the plane, Descartes' theorem *(see the box in the article "Tangent Circles")* gives a relation between the curvatures of four pairwise tangent circles, no three of which share the same tangent line. Does this result generalize to space? It does. More precisely, given five pairwise tangent spheres in space, no three of which share the same tangent plane, we have the following relation:
(∑i=15ki)2=3(∑i=15ki2),\left( \sum_{i=1}^5 k_i \right)^2 = 3 \left( \sum_{i=1}^5 k_i^2 \right) ,
where the values *ki, for i ranging from 1 to 5, are the curvatures* of the five spheres—that is, the reciprocals of their radii.
This result was stated in 1886 by the British mathematician Robert Lachlan (1861–1945). A proof appeared in 1916 in a book by the American mathematician Julian Lowell Coolidge (1873–1954). In 1936, Frederick Soddy mentioned it in the third stanza of his poem The Kiss Precise (see "Récrémaths").