Descartes' Theorem - Generalizations

Generalizations

The generalization to n dimensions is sometimes referred to as the Soddy–Gosset theorem, even though it was shown by R. Lachlan in 1886. In n-dimensional Euclidean space, the maximum number of mutually tangent (n − 1)-spheres is n + 2. For example, in 3-dimensional space, five spheres can be mutually tangent.The curvatures of the hyperspheres satisfy

with the case k_i = 0 corresponding to a flat hyperplane, in exact analogy to the 2-dimensional version of the theorem.

Although there is no 3-dimensional analogue of the complex numbers, the relationship between the positions of the centers can be re-expressed as a matrix equation, which also generalizes to n dimensions.