  We propose a simple direct-sum method for the efficient evaluation of lattice
sums in periodic solids. It consists of two main principles: i) the creation of
a supercell that has the topology of a Clifford torus, which is a flat, finite
and border-less manifold; ii) the renormalization of the distance between two
points on the Clifford torus by defining it as the Euclidean distance in the
embedding space of the Clifford torus. Our approach does not require any
integral transformations nor any renormalization of the charges. We illustrate
our approach by applying it to the calculation of the Madelung constants of
ionic crystals. We show that the convergence towards the system of infinite
size is monotonic, which allows for a straightforward extrapolation of the
Madelung constant. We are able to recover the Madelung constants with a
remarkable accuracy, and at an almost negligible computational cost, i.e., a
few seconds on a laptop computer.
