The previous chapter made precise the ideas of a lattice \(\mathcal {L}(d,h)\) , where we take the divergence d and the rise h as parameters, and the principal parastichy pair as the pair of integers corresponding to the two shortest informative vectors in the lattice. In this chapter we solve the problem of classifying every lattice by its principal parastichy pair. The answer is given in the van Iterson diagram of Fig. 5.1 and the rest of the chapter explains how this Figure is constructed and its implications. Since the classification changes at the points of lattice space where either the first and second, or the second and third, parastichy vectors are equal in length, we need to find the d and h at which this occurs. This requires us to identify touching-circle lattices, and we will first construct Fig. 5.1 by algebraically finding touching-circle lattices, and then see how a more powerful and more abstract approach based on lattice re-normalisation gives the same answer and explains much of the geometric structure of the diagram.