Integral binary Hamiltonian forms and their waterworlds

We give a graphical theory of integral indefinite binary Hamiltonian forms f analogous to the one of Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order \mathscr {O} in a definite quaternion algebra over \mathbb {Q}, we define the waterworld of f, analogous to Conway’s river and Bestvina-Savin’s ocean , and use it to give a combinatorial description of the values of f on \mathscr {O}\times \mathscr {O}. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), the \operatorname {SL}_{2}(\mathscr {O})-equivariant Ford-Voronoi cellulation of the real hyperbolic 5-space, and the conformal action of \operatorname {SL}_{2}(\mathscr {O}) on the Hamilton quaternions.