We give a graphical theory of integral indefinite binary Hamiltonian forms
f
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
O
\mathscr {O}
in a definite quaternion algebra over
Q
\mathbb {Q}
, we define the waterworld of
f
f
, analogous to Conway’s river and Bestvina-Savin’s ocean, and use it to give a combinatorial description of the values of
f
f
on
O
×
O
\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
SL
2
(
O
)
\operatorname {SL}_{2}(\mathscr {O})
-equivariant Ford-Voronoi cellulation of the real hyperbolic
5
5
-space, and the conformal action of
SL
2
(
O
)
\operatorname {SL}_{2}(\mathscr {O})
on the Hamilton quaternions.