  In the 1990s, J.H. Conway published a combinatorial-geometric method for
analyzing integer-valued binary quadratic forms (BQFs). Using a visualization
he named the "topograph," Conway revisited the reduction of BQFs and the
solution of quadratic Diophantine equations such as Pell's equation. It appears
that the crux of his method is the coincidence between the arithmetic group
$PGL_2({\mathbb Z})$ and the Coxeter group of type $(3,\infty)$. There are many
arithmetic Coxeter groups, and each may have unforeseen applications to
arithmetic. We introduce Conway's topograph, and generalizations to other
arithmetic Coxeter groups. This includes a study of "arithmetic flags" and
variants of binary quadratic forms.
