Each Coxeter group G with n generators has an n-dimensional representation which is orthogonal with respect to a canonical bilinear form B(x,y). The associated quadratic
form B(x,x) will be Diophantine when G is right-angled. By construction, for each

integer D, the Coxeter group G acts on the set Sol(D) of primitive integer solutions
to B(x,x)=D and this set admits a partial ordering whose components coincide with the
orbits of the G action.

All of these constructs can be directly defined simply in terms

of the Coxeter graph associated with G. The talk will focus on describing some low
dimensional cases of this construction. Examples tend to have interesting geometric
interpretations involving such concepts as Apollonian circle packings, Pythagorean triples,
and Conway's pictorial realizations of binary forms and their rivers.

Tea is available at 3:30pm in the lobby of Ritter Hall, with the talk to follow at 4:10pm in 202 Ritter Hall.