  The Khuri-Jones correction to the partial wave scattering amplitude at
threshold is an automorphic function for a dihedron. An expression for the
partial wave amplitude is obtained at the pole which the upper half-plane maps
on to the interior of semi-infinite strip. The Lehmann ellipse exists below
threshold for bound states. As the system goes from below to above threshold,
the discrete dihedral (elliptic) group of Type 1 transforms into a Type 3
group, whose loxodromic elements leave the fixed points 0 and $\infty$
invariant. The transformation of the indifferent fixed points from -1 and +1 to
the source-sink fixed points 0 and $\infty$ is the result of a finite resonance
width in the imaginary component of the angular momentum. The change in
symmetry of the groups, and consequently their tessellations, can be used to
distinguish bound states from resonances.
