  We develop a geometric approach to spin networks with Heisenberg or XX
coupling. Geometry is acquired by defining a distance on the discrete set of
spins. The key feature of the geometry of such networks is their Gauss
curvature $\kappa$, viewed here as the ability to isometrically embed the chain
in the standard Riemannian manifold of curvature $\kappa$. Here we focus on
spin rings. Even though their visual geometry is trivial, it turns out that the
geometry they acquire from the quantum mechanical distance is far from trivial.
