Exact Finite Size Results for the Ising Model on the Tetrahedron

We propose an approach to statistical systems on lattices with sphere-like topology. Focusing on the Ising model, we consider the thermodynamic limit along a sequence of lattices with a finite number of defects approaching the large scale geometry of a tetrahedron. The hypothesis of scaling appears to hold at criticality, pointing at a sensible definition of the continuum limit of the model in the polyhedron. Finite size scaling is shown to produce, however, an anomalous exponent for the critical behavior of the correlation length, which we determine alternatively by looking at the temperature dependence of the gap at large lattice size.