  The relation between the 2d Ising partition function and spin network
evaluations, reflecting a bulk-boundary duality between the 2d Ising model and
3d quantum gravity, promises an exchange of results and methods between
statistical physics and quantum geometry. We apply this relation to the case of
the tetrahedral graph. First, we find that the high/low temperature duality of
the 2d Ising model translates into a new self-duality formula for Wigner's
6j-symbol from the theory of spin recoupling. Second, we focus on the duality
between the large spin asymptotics of the 6j-symbol and Fisher zeros. Using the
Ponzano-Regge formula for the asymptotics for the 6j-symbol at large spins in
terms of the tetrahedron geometry, we obtain a geometric formula for the zeros
of the (inhomogeneous) Ising partition function in terms of triangle angles and
dihedral angles in the tetrahedron. While it is well-known that the 2d
intrinsic geometry can be used to parametrize the critical point of the Ising
model, e.g. on isoradial graphs, it is the first time to our knowledge that the
extrinsic geometry is found to also be relevant.This outlines a method towards
a more general geometric parametrization of the Fisher zeros for the 2d Ising
model on arbitrary graphs.
