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The relation between the 2d Ising partition function and spin networkevaluations, reflecting a bulk-boundary duality between the 2d Ising model and3d quantum gravity, promises an exchange of results and methods betweenstatistical physics and quantum geometry. We apply this relation to the case ofthe tetrahedral graph. First, we find that the high/low temperature duality ofthe 2d Ising model translates into a new self-duality formula for Wigner's6j-symbol from the theory of spin recoupling. Second, we focus on the dualitybetween the large spin asymptotics of the 6j-symbol and Fisher zeros. Using thePonzano-Regge formula for the asymptotics for the 6j-symbol at large spins interms of the tetrahedron geometry, we obtain a geometric formula for the zerosof the (inhomogeneous) Ising partition function in terms of triangle angles anddihedral angles in the tetrahedron. While it is well-known that the 2dintrinsic geometry can be used to parametrize the critical point of the Isingmodel, e.g. on isoradial graphs, it is the first time to our knowledge that theextrinsic geometry is found to also be relevant.This outlines a method towardsa more general geometric parametrization of the Fisher zeros for the 2d Isingmodel on arbitrary graphs.

Self-duality of the 6j-symbol and Fisher zeros for the Tetrahedron