The Doyle conjecture on circle packing

The Doyle spirals were first discovered by Gerrit van Iterson in 1907 in the models of phyllotaxis. Peter Doyle conjectured that a locally univalent circle packing on the hexagonal triangulation must be either the trivial packing or a Doyle spiral. Beardon, Dubejko, and Stephenson proved the conjecture in the case of coherent circle packings. In this talk, we present a partial resolution of the Doyle conjecture in the general setting under the assumption of a uniformly bounded ratio of the radii of adjacent circles. This is joint work with Puchun Zhou.