  We generalise the geometric analysis of square fishnet integrals in two
dimensions to the case of hexagonal fishnets with three-point vertices. Our
results support the conjecture that fishnet Feynman integrals in two
dimensions, together with their associated geometry, are completely fixed by
their Yangian and permutation symmetries. As a new feature for the hexagonal
fishnets, the star-triangle identity introduces an ambiguity in the graph
representation of a given Feynman integral. This translates into a map between
different geometric interpretations attached to a graph. We demonstrate
explicitly how these fishnet integrals can be understood as Calabi-Yau
varieties, whose Picard-Fuchs ideals are generated by the Yangian over the
conformal algebra. In analogy to elliptic curves, which represent the simplest
examples of fishnet integrals with four-point vertices, we find that the
simplest examples of three-point fishnets correspond to Picard curves with
natural generalisations at higher loop orders.
