  We analyse the family of Calabi-Yau varieties attached to four-point fishnet
integrals in two dimensions. We find that the Picard-Fuchs operators for
fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder
integrals. This implies that the periods of the Calabi-Yau varieties for
fishnet integrals can be written as determinants of periods for ladder
integrals. The representation theory of the geometric monodromy group plays an
important role in this context. We then show how the determinant form of the
periods immediately leads to the well-known Basso-Dixon formula for four-point
fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau
geometry implies that the volume is also expressible via a determinant formula
of Basso-Dixon type. Finally, we show how the fishnet integrals can be written
in terms of iterated integrals naturally attached to the Calabi-Yau varieties.
