The pressure in the incompressible three-dimensional Navier–Stokes and Euler equations is governed by Poisson's equation: this equation is studied using the geometry of three-forms in six dimensions. By studying the linear algebra of the vector space of three-forms Λ³W* where W is a six-dimensional real vector space, we relate the characterization of non-degenerate elements of Λ³W* to the sign of the Laplacian of the pressure—and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, Δp, satisfies Δp>0, the three-form associated with Poisson's equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When Δp<0, a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence.