A numerical method is implemented for computing the shape of an infinite, hexagonal, doubly periodic, hydrostatic meniscus originating from contact lines whose projections in the horizontal plane are circles. The contact lines themselves may lie on vertical cylinders or spherical objects. The Laplace–Young equation determining the meniscus shape is solved by a finite-difference method in orthogonal curvilinear coordinates generated by conformal mapping. The elevation of the contact lines is either prescribed or computed as part of the solution to ensure a specified contact angle. The results illustrate that the contact line spacing has an important effect on the contact angle or contact line distribution. Meniscus shapes exist only for a limited range of pressure differences between the upper and lower fluid that depend on the capillary length. Copyright © 2009 John Wiley & Sons, Ltd.