Square tessellation patterns on curved surfaces: In search of a parametric design method

Methods for Tessellating a flat surface with regular or semi-regular patterns of polygons have already been addressed in literature and can be easily parameterized. For the tessellation of curved surfaces using patterns of one or more regular polygons there is not a uniquely defined approach to the problem within the context of architectural research and applications. This paper is focused on the tessellation of curved surfaces with square tiles, where the tessellation pattern consists of four squares with partly overlapping sides. In this study double curvature surfaces were considered first, and subsequently surfaces of more complex geometry such as minimal surfaces. Specifically, a method for the square tessellation of two types of doubly curved surfaces, the spherical and the ellipsoidal, is discussed and presented in the paper. In addition, the square tessellation of two types of minimal surfaces, the catenoid and the helicoid, have also been examined and presented. For each one of the surfaces that have been considered, an algorithm that generates the distribution of the planar square surfaces on the surface and renders possible the parametric description of the problem, was developed and presented in the paper. A discussion on boundary conditions for each developed method is also included. The Grasshopper visual programming language has been used for the parametric description and display of the results in a graphic environment. The research discussed in this paper can find application in several real world problems including surface paneling, or space packing of polyhedral structural units on a curved surface.