A self-Airy membrane shell is a special type of shell structure whose shape coincides with the shell’s Airy stress surface. It provides the convenient property that any polyhedral discretization of such a surface will automatically generate a mesh in funicular equilibrium. A self-Airy shell designed for a uniform vertical load would simply have a constant isotropic Gaussian curvature. However, a challenge in implementing a self-Airy shell in architecture is the lack of a design method, especially in designing unreinforced boundaries. Those are singular planar curves, where the two principal curvatures approach 0 and individually. This paper presents methods for designing unreinforced boundaries of self-Airy shells, including both smooth and discrete methods. These methods work for both positively and negatively curved surfaces. The proposed methods work linearly without iteration. The preliminary results show that the seemingly very restrictive conditions admit a variety of non-trivial surfaces.

We introduce the new concept of C-mesh to capture kinetic structures that can be deployed from a collapsed state. Quadrilateral C-meshes enjoy rich geometry and surprising relations with differential geometry: A structure that collapses onto a flat and straight strip corresponds to a Chebyshev net of curves on a surface of constant Gaussian curvature, while structures collapsing onto a circular strip follow surfaces which enjoy the linear-Weingarten property. Interestingly, allowing more general collapses actually leads to a smaller class of shapes. Hexagonal C-meshes have more degrees of freedom, but a local analysis suggests that there is no such direct relation to smooth surfaces. Besides theory, this paper provides tools for exploring the shape space of C-meshes and for their design. We also present an application for freeform architectural skins, namely paneling with spherical panels of constant radius, which is an important fabrication-related constraint.