We introduce C-tubes , 3D tubular structures composed of developable surface strips. C-tubes can be understood as a generalization of Monge surfaces—a special class of sweep surfaces—towards the recently introduced conenets. This observation allows formulating a constructive algorithm to create tubular structures that ensures developability of the constituent surfaces, while significantly broadening the design space. Our novel form-finding tool enables design exploration by solving for the input variables of the constructive algorithm so that the C-tube best conforms to user-specified objectives. We discuss several case studies that illustrate the versatility of our approach for the design and fabrication of complex structures, with applications in architecture, furniture, and lighting design.

We study ensembles of elastic rods that are tensioned by a small set of inextensible cables. The cables induce forces that deform the initially straight, but flexible rods into 3D space curves at equilibrium. Rods can be open or closed, knotted, and arranged in arbitrary topologies. We specifically focus on equilibrium states with no contacts among rods. Our setup can thus be seen as a generalization of classical tensegrities that are composed of rigid rods and tensile cables, to also support rods that elastically deform. We show how this generalization leads to a rich design space, where complex target shapes can be achieved with a small set of elastic rods. To explore this space, we present an inverse design optimization algorithm that solves for the length and placement of cables such that the equilibrium state of the rod network best approximates a given set of input curves. We introduce appropriate sparsity terms to minimize the number of required cables, which significantly simplifies fabrication. Using our algorithm, we explore new classes of bending-active 3D structures, including elastic tensegrity knots that only require a few internal cables. We design and fabricate several physical models from basic materials that attain complex 3D shapes with unique structural properties.

We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded in 3-space. When endowed with the material thickness and bending resistance of a physical wire, these knots settle into equilibrium states that balance the forces induced by elastic deformation and self-contacts of the wire. In general, elastic knots can have many distinct equilibrium states, i.e. they are multistable mechanical systems. We propose a computational pipeline that combines randomized spatial sampling and physics simulation to efficiently find stable equilibrium states of elastic knots. Leveraging results from knot theory, we run our pipeline on thousands of different topological knot types to create an extensive data set of multistable knots. By applying a series of filters to this data, we discover new transformable knots with interesting geometric and physical properties. A further analysis across knot types reveals geometric and topological patterns, yielding constructive principles that generalize beyond the currently tabulated knot types. We show how multistable elastic knots can be used to design novel deployable structures and engaging recreational puzzles. Several physical prototypes at different scales highlight these applications and validate our simulation.