Recent advances in material science and digital fabrication provide promising opportunities for industrial and product design, engineering, architecture, art and science. To bring new innovations to fruition, effective computational tools are needed that link creative design exploration with material realisation. One approach being investigated by the Computer Graphics and Geometry Laboratory (LGG) at the EPF Lausanne is to abstract the material and fabrication constraints of a design into suitable geometric representations, and then solve the design using computational algorithms.

Most common sheet materials, such as metal, plastic or even paper are limited in how they can be formed because they are not elastic. The LGG is studying how a new class of surfaces can be realized by cutting a regular pattern of thin slits into such inextensible materials to enable their irregular deformation (see Figure 1). The polygons formed through this cutting process can rotate relative to their neighbors, effectively enabling the material to stretch up to a certain limit. A key motivation for studying such materials is that one can approximate doubly-curved surfaces such as the sphere by using only flat pieces, making it attractive for fabrication.

NCCR researcher Mina Konaković of the LGG is developing a computational method for interactive 3D design and the rationalization of surfaces vià auxetic materials, i.e. flat, flexible material that can stretch. Flexible surfaces are realized by introducing a cutting pattern which is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. This stretching in turn allows the surface to express a wide variety of curvatures, enabling a single surface to take forms not possible with traditionally developed sheet materials.

Figure 1, top: Samples of materials used in experiments. The leftmost photo shows the un-deformed configuration. Bottom: Geometric abstraction using a triangular linkage.

Auxetic materials are furthermore interesting because they can also be used as transformable surfaces, i.e. animated structures that can change their geometry over time. Reconfigurable surfaces have a variety of application fields, including shading and lighting systems in architecture (Figure 2).

Figure 2: A free-form facade with uniform equilateral triangles. As the surface is reconfigurable, it can be used as a dynamic, external shading system.

For computational design, a constraint-based optimization is used to find the best geometric configurations that closely approximate a target surface. A key insight from the research thus far is that one can leverage theory and algorithms from conformal geometry to facilitate the design process. Such global optimization methods also help to address challenging design decisions. Simply wrapping a piece of material around a target object is for example unlikely to result in a clean, well formed surface. In contrast, the computation process is able to identify the 2D regions that most easily approximate a target shape in 3D (see Figure 3), allowing the surface to account for both geometry and material rigidity.

Through a series of design studies and physical prototypes, the LGG has demonstrated that this approach can create a great variety of shapes with attractive material and functional properties. This approach inspires new fundamental questions in discrete differential geometry, but more pragmatically this research opens up new design opportunities in diverse fields such as in biomechanics, engineering, consumer goods and of course architecture.

Figure 3: Fabrication of the Max Planck model from a single sheet of perforated copper. Left: Flat, un-deformed perforated copper sheet. Middle and right: Two photographs of the final model.