Discrete geometric structures motivated by applications in architecture

The emergence of freeform structures in contemporary architecture raises numerous challenging research problems, most of which are related to the actual fabrication and are of a mathematical nature. The present project will take up those fundamental research problems raised by demands in architecture which cannot be addressed by specialized practitioners or current software, and which are also beyond the scope of application oriented research projects. While motivation comes from architecture, we focus on research topics which are also of substantial interest from a purely mathematical perspective. Research on discrete structures motivated by architecture has been strongly influenced by our work on planar quad meshes, meshes with offset properties and related optimized supporting beam layouts, multilayer constructions, architectural structures from single-curved panels based on semi-discrete representations, and others. The scientific background of this previous research as well as the planned work in the present project is in discrete differential geometry, conformal geometry, the geometry of webs, and other areas of classical geometry. We plan to work in close cooperation with application oriented projects and also benefit from recent advances in Geometric Computing. The main research directions are the following ones: Discrete webs. We are interested in such discrete hexagonal webs which are formed by either smooth or discrete curves of a specified type (planar, geodesic, principal curvature line, etc.). These problems have remarkable implications on architectural design, but are in fact very difficult. Even the seemingly simple problem of determining all hexagonal webs of circles in the plane or the sphere is unsolved. This indicates that nontrivial hexagonal meshes formed by planar curves on general surfaces are hard to find. We will address constructive approaches to this problem and expect to get the right conjectures from computational results. Moreover, we intend to perform analogous web geometric studies for 3D structures. Evolution of important classes of meshes. Here evolution means the embedding of a mesh into a discrete or continuous family of meshes of the same type. In particular, we will investigate planar quad meshes, circular meshes and circle packing meshes. This has applications in mesh design and editing. Discrete 3D structures motivated by architecture. So far, the relation between discrete differential geometry and architecture has been exploited in form of discretizations of surfaces. However, fully 3D structures are relevant as well. This is related to the previous topic, but now the evolution itself is discrete and restricted so that 3D structures with desirable properties, e.g. only planar faces, can arise.

Organic geometric shapes play an important role in contemporary architecture. In order to realize such shapes as architectural structures, one has to decompose curved facades into easily producible panels, to compute a geometrically and statically optimized support structure, to reduce the number of different parts or to minimize the amount of material to be used. The decomposition of a complicated geometric shape into simpler parts leads to a modern part of mathematics, called discrete differential geometry. It is remarkable that needs in the building industry have motivated the development of new mathematical concepts. In the present project, we aimed at deepening and extending these mathematical connections. We succeeded to develop a new method for interactive geometric modelling of a panelized shape which at the same time performs an optimization of its structural behaviour and maintains constraints from manufacturing as well as aesthetics. The applicability of this method is not confined to architecture. We see our results as an important step towards the development of new intelligent design systems. Those shall guide the designer during modelling of the shape so that constraints from function, stability and manufacturing are automatically satisfied. The project also aimed at purely mathematical studies of concepts which have been motivated by architecture. Here, we would like to mention constructions of surfaces with special curvature behaviour from planar quadrilaterals or developable surface strips. Moreover, we studied smooth surfaces which can be decomposed into very simple patches. Here, the main interest has been on the theoretical side, but there is the possibility that such structures get applied in architecture.