Andreas Groß, Peter Hilgers
Andreas Groß is a plant design engineer with a long term passion in 3D modelling and rendering. For more than twelve years he has been working in 3D CAD modeling and construction. Through this experience his passion for CG art came to life, and he started focussing on 3D modelling in creative fields. In CG art, Andreas' preference is to do photorealistic modelling and rendering.
Peter Hilgers is a retired engineer, amateur in mathematics and computer hobbyist. He has always been fascinated by geometry. Current computers and sophisticated programs make it possible to create, study and visualise complex structures. For him this is a constant source of challenges and delight.
Ammann non periodic 3D tilings use two basic building elements of rhombohedric shape. A rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares, but rhombi. Such tilings can be computed by a "dualisation method".
Here we use a so far unnoted structural feature of Ammann tilings, the neighbourhood relation of tiles. A core of 20 rhombohedra is covered face to face by a layer of tiles, called corona. The first corona is covered by a second one etc. Surprisingly the coronas approach the shape of an Archimedian solid, the Icosidodecahedron.
It is supposed that this property is independent of the shape of the core and exact in the limit. A proof is a topic of actual research.
Non-periodic tilings of the plane, e.g. the Penrose tiling, are fascinating and have gained some popularity in previous decades. Not so well known are non-periodic tilings of 3D space. Here we use the so called Danzer 3D tiling to build geometric structures. Danzer tilings use 4 types of tetrahedra, the prototiles. These can be enlarged or inflated in such a way that the prototiles can be collected in larger versions of themselves. This process may be continued ad infinitum.
Danzer tilings of 3D space have vertex configurations where 120 tiles of the same type meet in an isocahedral symmetric vertex star. Various symmetries are used here to shape the structure