Discrete geometric structures (points, lines, triangles, rectangles, polygons, etc.) are ubiquitous in everyday life, ranging from visible sophisticated freeform structures in contemporary architecture to hidden geometric algorithms in computer generated imagery. On the other hand, discrete geometric structures have also proven to be very useful in modeling and approximating continuous shapes (e.g. curves and surfaces) and real processes. This lecture provides a non-technical and pictorial introduction to the foundations of a new branch of mathematics which underpins these real world situations.
By way of simple and concrete examples, we will illustrate the paradigm of so-called structure-preserving discretizations. These include toy spinning tops, elastic rods, smoke rings and vortex lines in fluids, conformal texture mappings in computer graphics, free form glass and steel structures, and animations from Hollywood movies. We will also show excerpts from our new computer-animated movie entitled "conform!". This film has won Best Experimental Short Film at the Berlin Short Film Festival.
It will be demonstrated that the difference between the continuous and discrete models in geometry and dynamical systems theory is hardly noticeable.
Our aim is to convince you that this new branch of mathematics is both (literally) beautiful and useful.