A discrete time evolution model for fracture networks
Gábor Domokos, Krisztina Regős
Abstract: We examine geological crack patterns using the mean field theory of convex mosaics. We assign the pair (n̅∗, v̅∗) of average corner degrees to each crack pattern and we define two local, random evolutionary steps R0 and R1, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the (n̅∗, v̅∗) plane. We prove the existence of limit points for several types of trajectories. Also, we prove that cell density ρ̅=n̅∗/v̅∗ increases monotonically under any admissible trajectory.
On k-diametral point configurations in Minkowski spaces
Károly Bezdek, Zsolt Lángi
Abstract: The structure of k-diametral point configurations in Minkowski d-space is shown to be closely related to the properties of k-antipodal point configurations in ℝd. In particular, the maximum size of k-diametral point configurations of Minkowski d-spaces is obtained for given k≥2 and d≥2 generalizing Petty’s results on equilateral sets in Minkowski spaces. Furthermore, bounds are derived for the maximum size of k-diametral point configurations in given Minkowski d-space (resp., Euclidean d-space). Some of these results have analogues for point sets, which are discussed as well. In the proofs convexity methods are combined with volumetric estimates and combinatorial properties of diameter graphs.
An interview with Gábor Domokos, one of the discoverers of Gömböc has recently been published at perpal.hu.
Beyond the history of Gömböc, the talk highlights the mutual impact the discovery and scientific thinking may have on each other.