A two-vertex theorem for normal tilings
Gábor Domokos, Ákos G. Horváth, Krisztina Regős
Abstract: We regard a smooth, 𝑑=2-dimensional manifold ℳ and its normal tiling M, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by 𝑣¯⋆ and we prove that if M is periodic then 𝑣¯⋆≥2. We show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with 𝑣¯⋆=0.
Marjorie Senechal gives a talk “Penrose tilings and the Imagination” at BME Applied Mathematics Day 2022. Date and location: April 20th, 15:00, BME „K” building, 3rd floor, room K350.
Gábor Domokos will give a lecture “The invisible cube” at the University of Pennsylvania Stuart Weitzman School of Design on 16th February, at 6pm (Central European Time). The lecture gives insight into the work of the Morphology and Applied Geometry Department and the Morphodynamics research group. Follow the lecture online: zoom link.
The Morphodynamics Group had a session titled ‘Geometry’ in the 2021 TDK Conference. Five students participated in the session and received numerous awards. Congratulations!
Krisztina Regős (1st Prize + Rector’s Award), Anna Viczián (1st Prize), Ágoston Szesztay (3rd Prize + Csonka Pál Special Prize), Klaudia Nagy (Department’s Special Prize), Máté Szondi (Metszet Journal Special Prize).
Continue Reading “Institutional Scientific Students’ Associations Conference 2021”
Mono-unstable polyhedra with point masses have at least 8 vertices
Sándor Bozóki, Gábor Domokos, Flórián Kovács, Krisztina Regős
Abstract: The monostatic property of convex polyhedra (i.e., the property of having just one stable or unstable static equilibrium point) has been in the focus of research ever since Conway and Guy (1969) published the proof of the existence of the first such object, followed by the constructions of Bezdek (2011) and Reshetov (2014). These examples establish
as the respective upper bounds for the minimal number of faces and vertices for a homogeneous mono-stable polyhedron. By proving that no mono-stable homogeneous tetrahedron existed, Conway and Guy (1969) established for the same problem the lower bounds for the number of faces and vertices as
and the same lower bounds were also established for the mono-unstable case (Domokos et al., 2020b). It is also clear that the
bounds also apply for convex, homogeneous point sets with unit masses at each point (also called polyhedral 0-skeletons) and they are also valid for mono-monostatic polyhedra with exactly one stable and one unstable equilibrium point (both homogeneous and 0-skeletons). In this paper we draw on an unexpected source to extend the knowledge on mono-monostatic solids: we present an algorithm by which we improve the lower bound to
vertices on mono-unstable 0-skeletons. The problem is transformed into the (un)solvability of systems of polynomial inequalities, which is shown by convex optimization. Our algorithm appears to be less well suited to compute the lower bounds for mono-stability. We point out these difficulties in connection with the work of Dawson, Finbow and Mak (Dawson, 1985, Dawson et al., 1998, Dawson and Finbow, 2001) who explored the monostatic property of simplices in higher dimensions.
Curvature flows, scaling laws and the geometry of attrition under impacts
Gergő Pál, Gábor Domokos & Ferenc Kun
Abstract: Impact induced attrition processes are, beyond being essential models of industrial ore processing, broadly regarded as the key to decipher the provenance of sedimentary particles. Here we establish the first link between microscopic, particle-based models and the mean field theory for these processes. Based on realistic computer simulations of particle-wall collision sequences we first identify the well-known damage and fragmentation energy phases, then we show that the former is split into the abrasion phase with infinite sample lifetime (analogous to Sternberg’s Law) at finite asymptotic mass and the cleavage phase with finite sample lifetime, decreasing as a power law of the impact velocity (analogous to Basquin’s Law). This splitting establishes the link between mean field models (curvature-driven partial differential equations) and particle-based models: only in the abrasion phase does shape evolution emerging in the latter reproduce with startling accuracy the spatio-temporal patterns (two geometric phases) predicted by the former.
Congratulations to Sára Lévay, who successfully defended her PhD thesis: Self-organizing processes in granular materials (supervisor: János Török).
An analogue of a theorem of Steinitz for ball polyhedra in R-3
Sami Mezal Almohammad, Zsolt Lángi, Márton Naszódi
Abstract: Steinitz’s theorem states that a graph G is the edge-graph of a 3-dimensional convex polyhedron if and only if, G is simple, plane and 3-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in ℝ3.
After our short visit and consultation in Sienna, Italy, before the global pandemic breakout, our colleagues finally had the opportunity to construct their own scanning equipment. We are looking forward to see their scanning results and experiences with the newly built devices.
The Gomboc simulation software – whose name was chosen after the Gömböc of Gábor Domokos and Péter Várkonyi – played a crucial role in the the 36th America’s cup.
Read more (in Hungarian): A MAGYAR GÖMBÖCRŐL ELNEVEZETT GOMBOC SZOFTVERREL DIADALMASKODOTT A GYŐZTES CSAPAT A VILÁG LEGELITEBB VITORLÁSVERSENYÉN
An article about the Gomboc software in the 35th America’s cup (in English): Gomboc: A design high-flier for ETNZ