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Douglas J. Jerolmack, University of Pennsylvania
April 28, 2023
Our guest lecturer, second time since 2019, gave a talk on various aspects of fragmentation in Nature.
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The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices
Dávid Papp, Krisztina Regős, Gábor Domokos, Sándor Bozóki
Abstract: In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question is to construct such an object with the minimal number of faces and vertices. By distinguishing between various material distributions and stability types, this expands into a small family of related questions. While many upper and lower bounds on the necessary numbers of faces and vertices have been established, none of these questions has been so far resolved. Adapting an algorithm presented in Bozóki et al. (2022), here we offer the first complete answer to a question from this family: by using the toolbox of semidefinite optimization to efficiently generate the hundreds of thousands of infeasibility certificates, we provide the first-ever proof for the existence of a monostatic polyhedron with point masses, having minimal number (V=11) of vertices (Theorem 3) and a minimal number (F=8) of faces. We also show that V=11 is the smallest number of vertices that a mono-unstable polyhedron can have in all dimensions greater than 1.
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The “Gömböc” is on permanent display since 17th April, 2023 at Pompidou Centre, Paris. The largest copy of Gömböc ever made of a single piece of material was introduced in the presence of one of its inventors, Gábor Domokos.
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Several presentations were held in the session for Mathematics, Physics and Geosciences of the 36th National Scientific Students’ Associations Conference in relation with the Morphodynamics Research Group. Gergő Almádi has been awarded by 3rd prize for his presentation entitled Inhomogén politópok mechanikai komplexitása – avagy van-e egy tetraédernek lelke?, under the supervision of Gábor Domokos and Krisztina Regős. Ágoston Szesztay (Iteratív módon csonkolt poliéderek statikai egyenúlyáról) and Máté Szondi (A kvantummechanikai állapottér egy felbontása által indukált geometria) got special prizes.
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Polygonal tessellations as predictive models of molecular monolayers
Krisztina Regős et al.
Abstract: Molecular self-assembly plays a very important role in various aspects of technology as well as in biological systems. Governed by covalent, hydrogen or van der Waals interactions–self-assembly of alike molecules results in a large variety of complex patterns even in two dimensions (2D). Prediction of pattern formation for 2D molecular networks is extremely important, though very challenging, and so far, relied on computationally involved approaches such as density functional theory, classical molecular dynamics, Monte Carlo, or machine learning. Such methods, however, do not guarantee that all possible patterns will be considered and often rely on intuition. Here, we introduce a much simpler, though rigorous, hierarchical geometric model founded on the mean-field theory of 2D polygonal tessellations to predict extended network patterns based on molecular-level information. Based on graph theory, this approach yields pattern classification and pattern prediction within well-defined ranges. When applied to existing experimental data, our model provides a different view of self-assembled molecular patterns, leading to interesting predictions on admissible patterns and potential additional phases. While developed for hydrogen-bonded systems, an extension to covalently bonded graphene-derived materials or 3D structures such as fullerenes is possible, significantly opening the range of potential future applications.
