MIMS第7回トポロジーとその応用融合研究セミナー

※ 講演は英語で行われます。

講演者： Gabor Domokos氏（Budapest University of Technology and Economics）

講演タイトル：Plato’s cube and the natural geometry of fragmentation

Abstract:
If we approximate natural fragments by convex polyhedra and count the respective numbers for faces, vertices and edges then, in most cases, we find averages remarkably close to 6,8,12, the values corresponding to the cube. Not only can this observation be translated into a simple Lemma about hyperplane convex mosaics, we also find that general convex mosaics may serve as particularly fitting models for natural fragmentation patterns. This approach not only offers a complete, global catalog of natural tilings, ranging from atomic lattices to the geometry of tectonic plates, but also leads to a rigorous dynamical theory on the evolution of tilings where cubic averages can appear as global attractors.
To verify our geophysical claims, we vetted field data from over 4000 natural fragments against computer simulations and found not only good agreement for the aforementioned cuboid averages, but we could also reproduce full distributions for many geophysical shape descriptors with remarkable accuracy.
The appearance of the cube (albeit in an averaged sense) in this context may remind one of Plato’s theory of the Element Earth. I will briefly comment on some purely geometric aspects of this connection.
1. https://www.pnas.org/content/117/31/18178
2. https://www.quantamagazine.org/geometry-reveals-how-the-world-is-assembled-from-cubes-20201119/
3. https://www.pnas.org/doi/10.1073/pnas.2300049120
4. https://www.quantamagazine.org/the-simple-geometry-that-predicts-molecular-mosaics-20230621/
5. https://link.springer.com/article/10.1007/s10955-02
This is joint work with Doug Jerolmack (U. Pennsylvania), Ferenc Kun (U. Debrecen).
János Török, Péter Bálint and Krisztina Regős (Budapest U. of Technology and Economics).

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