I think one of the roles we play as mathematicians in society is to help people become acquainted with the underlying secret patterns. I have been working for several years on projects in crystallography, where we study crystal structures. But few people know that such periodic patterns come with severe constraints on their symmetry. In the plane, there are 17 different symmetry types, a fact well known even to the designers of the mesmerizing patterns of the Alhambra. Here you can find and experiment with such tilings in the plane, gaining insight into the intrinsic beauty of the so-called wallpaper groups, the crystal symmetries in dimension 2. The app interactively helps you design symmetric patterns with colors and shows how changes in the structure of the unit cell propagate via symmetry. nasqret.github.io/symm/ In dimension 3, if you look into International Tables for Crystallography, Vol. 1, you will find a theorem due to Schoenflies and Fedorov stating that there are 230 such symmetry types, a cornerstone of modern chemistry. Beyond that, in dimensions 4 and higher, a count can be made, but it requires a proof of the general theorem due to Frobenius and Bieberbach. This was an answer to the first part of Hilbert’s famous eighteenth problem. One of the fun consequences of such a classification is that in dimensions 2 and 3, 5-fold symmetry is forbidden in regular periodic arrangements. Intrinsically, this fact is related to the existence of matrices with a fifth-root-of-unity eigenvalue. For integral matrices, this is possible only in dimensions 4 and higher. If you generalize the square and cube tilings to dimensions 4 and 5, obtaining hypercubic tilings, the 5-fold symmetry pattern emerges. Skew projections of the 5D hypercubic tiling onto a 2-dimensional plane give rise to a quasicrystalline tiling known as the Penrose tiling. You can find such patterns in front of the Andrew Wiles Building at the Oxford Mathematical Institute. In later posts this summer, I will take a deep dive into group homology, a modern tool for studying the geometry of crystals. There are still many open questions, for example, how many symmetry types exist exactly in dimensions beyond 6. This is still largely unknown; at present, we only have asymptotic lower bounds.

Bye-bye 👋

Le immagini del ministro israeliano Ben Gvir sono inaccettabili. È inammissibile che questi manifestanti, fra cui molti cittadini italiani, vengano sottoposti a questo trattamento lesivo della dignità della persona. Il Governo italiano sta immediatamente compiendo, ai più alti livelli istituzionali, tutti i passi necessari per ottenere la liberazione immediata dei cittadini italiani coinvolti. L’Italia pretende inoltre le scuse per il trattamento riservato a questi manifestanti e per il totale disprezzo dimostrato nei confronti delle esplicite richieste del Governo italiano. Per questi motivi, il Ministero degli Affari Esteri e della Cooperazione Internazionale convocherà immediatamente l’ambasciatore israeliano per chiedere chiarimenti formali su quanto accaduto.

an explicit drawing doesnt seem possible, but maybe the last paragraph satisfies your request. (its essentially a projection of the lattice construction in another field into R^2)

Here are Hussein's cousins who were killed by Israel Western mainstream media refers to them as "Hezbollah targets"

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I've recently got in on the act of getting AI to solve open problems in mathematics. More precisely, I gave some questions asked by Melvyn Nathanson to ChatGPT 5.5 Pro, to which I have been given access, and it answered them. 🧵