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1. Evolution of adaptive force chains in reconfigurable granular metamaterials

   https://doi.org/10.1039/d4sm00965g  

   Witthaus, Sven; Parsa, Atoosa; Wang, Dong; Pashine, Nidhi; Zhang, Jerry; MacKeith, Arthur K; Shattuck, Mark D; Bongard, Josh; O’Hern, Corey S; Kramer-Bottiglio, Rebecca (July 2025, Soft Matter)

   We combine discrete element method simulations, evolutionary algorithms, and experiments to search for granular packings of variable modulus (VM) particles arranged in a triangular lattice with optimal bulk mechanical properties. 

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2. Local-global principles in circle packings

   https://doi.org/10.1112/S0010437X19007139  

   Fuchs, Elena; Stange, Katherine E.; Zhang, Xin (June 2019, Compositio Mathematica)

   We generalize work by Bourgain and Kontorovich [ On the local-global conjecture for integral Apollonian gaskets , Invent. Math. 196 (2014), 589–650] and Zhang [ On the local-global principle for integral Apollonian 3-circle packings , J. Reine Angew. Math. 737 , (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group $${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$$ satisfying certain conditions, where $$K$$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that $${\mathcal{A}}$$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$$ containing a Zariski dense subgroup of $$\operatorname{PSL}_{2}(\mathbb{Z})$$ . 

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3. Geometry and arithmetic of crystallographic sphere packings

   https://doi.org/10.1073/pnas.1721104116  

   Kontorovich, Alex; Nakamura, Kei (December 2018, Proceedings of the National Academy of Sciences)

   We introduce the notion of a “crystallographic sphere packing,” defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit an infinite family of conformally inequivalent crystallographic packings with all radii being reciprocals of integers. We then prove a result in the opposite direction: the “superintegral” ones exist only in finitely many “commensurability classes,” all in, at most, 20 dimensions. 

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4. Kleinian sphere packings, reflection groups, and arithmeticity

   https://doi.org/10.1090/mcom/3858  

   Bogachev, Nikolay; Kolpakov, Alexander; Kontorovich, Alex (January 2024, Mathematics of Computation)

   In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study “properly integral” packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher–dimensional hyperbolic polyhedra. 

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5. The Apollonian Staircase

   https://doi.org/10.1093/imrn/rnad007  

   Rickards, James (March 2023, International Mathematics Research Notices)

   Abstract A circle of curvature $$n\in \mathbb{Z}^+$$ is a part of finitely many primitive integral Apollonian circle packings. Each such packing has a circle of minimal curvature $$-c\leq 0$$, and we study the distribution of $c/n$ across all primitive integral packings containing a circle of curvature $$n$$. As $$n\rightarrow \infty $$, the distribution is shown to tend towards a picture we name the Apollonian staircase. A consequence of the staircase is that if we choose a random circle packing containing a circle $$C$$ of curvature $$n$$, then the probability that $$C$$ is tangent to the outermost circle tends towards $$3/\pi $$. These results are found by using positive semidefinite quadratic forms to make $$\mathbb{P}^1(\mathbb{C})$$ a parameter space for (not necessarily integral) circle packings. Finally, we examine an aspect of the integral theory known as spikes. When $$n$$ is prime, the distribution of $c/n$ is extremely smooth, whereas when $$n$$ is composite, there are certain spikes that correspond to prime divisors of $$n$$ that are at most $$\sqrt{n}$$. 

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