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1. Magnetic Sphere Constructions

   Segerman, Henry; Zwier, Rosa (July 2017, Bridges 2017 Conference Proceedings)

   We investigate constructions made from magnetic spheres. We give heuristic rules for making stable constructions of polyhedra and planar tilings from loops and saddles of magnetic spheres, and give a theoretical restriction on possible configurations, derived from the Poincaré-Hopf theorem. Based on our heuristic rules, we build relatively stable new planar tilings, and, with the aid of a 3D printed scaffold, a construction of the buckyball. From our restriction, we argue that the dodecahedron is probably impossible to construct. We finish with a simplified physical model, within which we show that a hexagonal loop is in static equilibrium. 

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2. Skew and sphere fibrations

   https://doi.org/10.1090/tran/8953  

   Harrison, Michael (July 2023, Transactions of the American Mathematical Society)

   A great sphere fibration is a sphere bundle with total space S n S^n and fibers which are great k k -spheres. Given a smooth great sphere fibration, the central projection to any tangent hyperplane yields a nondegenerate fibration of R n \mathbb {R}^n by pairwise skew, affine copies of R k \mathbb {R}^k (though not all nondegenerate fibrations can arise in this way). Here we study the topology and geometry of nondegenerate fibrations, we show that every nondegenerate fibration satisfies a notion of Continuity at Infinity, and we prove several classification results. These results allow us to determine, in certain dimensions, precisely which nondegenerate fibrations correspond to great sphere fibrations via the central projection. We use this correspondence to reprove a number of recent results about sphere fibrations in the simpler, more explicit setting of nondegenerate fibrations. For example, we show that every germ of a nondegenerate fibration extends to a global fibration, and we study the relationship between nondegenerate line fibrations and contact structures in odd-dimensional Euclidean space. We conclude with a number of partial results, in hopes that the continued study of nondegenerate fibrations, together with their correspondence with sphere fibrations, will yield new insights towards the unsolved classification problems for sphere fibrations. 

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3. Sphere packing and quantum gravity

   https://doi.org/10.1007/JHEP12(2019)048  

   Hartman, Thomas; Mazáč, Dalimil; Rastelli, Leonardo (December 2019, Journal of High Energy Physics)
4. Drawing Graphs on the Sphere

   https://doi.org/10.1145/3399715.3399915  

   Perry, S; Gray, K; Yin, M; Kobourov, S (October 2020, 13th International Conference on Advanced Visual Interfaces (AVI))

   Graphs are most often visualized in the two dimensional Euclidean plane, but spherical space offers several advantages when visualizing graphs. First, some graphs such as skeletons of three dimensional polytopes (tetrahedron, cube, icosahedron) have spherical realizations that capture their 3D structure, which cannot be visualized as well in the Euclidean plane. Second, the sphere makes possible a natural “focus + context" visualization with more detail in the center of the view and less details away from the center. Finally, whereas layouts in the Euclidean plane implicitly define notions of “central" and “peripheral" nodes, this issue is reduced on the sphere, where the layout can be centered at any node of interest. We first consider a projection-reprojection method that relies on transformations often seen in cartography and describe the implementation of this method in the GMap visualization system. This approach allows many different types of 2D graph visualizations, such as node-link diagrams, LineSets, BubbleSets and MapSets, to be converted into spherical web browser visualizations. Next we consider an approach based on spherical multidimensional scaling, which performs graph layout directly on the sphere. This approach supports node-link diagrams and GMap-style visualizations, rendered in the web browser using WebGL. 

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5. Active smectics on a sphere

   https://doi.org/10.1088/1361-648X/ad21a7  

   Nestler, Michael; Praetorius, Simon; Huang, Zhi-Feng; Löwen, Hartmut; Voigt, Axel (February 2024, Journal of Physics: Condensed Matter)

   Abstract The dynamics of active smectic liquid crystals confined on a spherical surface is explored through an active phase field crystal model. Starting from an initially randomly perturbed isotropic phase, several types of topological defects are spontaneously formed, and then annihilate during a coarsening process until a steady state is achieved. The coarsening process is highly complex involving several scaling laws of defect densities as a function of time where different dynamical exponents can be identified. In general the exponent for the final stage towards the steady state is significantly larger than that in the passive and in the planar case, i.e. the coarsening is getting accelerated both by activity and by the topological and geometrical properties of the sphere. A defect type characteristic for this active system is a rotating spiral of evolving smectic layering lines. On a sphere this defect type also determines the steady state. Our results can in principle be confirmed by dense systems of synthetic or biological active particles. 

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