Upper bound on density of congruent hyperball packings
Upper bound on density of congruent hyperball packings in hyperbolic 3−space Jenő Szirmai Budapest University of Technology and Economics, Hungary 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Kepler conjecture n What is the most efficient way to pack spheres in three dimensional space? n The conjecture was first stated by Johannes Kepler (1611) in his paper 'On the six-cornered snowflake'. No packing of spheres of the same radius has a density greather than the face-centered cubic packing. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Results in Euclidean space n In 1953 László Fejes Tóth reduced the Kepler conjecture to an enormous calculation that involved specific cases, and later suggested that computers might be helpful for solving the problem and in this way the above four hundred year mathematical problem has finally been solved by Mathematician Thomas Hales of the University of Michigan. He had proved that the guess Kepler made back in 1611 was correct. (http: //www. math. lsa. umich. edu/~hales/countdown). T. C. Hales, Sphere Packings I, Discrete Comput. Geom. 17 (1997), 1 – 51, Sphere Packings II, Discrete Comput. Geom. 18 (1997), 135 – 149. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Spaces of constant curvature L. Fejes Tóth - Coxeter conjecture In an n-dimensionalen space of constant curvature let dn(r) be the density of n+1 spheres of radius r mutually touch one another with respect to the simplex spanned by the centres of the spheres. Then the density of packing spheres Rogers, C. A. (1958), "The packing of equal spheres", Proceedings of the London of Mathematical Society. Third Series 8: 609– 620 radius r can not exceed dn(r): . d (r) dn(r). 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Spaces of constant curvature The 2 -dimensional spherical and hyperbolic space was formerly settled by L. Fejes Tóth. In 1964 K. Böröczky and A. Florian proved this conjecture in the 3 -dimensional hyperbolic space. K. Böröczky, und A. Florian Über die dichteste Kugelpackung im hyperbolischen Raum, Acta Math. Hungar. (1964) 15, 237 --245. K. Böröczky proved the above conjecture for the n- dimensional spaces of constant curvature in 1978. K. Böröczky Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978), 243 -261. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Problems 1. What are the optimal ball packing and covering configurations of usual spheres and what are their densities? (n>2) 2. What are the optimal horoball packing and covering configurations and what are their densities allowing horoballs of different types? (n>3) 3. What are the optimal hyperball packing and covering configurations and what are their densities? (n>2) 4. What are the optimal so – called hyp-hor arrangements? (n>2) 5. What are the optimal packing and covering arrangements in other Thurston geometries? 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Cycles in the hyperbolic plane There are three kinds of pencils in the hyperbolic plane, depending on the mutual intersection between arbitrary two lines of the family. The orthogonal trajectories to elements of a pencil are called cycles. Circle 2021. 02. 26. Horocycle Discrete Geometry Days, 2019, Budapest Hypercycle
Previous results on hypercycle packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
How to imagine a hypersphere packing? 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Complete orthoschemes, d=1. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Complete orthoschemes, d=1. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
How to imagine a hypersphere packing? 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
How to imagine a hypersphere packing? 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings (R. Kellerhals) 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings and coverings related to prism tilings n n J. Szirmai, The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3 -space, Acta Mathematica Hungarica 111 (12) (2006), 65 -76. J. Szirmai: The regular prism tilings and their optimal hyperball packings in the hyperbolic n-space, Publ. Math. Debrecen, 69 (1 -2) (2006), 195 -207. J. Szirmai, The optimal hyperball packings related to the smallest compact arithmetic 5 -orbifolds, Kragujevac Journal of Mathematics, 40(2), (2016), 260 -270. J. Szirmai, The least dense hyperball covering to the regular prism tilings in the hyperbolic n-space, Annali di Matematica Pura ed Applicata, 195, (2016) 235 -248. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings We described to each saturated congruent hyperball packing a procedure to get a decomposition of the 3 dimensional hyperbolic space into truncated tetrahedra. Therefore, in order to get a density upper bound to hyperball packings it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. J. Szirmai, Decomposition method related to saturated hyperball packings, Ars Math. Contemp. [2019], 16/2, 349 -358, ar. Xiv: 1709. 04369. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Hypersphere packings in truncated regular octahedra {3, 4, p} 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
2021. 02. 26. Discrete Geometry Days, 2019, Budapest
2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Congruent hypersphere packings in truncated regular tetrahedra 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Congruent hypersphere packings in truncated regular tetrahedra J. Szirmai, Hyperball packings in hyperbolic 3 -space, Matematicki Vesnik, 70/3 (2018), 211221, ar. Xiv: 1405. 0248. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Congruent hypersphere packings in truncated regular tetrahedra 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
On monotony of density function 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
2021. 02. 26. Discrete Geometry Days, 2019, Budapest
n n n J. Szirmai, Hyperball packings in hyperbolic 3 -space, Matematicki Vesnik, 70/3 (2018), 211 -221, ar. Xiv: 1405. 0248. J. Szirmai, Density upper bound of congruent and non-congruent hyperball packings generated by truncated regular simplex tilings, Rendiconti del Circolo Matematico di Palermo Series 2, 67 [2018], 307 -322, DOI: 10. 1007/s 12215 -017 -0316 -8, ar. Xiv: 1510. 03208. J. Szirmai, Packings with horo- and hyperballs generated by simple frustum orthoschemes, Acta Mathematica Hungarica, 152 (2), (2017), 365– 382, DOI: 10. 1007/s 10474 -017 -0728 -0, ar. Xiv: 1505. 03338. J. Szirmai, Hyperball packings related to octahedron and cube tilings in hyperbolic space, (Submitted Manuscript), (2019), ar. Xiv: 1709. 04369. J. Szirmai, Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic 3 -space, (Submitted Manuscript) (2019). J. Szirmai, Upper bound of density for packing of congruent hyperballs in hyperbolic 3 -space, (Submitted Manuscript) (2019). 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
J. Szirmai: Horoball packings for the Lambert-cube tilings in the hyperbolic 3 -space, Beiträge zur Algebra und Geometrie (Contributions to Algebra und Geometry) 46 No. 1 (2005), 43 -60. R. T. Kozma – J. Szirmai, Optimally Dense Packings for Fully Asymptotic Coxeter Tilings by Horoballs of Different Types, Monatshefte für Mathematik, 168, [2012], 27 -47 DOI: 10. 1007/s 00605 -012 -0393 -x, ar. Xiv: 1007. 0722. J. Szirmai, Horoball packings and their densities by generalized simplicial density function in the hyperbolic space, Mathematica Hungarica, 136/1 -2, DOI: R. Acta T. Kozma – J. Szirmai, New Lower Bound[2012] for the, 39 -55, Optimal Ball 10. 1007/s 10474 -012 -0205 -8, Packing Density of Hyperbolic 4 -space, ar. Xiv: 1105. 4315 Discrete and Computational Geometry (2015) 53, 182 --198, DOI: 10. 1007/s 00454 -014 -9634 -1, ar. Xiv: 1401. 6084 J. Szirmai, Horoball packings to the totally asymptotic regular simplex in the hyperbolic n-space, Aequationes mathematicae, 85 (2013), 471– 482, DOI: 10. 1007/s 00010 -012 -0158 -6, r. Xiv: 1112. 1969. R. T. Kozma - J. Szirmai, Horoball Packing Density Lower Bounds in Higher Dimensional Hyperbolic nspace for 6 ≤ n ≤ 9, , (Submitted Manscript) ar. Xiv: 1907. 00595. R. T. Kozma – J. Szirmai, , The structure and visualization of optimal horoball packings in 3 -dimensional hyperbolic space, Manuscript [2017]. ar. Xiv: 1601. 03620. J. Szirmai, Horoball packings related to the 4 -dimensional hyperbolic 24 cell honeycomb {3, 4, 3, 4}, Filomat [2018], 32/1, 87 -100, DOI: 10. 2298/FIL 1801087 S, ar. Xiv: 1502. 02107. R. T. Kozma – J. Szirmai, New horoball packing density lower bounds in hyperbolic 5 -space, Geometriae Dedicata (to appear) (2019), ar. Xiv: 1809. 05411. 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
Thank You 2021. 02. 26. Discrete Geometry Days, 2019, Budapest
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