Upper bound on density of congruent hyperball packings in hyperbolic 3−space

1. Upper bound on density of congruent hyperball packings in hyperbolic 3−space Jenő Szirmai Budapest University of Technology and Economics, Hungary Discrete Geometry Days, 2019, Budapest

2. Kepler conjecture • What is the most efficient way to pack spheres in three dimensional space? • 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. Discrete Geometry Days, 2019, Budapest

3. Results in Euclidean space • 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 waythe 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. Discrete Geometry Days, 2019, Budapest

4. 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 of radius r can not exceed dn(r):. d (r)  dn(r) . Rogers, C. A. (1958), "The packing of equal spheres", Proceedings of the London Mathematical Society. Third Series 8: 609–620 Discrete Geometry Days, 2019, Budapest

5. 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 proved the above conjecture for the n- dimensional spaces of constant curvature in 1978. K. Böröczky, und A. Florian Über die dichteste Kugelpackung im hyperbolischen Raum, Acta Math. Hungar. (1964) 15, 237--245. K. Böröczky Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978), 243--261. Discrete Geometry Days, 2019, Budapest

6. 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? Discrete Geometry Days, 2019, Budapest

7. 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. Horocycle Circle Hypercycle Discrete Geometry Days, 2019, Budapest

8. Previous results on hypercycle packings 2019.10.24. Discrete Geometry Days, 2019, Budapest

9. How to imagine a hypersphere packing? Discrete Geometry Days, 2019, Budapest

10. Complete orthoschemes, d=1. Discrete Geometry Days, 2019, Budapest

11. Hypersphere packings Discrete Geometry Days, 2019, Budapest

12. Complete orthoschemes, d=1. Discrete Geometry Days, 2019, Budapest

13. How to imagine a hypersphere packing? Discrete Geometry Days, 2019, Budapest

14. How to imagine a hypersphere packing? Discrete Geometry Days, 2019, Budapest

15. Hypersphere packings Discrete Geometry Days, 2019, Budapest

16. Hypersphere packings Discrete Geometry Days, 2019, Budapest

17. Hypersphere packings Discrete Geometry Days, 2019, Budapest

18. Hypersphere packings Discrete Geometry Days, 2019, Budapest

19. Hypersphere packings Discrete Geometry Days, 2019, Budapest

20. Hypersphere packings Discrete Geometry Days, 2019, Budapest

21. Hypersphere packings (R. Kellerhals) Discrete Geometry Days, 2019, Budapest

22. Hypersphere packings Discrete Geometry Days, 2019, Budapest

23. Hypersphere packings Discrete Geometry Days, 2019, Budapest

24. Hypersphere packings and coverings related to prism tilings • J. Szirmai, The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3-space, Acta MathematicaHungarica 111 (1-2) (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. Discrete Geometry Days, 2019, Budapest

25. Hypersphere packings We described to each saturated congruent hyperball packinga 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 determinethe density upper bound of hyperball packings in truncated simplices. J. Szirmai, Decompositionmethodrelatedtosaturatedhyperballpackings, Ars Math. Contemp. [2019], 16/2, 349-358, arXiv: 1709.04369. Discrete Geometry Days, 2019, Budapest

26. Hypersphere packings in truncated regular octahedra {3,4,p} Discrete Geometry Days, 2019, Budapest

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29. Congruent hypersphere packings in truncated regular tetrahedra Discrete Geometry Days, 2019, Budapest

30. Congruent hypersphere packings in truncated regular tetrahedra J. Szirmai, Hyperball packings in hyperbolic 3-space, Matematicki Vesnik, 70/3 (2018), 211-221,arXiv: 1405.0248. Discrete Geometry Days, 2019, Budapest

31. Congruent hypersphere packings in truncated regular tetrahedra Discrete Geometry Days, 2019, Budapest

32. On monotony of density function Discrete Geometry Days, 2019, Budapest

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34. J. Szirmai, Hyperballpackings in hyperbolic 3-space, MatematickiVesnik, 70/3 (2018), 211-221,arXiv: 1405.0248. • J. Szirmai, Densityupperbound of congruent and non-congruenthyperballpackingsgeneratedbytruncatedregularsimplextilings, RendicontidelCircoloMatematico di Palermo Series 2, 67 [2018], 307-322, DOI: 10.1007/s12215-017-0316-8, arXiv:1510.03208. • J. Szirmai, Packingswithhoro- and hyperballsgeneratedbysimplefrustumorthoschemes, ActaMathematica Hungarica,152 (2), (2017), 365–382, DOI:10.1007/s10474-017-0728-0, arXiv: 1505.03338. • J. Szirmai, Hyperball packings related to octahedron and cube tilings in hyperbolic space, (SubmittedManuscript), (2019), arXiv: 1709.04369. • J. Szirmai, Congruent and non-congruent hyperball packings related to doubly truncated Coxeterorthoschemes in hyperbolic 3-space, (SubmittedManuscript) (2019). • J. Szirmai, Upper bound of density for packing ofcongruent hyperballs in hyperbolic 3-space, (SubmittedManuscript) (2019). Discrete Geometry Days, 2019, Budapest

35. J. Szirmai: Horoballpackingsforthe Lambert-cubetilings in thehyperbolic 3-space, Beiträgezur Algebra und Geometrie (Contributionsto 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],27-47DOI: 10.1007/s00605-012-0393-x,arXiv:1007.0722. J. Szirmai, Horoball packings and their densities by generalized simplicial density function in the hyperbolic space, Acta Mathematica Hungarica,136/1-2, [2012], 39-55, DOI: 10.1007/s10474-012-0205-8, arXiv:1105.4315 R. T. Kozma – J. Szirmai, New Lower Bound for the Optimal Ball Packing Density of Hyperbolic 4-space, Discrete and Computational Geometry (2015) 53, 182--198, DOI: 10.1007/s00454-014-9634-1, arXiv: 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/s00010-012-0158-6,arXiv:1112.1969. R.T. Kozma - J. Szirmai, Horoball Packing Density Lower Bounds in Higher Dimensional Hyperbolic n-space for 6 ≤ n ≤9, , (Submitted Manscript)arXiv: 1907.00595 . R. T. Kozma – J. Szirmai,, The structure and visualization of optimal horoball packings in 3-dimensional hyperbolic space, Manuscript [2017]. arXiv: 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/FIL1801087S,arXiv: 1502.02107. R. T. Kozma – J. Szirmai, New horoball packing density lower bounds in hyperbolic 5-space, Geometriae Dedicata (to appear) (2019), arXiv:1809.05411. Discrete Geometry Days, 2019, Budapest

36. Thank You Discrete Geometry Days, 2019, Budapest