(joint work with  Jenő SZIRMAI)

The famous KEPLER Conjecture on the densest packing the Euclidean 3-space  E³ with equal balls, has recently been solved by Thomas HALES by computer. His procedure (in more than 200 pages) followed the strategy of László FEJES TÓTH.

In this presentation we report the analogous problems in the other homogeneous geometries, the 8 THURSTON spaces:  E³, S³, H³, S²×R, H²×R, ~SL₂R, Nil and Sol.

There many problems are open, e.g in János BOLYAI’s hyperbolic space H³ as well. Our method is based on the intensive use of computer procedures, developed in recent works of the authors below.

References

[1] Molnár, E., Szirmai, J.: Classification of Sol lattices. Geom. Dedicata 161, 251–275 (2012).

[2] Molnár, E., Szirmai, J.: Volumes and geodesic ball packings to the regular prism tilings in ~SL₂R space, to appear in Publ. Math. Debrecen. [2014]. arXiv.1304.0546.

[3] Szirmai, J.: The densest geodesic ball packing by a type of Nil lattices. Beitr. Algebra Geom. 48(2), 383–398 (2007).

[4] Szirmai, J.: Geodesic ball packing in S2 × R space for generalized Coxeter space groups. Beitr. Algebra Geom. 52, 413–430 (2011).

[5] Szirmai, J.: Geodesic ball packing in H2 × R space for generalized Coxeter space groups. Math. Commun. 17, 151–170 (2012).

[6] Szirmai, J.: Lattice-like translation ball packings in Nil space. Publ. Math. Debrecen 80, 427–440 (2012).

[7] Szirmai, J.: A candidate to the densest packing with equal balls in the Thurston geometries. Beitr. Algebra Geom. (2013). doi:10.1007/s13366-013-0158-2