Post on 20-May-2020
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Top dense ball packings and
coverings in hyperbolic space XIX. Geometrical Seminar
2016, Zlatibor, Serbia
Emil Molnár and Jenő Szirmai
Budapest University of Technology and Economics,
Hungary
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
XIX. Geometrical Seminar, August 28 – September 4, 2016, Zlatibor/Serbia
Top dense ball packings and coverings in hyperbolic space
Emil Molnár and Jenő Szirmai
Budapest University of Technology and Economics, Institute of Mathematics, Department of Geometry
emolnar@math.bme.hu, szirmai@math.bme.hu
Keywords: Ball packing and covering in homogeneous 3-geometry, density, complete Coxeter orthoscheme groups and their ball arrangements in hyperbolic 3-space.
In the classical Euclidean 3-space the so-called Kepler conjecture on the densest packing E³
with congruent balls (with density 0.74…) has been recently solved by Thomas Hales by
computer, following the strategy of László Fejes Tóth (1953). In the Bolyai-Lobachevsky
hyperbolic space we know only a density upper bound (K. Böröczky and A. Florian (1964)),
realized (only) by horoballs in ideal regular simplex arrangement with density 0.85…, and the
realization is not unique (R.T. Kozma and J. Szirmai [2]). With proper balls we are far from
this packing upper bound, and there is no real chance yet for the more difficult ball covering
problem in H³.
Our aim in this work is a systematic computer experiment to attacking both problems for
packing and covering by a construction scheme. These ball arrangements will be based on
complete (or extended) Coxeter orthoscheme groups, generated by plane reflections.
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
E.g. the Coxeter-Schläfli symbol (u, v, w) = (5, 3, 5) describes first the characteristic orthoscheme of a regular dodecahedron (as (5, 3, . )) refers to it) with dihedral face angle 2/5 (as (., ., 5) indicates it). This dodecahedron – by its congruent copies – fills H³ just by the reflections in the side faces of the above
orthoscheme (characterized also by a Coxeter-Schläfli matrix, scalar product of signature (+++), etc.). This orthoscheme A0 A1 A2 A3 has also a half-turn symmetry 03, 1 2 that extends to the complete symmetry group of the H³ tiling. Not surprisingly, we get the H³ tiling with the hyperbolic football (the Archimedean solid {5, 6, 6}) as in the earlier works [3, 4] of the first author. The central ball (centred in A3 or in A0) in the above football solid has the packing density 0.771…, as a maximal density so far, just discovered now. This hyperbolic football provides also the covering density 1.369…, minimal so far, as brand new observation.
For this and the analogous generalized further series, a volume formula of orthoscheme by N.I. Lobachevsky
(1837) was needed, that has been extended to complete (or truncated) orthoschemes by R. Kellerhals [1]. The
second author intensively worked on its computer program (see e.g. [5]). Thus we get a large list of good (top!?)
constructions as for packing densities as for covering ones as well, together with their metric data. For these the
ball centre also varies on the surface of the (truncated) orthoscheme, together with the ball radius. So we have to
implement large computations, indeed.
References [1] R. Kellerhals, On the volume of hyperbolic polyhedra, Math. Ann., (1989) 245, 541-569. [2] R.T. Kozma and J. Szirmai, Optimally dense packings for fully asymptotic Coxeter tilings by
horoballs of different types, Monatsh. Math., 168/1 (2012), 27-47. [3] E. Molnár, Two hyperbolic football manifolds. In: Proceedings of International Conference
on Differential Geometry and Its Applications, Dubrovnik Yugoslavia, 1988. 217–241. [4] E. Molnár, On non-Euclidean crystallography, some football manifolds, Struct Chem (2012)
23:1057–1069 DOI 10.1007/s11224-012-0041-z [5] J. Szirmai, The optimal hyperball packings related to the smallest compact arithmetic 5-
orbifolds, Kragujevac J. Math. (to appear) (2016).
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
2016.09.08.
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.
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
2016.09.08.
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 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.
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Spaces of constant curvature - 1
Rogers – 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
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Spaces of constant curvature - 2
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 claimed 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. 15 (1964), 237--245.
K. Böröczky Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978), 243--261.
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
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/s00605-012-0393-x, arXiv:1007.0722.
Optimal arrangements
Hyperbolic space, Beltami-Cayley-Klein model,
(3,3,6) Tetrahedron tiling
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
2016.09.08.
Cycles in hyperbolic plane,
spheres in hyperbolic space
U
V
P
X
X
V
t
O Ot
t
U
t
t
U
V
V
t
O
Ut
t
=
x
y
Ot
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 Horocycle Hypercycle
X
X
O Ot
t
t
=
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Complete orthoschemes - 1
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XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Complete orthoschemes - 2
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Complete orthoschemes - 3
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Coxeter-Schläfli matrix and its inverse
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Main cases
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Some essential points - 1
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Some essential points - 2
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
The volume of the orthoscheme
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Volume of the ball; packing and covering
densities
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Ball packings and coverings
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Ball packings and coverings
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Ball packings and coverings
Case 1.i.a
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Ball packings and coverings
Case 1.i.a
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Ball packings and coverings
Case 1.s.i.a
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Ball packings and coverings
Case 1.s.i.a
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
E. Molnár, Two hyperbolic football manifolds. In: Proceedings of International Conference on Differential Geometry and Its Applications, Dubrovnik Yugoslavia, 1988. 217–241.
E. Molnár, On non-Euclidean crystallography, some football manifolds, Struct Chem (2012) 23:1057–1069 DOI 10.1007/s11224-012-0041-z
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Ball packings and coverings
Case 1.i.b
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Ball packings and coverings
Case 1.i.b
2016.09.08.
XIX. Geometrical Seminar 2016,
Zlatibor/Serbia
Thank you