top dense ball packings and coverings in hyperbolic...

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.


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

[email protected], [email protected]

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.

mailto:[email protected]



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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).

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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.

http://en.wikipedia.org/wiki/Johannes_Kepler

http://en.wikipedia.org/wiki/Johannes_Kepler

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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.

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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

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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.

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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

http://arxiv.org/abs/1007.0722

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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

=

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Complete orthoschemes - 1

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Coxeter-Schläfli matrix and its inverse

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Main cases

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Some essential points - 1

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Some essential points - 2

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The volume of the orthoscheme

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Volume of the ball; packing and covering

densities

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Ball packings and coverings

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Case 1.i.a

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Case 1.s.i.a

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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.


Zlatibor/Serbia


Case 1.i.b

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Thank you

top dense ball packings and coverings in hyperbolic...

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