Space-filling packings

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The figure on the left shows a 2d Apollonian packing. With S.S. Manna we calculated its fractal dimension to be 1.3057 +/- 0.0001 as published in J. Phys. A Vol. 24, L481-L490 (1991).
In 1989 I constructed a much larger family of such packings of different topology and fractal dimension that also could serve as space-filling bearings as explained in detail in the proceedings of the Cargese School of July 1990. In an article published in Phys. Rev. Lett. Vol. 65, p.3223-3226 (1990) we showed with Giorgio Mantica and Daniel Bessis that there exist only two discrete families of different solutions for packings having only loops of four discs. The generalization to arbitrary even loop length was published with Gadi Oron in a paper published in J.Phys.A Vol. 33, 1417-1434 (2000).

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The picture on the right shows a configuration in the strip geometry (goes to infinity in horizontal direction) of the first family having loops of length 6 and the parameters n=m=0. It is obtained by a conformal map that consists iteratively of an inversion and a reflection. Only very specific radia for the original circles as well as for the inversion circles are allowed which gives the condition for the discrete families of possible solutions. Some artistic pictures were also made by Jos Leys. A popular view also appeared in Pour la Science , in Physics World of February 1991, in La Recherche of April 1991 and even as cover of "Physik in unserer Zeit".

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The most prominent application of these bearings in Nature seem to be shear bands as the one shown in the figure beside. Indeed rotations have been measured and simulated within shear bands in several occasions. Famous shear bands on a gigantic scale are faults between tectonic plates, like the San Andreas fault. They have so called "sismic gaps". i.e. regions without earthquakes or measurable frictional heat. One way to measure rotations in geological faults is through the remnant magnetization of the earth as explained in a paper published with Stephane Roux, Alex Hansen and Jean Pierre Vilotte in Geophysical Research Letters, Vol. 20, 1499-1502 (1993).

That roller bearings do in fact spontaneously appear in shear bands has in fact also been observed in simulations of packings of disks of similar size under uniaxial load as we showed in a paper with Jan Astrom and Jussi Timonen published in Physical Review Letters, Vol.84, 638-641 (2000). In a more recent paper published in Physica A, Vol.344, p.516-522 (2004) I combine this with the existence of three-dimensional bearings as discussed below.

Recently with Reza Mahmoodi we made some progress in three dimensions. We could construct five different topologies using conformal mappings as described in our paper that has been published in Fractals, Vol.12, p.293-301 (2004). Together with Nick Rivier we could prove that in one case which is bi-chromatic (see figure) it acts as a slip-free rolling, space-filling bearing as described in the paper published in Physical Review Letters Vol.92, 044301 (2004) and as you can see in this movie . Jos Leys has produced beautiful pictures of our packings.


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In the figure one sees the new space-filling bearing based on an octahedron instead of a tetrahedron as usually used in Apollonian packings. Its fractal dimension is 2.58 as compared to 2.47 for the classical case. See also the paper published in Physica A, Vol.330, p.77-82 (2003) or in an article the February 6 2004 edition of Science Now, an article in the April 2004 edition of Scientific American or some German newspaper texts. You can also see some movies of the rolling bearings on Reza's homepage .


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One up to now not really understood particularity of our conformal mapping method to construct self-similar packings is that in some specific cases of the second family of solutions the space is not completely filled but holes appear as seen in the picture besides. These holes are surrounded by a powder of infinitely small disks that mechanically decouple the hole from the rest of the packing. Due to self similarity one never finds just one hole but if there is one hole there are also infinitely many smaller holes in the system.


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As seen from the picture at left it is also possible to construct random space-filling bearings. There blue circles only touch grey ones and grey only blue ones. At some places one might think that two blue do touch in the figure but in fact there is a layer of very small grey particles in between. This configuration is obtained numerically by placing one by one circles (or spheres) onto random positions into the not yet occupied space and repositioning and growing them until an even loop is created optimizing at the same time towards maximum filling. A paper about this together with Reza Mahmoodi has been published in Physical Review Letters, Vol.95, 224303 (2005).

You can also download a talk that I gave in several occasions on the subject.

Finally we should mention that one can also study the network formed by the connections between the centers of mass of an Apollonian packing. We call them "Apollonian networks" (see figure below). They can be applied to porous media, polydisperse packings, road networks or electrical supply systems and they have interesting properties like being scale-free, ultrasmall world, Euclidean, matching and space-filling. With Jose Soares Andrade, Roberto Andrade and Luciano da Silva we recently published in Physical Review Letters, Vol.94, p.018702 (2005). In fact this network already forms part of Wolfram's Mathematica . Photo

A calculation of coupled maps on such a network has recently been achieved with Pedro Lind and Jason Gallas as can be seen in our paper published in Physical Review E, Vol.70, p.056207 (2004). With Roberto Andrade we also wrote a extensive paper published in Phys. Rev. E, Vol. 71, p.056131 (2005) on the properties of the ferro- and antiferromagnetic Ising model on the Apollonian network.