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Hans Herrmann Apollonian variations Computational Physics IfB, ETH Zürich Switzerland DISCO Dynamics of Complex Systems Valparaiso Valparaiso November 24-26, 2011 Feliz Cumpleaños !

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 The art of packing densely Dense packings of granular systems are of fundamental importance in the manufacture of hard ceramics and ultra strong concrete. The key ingredient lies in the size distribution of grains. In the extreme case of perfect filling of spherical beads (density one), one has Apollonian tilings with a powerlaw distribution of sizes.

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 (Christian Vernet, Bouygues) High performance cement (HPC) (Christian Vernet, Bouygues)

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 San Andreas fault tectonic plate 2 tectonic plate 1 gouge

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Roller bearing ?

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Apollonian packings

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Apollonian packing Space between disks is fractal (Mandelbrot: self-inverse fractal) of dimension Boyd (73): bounds: < < numerical: =

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Example for space filling bearing

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 construction by inversion D D C C C

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 D D C C C construction by inversion

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 D D C C C construction by inversion

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 D D C C C construction by inversion

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 construction by inversion

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Construction of space filling bearing

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Möbius transformations mapping that maps circles into circles (in d=2) z = point in complex plane mapping is conformal, ie preserves angles

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Solution of coordination 4 without loss of generality consider only largest disks in a strip geometry x x center of inversion to fill largest wedge x

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, a 17 Solution of coordination 4 invariance under reflexion 2a disks touching periodicity 1st family2nd family

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Inversion inversions: x = radial distance from Inversion center

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Total transformation reflexion around a: consider B: 0th disk: mth disk: m times

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Solving the odd case m odd last disk: symmetric under T, ie at a m

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Solving the even case m even last disk: is fixed point, ie at m

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Continuous fraction equations m oddm even

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Result For four-fold loops one has two families: (n,m) 1st family2nd family half-period radius of upper circle radius of lower circle

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Examples for z m

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, First family touching of largest spheres: case n=2, m=1 :

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Classification of space filling bearing n=1 m=1n=2 m=1 n=3 m=1 n= m=1

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, First family

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Second family Exists additional symmetry: On strip: A 0 2a A is fixed point of both inversions

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Second family

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Second family n = m = 0

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Second family n = 1, m = 0

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Second family n = 4, m = 1

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Second family n = m = 3

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Loop 6

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, Loop 8

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Scaling laws Fractal dimension Disk-size distribution

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Scaling laws suppose r = Radius of disk

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Fractal dimensions ,42 1,4321 (1) 1,40567 (10) 1,30 1 1,41 1,4123 (2) 1,38 3 1,36 1,30 1, (1) 2 1,33967 (5) ,721, ,67 m m n n First family

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Mahmoodi packing Reza Mahmoodi Baram

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Rolling space-filling bearings See movie on:

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Three-dimensional loop

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Rotation of spheres without frustration To avoid friction the tangent velocity at any contact point must be the same:

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Rotation of spheres without frustration For a loop of n spheres, the consistency condition is: which implies if we choose we have Therefore, under the following condition we have rotating spheres without any sliding friction: and

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Apollonian packing

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Apollonian network scale-free small world Euclidean space-filling matching with J.S. Andrade, R. Andrade and L. Da Silva Phys. Rev. Lett., 94, (2005) Phys. Rev. Lett., 94, (2005)

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Applications Systems of electrical supply lines Friendship networks Computer networks Force networks in polydisperse packings Highly fractured porous media Networks of roads

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Degree distribution

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Small-world properties Z. Zhang et al PRE 77, (2008)

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Ising model opinion with Roberto Andrade

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DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 Feliz Cumpleaños, Eric !......

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