Apollonian Packings and Dense Granular Systems

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

2. 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.

3. High performance cement (HPC)(Christian Vernet, Bouygues)

4. San Andreas fault tectonic plate 2 tectonic plate 1 gouge

5. Roller bearing ?

6. Apollonian packings

7. Apollonian packing Space between disks is fractal(Mandelbrot: „self-inverse“ fractal) of dimension Boyd (73): bounds: 1.300197 < < 1.314534 numerical: = 1.3058 7

8. Example for space filling bearing

9. construction by inversion C D D‘ C‘ C‘‘

10. C‘‘ C‘ D‘ D C construction by inversion

11. C D D‘ C‘ C‘‘ construction by inversion

12. C‘‘ C‘ D‘ D C construction by inversion

13. construction by inversion

14. Construction of space filling bearing

15. Möbius transformations mapping that maps circles into circles (in d=2) z = point in complex plane mapping is conformal, ie preserves angles 15

16. Solution of coordination 4 without loss of generality consider only largest disks in a strip geometry 1 3 4 4 x 2 3 1 x 2 x center of inversion to fill largest wedge 16

17. Solution of coordination 4 invariance under reflexion 1st family 2nd family 2a 2a periodicity disks touching 17

18. Inversion inversions: x = radial distance from Inversion center 18

19. Total transformation reflexion around a: consider B: 0th disk: mth disk: m times 19

20. Solving the odd case m odd last disk: symmetric under T, ie at a m 20

21. Solving the even case m even last disk: is fixed point, ie at m 21

22. Continuous fraction equations m odd m even 22

23. Result For four-fold loops one has two families: (n,m) 23

24. Examples for zm 24

25. First family touching of largest spheres: case n=2, m=1 : 25

26. Classification of space filling bearing n=1 m=1 n=2 m=1 n=∞ m=1 n=3 m=1

27. First family 27

28. Second family A 2a 0 Exists additional symmetry: On strip: A is fixed point of both inversions 28

29. Second family 29

30. Second family n = m = 0 30

31. Second family n = 1, m = 0 31

32. Second family n = 4, m = 1 32

33. Second family n = m = 3 33

34. Loop 6 34

35. Loop 8 35

36. Scaling laws Fractal dimension Disk-size distribution

37. Scaling laws r = Radius of disk suppose

38. Fractal dimensions m First family n m 2 1,33967 (5) n

39. Mahmoodi packing • Reza Mahmoodi Baram

40. Rolling space-filling bearings See movie on: http://www.comphys.ethz.ch/hans/appo.html

41. Three-dimensional loop

42. Rotation of spheres without frustration To avoid friction the tangent velocity at any contact point must be the same:

43. Rotation of spheres without frustration For a loop of n spheres, the consistency condition is: which implies if we choose and wehave Therefore, under the following condition we have rotating spheres without any sliding friction:

44. Apollonian packing

45. Apollonian network • scale-free • small world • Euclidean • space-filling • matching with J.S. Andrade, R. Andrade and L. Da Silva Phys. Rev. Lett., 94, 018702 (2005)

46. Systems of electrical supply lines Friendship networks Computer networks Force networks in polydisperse packings Highly fractured porous media Networks of roads Applications

47. Degree distribution

48. Small-world properties Z. Zhang et al PRE 77, 017102 (2008)

49. Ising model opinion with Roberto Andrade

50. Feliz Cumpleaños, Eric !......