Apollonian variations
Hans Herrmann
Computational Physics
IfB, ETH Zürich
Switzerland
DISCO
Dynamics of Complex Systems
Valparaiso
November 24-26, 2011
Feliz Cumpleaños !
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.
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
High performance cement (HPC)
(Christian Vernet, Bouygues)
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
San Andreas fault
tectonic
plate 2
tectonic
plate 1
gouge
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Roller bearing ?
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Apollonian packings
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Apollonian packing
Space between disks is fractal
(Mandelbrot: „self-inverse“ fractal)
of dimension df
Boyd (73):
bounds: 1.300197 < df < 1.314534
df = 1.3058
numerical:
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
7
Example for space filling bearing
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
construction by inversion
C
C‘
D
D‘
C‘‘
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
construction by inversion
C‘
C‘‘
C
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
construction by inversion
C
C‘
D
D‘
C‘‘
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
construction by inversion
C‘
C‘‘
C
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
construction by inversion
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Construction of space filling bearing
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Möbius transformations
mapping that maps circles into circles (in d=2)
z = point in complex plane
az  b
A: z 
az  d
with ad - bc  1
mapping is conformal, ie preserves angles
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
15
Solution of coordination 4
4
1
1
3
4
2
x
3
2
without loss of generality consider only largest
disks in a strip geometry
center of inversion to fill largest wedge
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Solution of coordination 4
invariance under reflexion
1st family
2nd family
2a
disks touching
2a
periodicity
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Inversion
inversions:
2
rA
A: x 
x
rB 2
B:x 
x
x = radial distance from Inversion center
rA2  2RA
rB 2  2RB
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Total transformation
TB : x  2a  x
reflexion around a:
consider B:
mth disk:
0th disk: b0
bm  TBTBTBb0
m times
2
rB
BT ( x) 
2a  x
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Solving the odd case
m odd
last disk: bm  BT ...BTBb0
m
symmetric under T, ie at a
rB 2
a
rB 2
2a 
rB 2
2a 
..
rB 2
2a
rB 2
Z 2
a
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Solving the even case
m even
last disk: bm  TBT ...TBb0
m
is fixed point, ie at rB
rB 2
rB  2 a 
rB 2
2a 
rB 2
2a 
..
rB 2
2a
rB 2
Z 2
a
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
21
Continuous fraction equations
m odd
1
m even
Zm
Zm
2
Zm
2
Zm
2
2
.
2  Zm 
m 1
 times
2
Zm
Zm
2
Zm
2
Zm
2
2
.
.
.
.
.
Zm
2
Zm
2
m
 times
2
z  1
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Result
zn  cos 2

 radius of inversion 
zn  

half-period


n3
2
For four-fold loops one has two families: (n,m)
1st family
half-period a
2nd family
2
a  zn  zm  1 a
2
 zn  zm
radius of upper circle RA
RA 
1
zn
2 zn  zm  1
RA 
1 zn
2 zn  zm
radius of lower circle RB
RB 
1
zm
2 z n  zm  1
RB 
1 zm
2 z n  zm
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Examples for zm
m
0
1
zm
4
2
2
3
5

62 5
4
3
42 2
1
z4  x 2 : x 3  4 x 2  4 x  8  0
z6  x 2 : x 4  4 x3  12 x 2  8 x  16  0
a  r  2a
r 1
1  zm  4
1
 a 1
2
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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First family
touching of largest spheres:
( RA  RB ) 2  a 2  (1  RA  RB ) 2
a 2  rA2  rB 2  1
a 2  zn  zm  1
zm
2 RB  rB 
zn  zm  1
zn
2 RA  rA 
zn  zm  1
2
2
case n=2, m=1 :
a
1
72 5
,
rA 
62 5
, rB 
72 5
2
72 5
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Classification of space filling bearing
n=1 m=1
n=2 m=1
n=3 m=1
n=∞ m=1
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
First family
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
27
Second family
Exists additional symmetry:
On strip:
A is fixed point of both
inversions
A
2a
0
A  ( x, 2 RB )
rA, B 2  2RA, B
rB 2  x 2  (2 RB ) 2
rA 2  x 2  (2 RA ) 2
n  m : rA2  rA4  rB 2  rB 4
 zn ( zn a 2  1)  zm ( zm a 2  1)
n  m : RA  RB 
1
1
 rA 2  rB  ,
4
2
x
1
1
 zn  2
2
2a
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Second family
a2 
1
for n  m
zn  zm
rA2 
rA 
zn
,
zn  zm
1
4
,
a2 
1
for n  m
2 zn
rB 2 
zm
for n  m
zn  zm
rB 
1
4
for n  m
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Second family
n=m=0
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
30
Second family
n = 1, m = 0
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Second family
n = 4, m = 1
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
32
Second family
n=m=3
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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Loop 6
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
34
Loop 8
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
35
Scaling laws
Fractal dimension d f
cut off  = radius of smallest disk
covered area
- Porosity p  1 
total area
p~
2d f
s ~
1d f
- Surface s  sum of perimeters per unit area
Disk-size distribution ns
ns  # of disks of area s per unit area
N   

 n ds  # of disks of radius
s
  per unit area
 2
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Scaling laws
suppose
ns ~ s
nr  ~ r

2
r = Radius of disk

N     nr dr ~  12


s   2rnr dr ~  22


3 2


p  1   r n r dr ~ 
2

d f  2  1
1.305768  d f  1.85
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Fractal dimensions
m
df
0
n
0
1
3
1,42
1,40567
(10)

First family
1,30
1,4321 (1)
1,41
1
1,38
1,4123 (2)
1,36
3


1,30
1,305768
(1)
2
1,33967 (5)
n
0
1
3

m
0
1
3
1,72

1,71
1,71
1,71
1,71
1,67
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Mahmoodi packing
• Reza Mahmoodi Baram
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Rolling space-filling bearings
See movie on:
http://www.comphys.ethz.ch/hans/appo.html
c  0.0
c  0.5
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Three-dimensional loop
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:
 
v1  v2 


R1rˆ12  1   R2 rˆ12  2 


R11  R22  rˆ12  0


R22   R11  12rˆ12


j 1
R j j   1 R11    1  i ,i 1rˆi ,i 1
j 1
j i
i 1
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:


n
R11   1 R11    1
which implies
n

n
n i 1
i 1
 i ,i 1rˆi ,i 1
is even
n
n i 1
1
i 1 
i ,i 1rˆi ,i 1 0

i
if we choose  i ,i 1  c  1  Ri  Ri 1  and R
ˆ
i ,i 1  Ri  Ri 1 ri ,i 1
n
n

n i 1
n 1
we have   1
 i ,i 1rˆi ,i 1  c 1  Ri ,i 1  0
i 1
i 1
Therefore, under the following condition we have rotating spheres without any
sliding friction:
j 1

R j j   1
j 1

R11  c 1  Ri ,i 1

j
i 1
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Apollonian packing
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, 018702 (2005)
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
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Degree distribution
scale-free: P(k )  k 
 W (k )  k 1
k
m( k , n )
3n
3
3n-1
3 2
3n-2
3  22
N n  number of sites at generation n
m(k , n)  number of vertices of degree k
cummulative distribution W (k )   m(k , n) / N n
k  k
3  2n-1
3  2n
2n+1
32
3
1
 
ln 3
 1.585
ln 2
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Small-world properties
Z. Zhang et al PRE 77, 017102 (2008)
clustering coefficient
2
C
 number of connections between neighbors
k ( k  1)
C  0.828
shortest path
l  chemical distance between two sites
l  ln N
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Ising model
opinion
coupling constant J n  n 
correlation length J n diverges at Tc
free energy, entropy, specific heat are smooth
magnetization m  e  T

T 
with Roberto Andrade
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Feliz Cumpleaños,
Eric !......
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011