hans herrmann apollonian variations computational physics ifb, eth zürich switzerland disco...
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Hans HerrmannHans Herrmann
Apollonian variationsApollonian variations
Computational PhysicsIfB, ETH Zürich
Switzerland
DISCO DISCO Dynamics of Complex SystemsDynamics of Complex Systems
Valparaiso Valparaiso November 24-26, 2011November 24-26, 2011
Feliz Cumpleaños !
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: 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.
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
High performance cement (HPC)(Christian Vernet, Bouygues)(Christian Vernet, Bouygues)
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: 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, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Roller bearing ?
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Apollonian packings
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 7
Apollonian packing
Space between disks is fractal(Mandelbrot: „self-inverse“ fractal)
of dimension
Boyd (73):
bounds: 1.300197 < < 1.314534
numerical: = 1.3058
fd
fdfd
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Example for space filling bearing
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
construction by inversionconstruction by inversion
DD‘
C
C‘ C‘‘
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
DD‘
C
C‘ C‘‘
construction by inversionconstruction by inversion
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
DD‘
C
C‘ C‘‘
construction by inversionconstruction by inversion
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
DD‘
C
C‘ C‘‘
construction by inversionconstruction by inversion
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
construction by inversionconstruction by inversion
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Construction of space filling bearing
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 15
Möbius transformations
mapping that maps circles into circles (in d=2)
z = point in complex plane
mapping is conformal, ie preserves angles
: with 1a b
adz
z - bz
cA a d
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 16
Solution of coordination 4
without loss of generality consider only largest disks in a strip geometry
12
34
x 3
41
2
x
center of inversion to fill largest wedgex
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
2a
17
Solution of coordination 4
invariance under reflexion
2a
disks touching periodicity
1st family 2nd family
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 18
Inversion
inversions:
x = radial distance from Inversion center
2
:A
A xx
r
2
:B
B xx
r
2 2A Ar R 2 2B Br R
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 19
Total transformation
reflexion around a:
consider B: 0th disk:
mth disk:
: 2B x xT a
0b
0m T TBb BTBbm times
2
( )2
BrB x
xT
a
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 20
Solving the odd case
m odd
last disk:
symmetric under T, ie at a
0...m B Bb T TBbm
2
2
2
2
22
..
2
B
B
B
B
a
aa
r
a
r
r
r
2
2
BZ
r
a
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 21
Solving the even case
m even
last disk:
is fixed point, ie at
0...m Tb TB TBbm
2
2
2
2
22
2..
2
BB
B
B
B
rr
rr
aa
a
ar
2
2
BZ
r
a
Br
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 22
Continuous fraction equations
m odd m even
12
22
2
.
.
.
2
m
m
m
m
m
ZZ
ZZ
Z
1
2
mtimes
22
22
2
.
.
.
2
mm
m
m
m
m
ZZ
ZZ
Z
Z
2
mtimes
1z
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 23
ResultResult
For four-fold loops one has two families: (n,m)
2cos3n n
z
2radius of inversion
half-periodnz
1st family 2nd family
half-period
radius of upper circle
radius of lower circle
AR
BR
a 2 1n mza z 2n mza z
1
2 1nA
n
m
z
zR
z
1
2
n
nA
m
z
z zR
1
2 1nB
m
m
z
zR
z
1
2
m
nB
m
z
z zR
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 24
Examples for zm
0 1 2 3 5
4 2 1
m
mz 6 2 54
34 2 2
24
3 2: 4 4 8 0x x xz x 2 4 3 2
6 : 4 12 8 16 0x x x xz x
2a r a
1 4mz
1r
11
2a
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 25
First family
touching of largest spheres:2 2 2( ) (1 )A B A BR R a R R 2 2 2 1A Ba r r 2 1n mza z
221A
n mA
nz
z zrR
22
1Bn m
Bmz
z zrR
case n=2, m=1 :1 6 2 5 2
, , 7 2 5 7 2 57 2 5
A Ba r r
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
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, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 27
First family
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 28
Second family
Exists additional symmetry:
On strip:A
0 2a
A is fixed point of bothinversions
( , 2 )BxA R 2, ,2A B A Br R
2 22 (2 )B Br Rx 2 22 (2 )A Ar Rx
2 4 2 4: A A B Br r r rn m 2 2( 1) ( 1)n n m mz z z za a
22
1 1 1 1, :
4 2 2 2A B A nBn m R R r r x za
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 29
Second familySecond family
2 1
forn m
az
nz
m
2 1 for
2 nzma n
2 2, for A Bn m
n m n m
nz z
z zm
z zr r
, 1 1
4 4 for A B mr r n
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 30
Second family
n = m = 0
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 31
Second family
n = 1, m = 0
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 32
Second family
n = 4, m = 1
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 33
Second family
n = m = 3
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 34
Loop 6
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011 35
Loop 8
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Scaling laws
fdcut off = radius of smallest disk
covered area - Porosity 1
total area
- Surface sum of perimeters per unit area
p
s
Fractal dimensionFractal dimension
Disk-size distributionDisk-size distribution
2
# of disks of area per unit area
# of disks of radius per unit area
s
s
n s
N n ds
sn
fdp 2~ fds 1~
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Scaling laws
suppose
2 1fd
1.305768 1.85fd
r = Radius of disksns ~ 2~ rrn
21~
drrnN
22~2
drrrns
232 ~1
drrnrp
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Fractal dimensions
0 1 3
0 1,421,4321 (1)
1,40567 (10) 1,30
1 1,411,4123 (2)
1,38
3 1,36
1,301,305768
(1)
fd
2
1,33967 (5)
0 1 3
0 1,72 1,71
1 1,71 1,71
3 1,71
1,67
m
m
n
n
First family
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Mahmoodi packing
• Reza Mahmoodi Baram
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Rolling space-filling bearings
0.5c 0.0c
http://www.comphys.ethz.ch/hans/appo.html
See movie on:
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Three-dimensional loop
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: 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:
0ˆˆˆ
122211
21221121
21
rRRrRrR
vv
1
11,1,11
1
12121122
ˆ11
ˆ
j
iiiii
ijjjj rRR
rRR
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: 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:
is evenn
1, 1 , 11
ˆ1 0n n i
i i i ii r
, 1 11i
i i i ic R R and
n
iiiii
inn rRR1
1,1,1
1111 ˆ11
1,11, ˆ iiiiii rRRR
n
i
n
iii
niiii
in Rcr1 1
1,1
1,1,1 01ˆ1
1
11,11
1 11j
iii
jjjj RcRR
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Apollonian packingApollonian packing
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Apollonian networkApollonian network
• scale-free
• small world
• Euclidean
• space-filling
• matching
with J.S. Andrade, R. Andrade and L. Da Silvawith J.S. Andrade, R. Andrade and L. Da Silva Phys. Rev. Lett., Phys. Rev. Lett., 94 94, 018702 (2005), 018702 (2005)
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
ApplicationsApplications
• 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, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Degree distributionDegree distribution
n-1
n- 2
2
1
2
3 3
3 3 2
3
scale
3 2
-free: (
3
)
( )
( )
,
n
k
P k k
W k km k n
n-1
n
n+1
3 2
3 3 2
1 ln 3
1.585l
2
n 2
number of sites at generation ( , ) number of vertices of degree
cummulative distribution ( ) ( , ) /
n
nk k
N nm k n k
W k m k n N
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Small-world propertiesSmall-world properties
clustering coefficient
2number of connections between neighbors
( 1)C
k k
shortest path
chemical distance between two sitesl
0.828C lnl N
Z. Zhang et al PRE 77, 017102 (2008)Z. Zhang et al PRE 77, 017102 (2008)
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Ising modelIsing model
coupling constant
correlation length diverges at
free energy, entropy, specific heat are smooth
magnetization
n
n c
T
J n
J T
m e T
opinionopinion
with Roberto Andradewith Roberto Andrade
DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011DISCO: Dynamics of Complex Systems, Valparaiso, November 24-26, 2011
Feliz Cumpleaños,
Eric !......