hans herrmann

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.


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


San Andreas fault

tectonic plate 2

tectonic plate 1

gouge


Roller bearing ?


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


Example for space filling bearing


construction by inversionconstruction by inversion

DD‘

C

C‘ C‘‘


DD‘

C

C‘ C‘‘



Construction of space filling bearing


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


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


2a

17

Solution of coordination 4

invariance under reflexion

2a

disks touching periodicity

1st family 2nd family


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


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


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


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


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


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


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


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


Classification of space filling bearing

n=1 m=1 n=2 m=1

n=3 m=1 n=∞ m=1


First family


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


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


Second family

n = m = 0


Second family

n = 1, m = 0


Second family

n = 4, m = 1


Second family

n = m = 3


Loop 6


Loop 8


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~


Scaling laws

suppose

2 1fd

1.305768 1.85fd

r = Radius of disksns ~ 2~ rrn

21~

drrnN

22~2

drrrns

232 ~1

drrnrp


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


Mahmoodi packing

• Reza Mahmoodi Baram


Rolling space-filling bearings

0.5c 0.0c

http://www.comphys.ethz.ch/hans/appo.html

See movie on:


Three-dimensional loop


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


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


Apollonian packingApollonian packing


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)


ApplicationsApplications

• Systems of electrical supply lines

• Friendship networks

• Computer networks

• Force networks in polydisperse packings

• Highly fractured porous media

• Networks of roads


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


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)


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


Feliz Cumpleaños,

Eric !......

hans herrmann

Documents

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