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

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

Example for space filling bearing

construction by inversionconstruction by inversion

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

z - bz

cA a d

Solution of coordination 4

without loss of generality consider only largest disks in a strip geometry

center of inversion to fill largest wedgex

Solution of coordination 4

invariance under reflexion

disks touching periodicity

1st family 2nd family

Inversion

inversions:

x = radial distance from Inversion center

2 2A Ar R 2 2B Br R

Total transformation

reflexion around a:

consider B: 0th disk:

mth disk:

: 2B x xT a

0m T TBb BTBbm times

Solving the odd case

last disk:

symmetric under T, ie at a

0...m B Bb T TBbm

Solving the even case

m even

last disk:

is fixed point, ie at

0...m Tb TB TBbm

Continuous fraction equations

m odd m even

mtimes

ResultResult

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

2cos3n n

2radius of inversion

half-periodnz

1st family 2nd family

half-period

radius of upper circle

radius of lower circle

a 2 1n mza z 2n mza z

Examples for zm

0 1 2 3 5

mz 6 2 54

34 2 2

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

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

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

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

1 1 1 1, :

4 2 2 2A B A nBn m R R r r x za

Second familySecond family

forn m

2 1 for

2 nzma n

2 2, for A Bn m

n m n m

z zr r

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

Fractal dimensionFractal dimension

Disk-size distributionDisk-size distribution

# of disks of area per unit area

# of disks of radius per unit area

N n ds

fdp 2~ fds 1~

Scaling laws

suppose

1.305768 1.85fd

r = Radius of disksns ~ 2~ rrn

drrrns

232 ~1

drrnrp

Fractal dimensions

0 1,421,4321 (1)

1,40567 (10) 1,30

1 1,411,4123 (2)

3 1,36

1,301,305768

1,33967 (5)

0 1,72 1,71

1 1,71 1,71

3 1,71

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

rRRrRrR

11,1,11

12121122

ijjjj 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

inn rRR1

1111 ˆ11

1,11, ˆ iiiiii rRRR

in Rcr1 1

1,1,1 01ˆ1

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

-free: (

W k km k n

1 ln 3

1.585l

number of sites at generation ( , ) number of vertices of degree

cummulative distribution ( ) ( , ) /

N nm k n k

W k m k n N

Small-world propertiesSmall-world properties

clustering coefficient

2number of connections between neighbors

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

opinionopinion

with Roberto Andradewith Roberto Andrade

Feliz Cumpleaños,

Eric !......

hans herrmann

complex planemapping

mth disk

largest disks

apollonian tilings

apollonian packingspace

apollonian packingsdisco

solution of coordination

inversion centerdisco

Documents

and house herrmann

nuno a. m. araujo, jose s. andrade jr., hans j. herrmann ·...

Öffentliche sitzung · 2020. 6. 2. · frau pia aßmann...

jürgen herrmann, mdb deutschland (k)ein sozialstaat !?...

tics modelling damy herrmann

hans herrmann apollonian variations computational physics...

diktate in der grundschule susanne herrmann susanne herrmann...

kinofilm - mathias herrmann

menaker herrmann - uscourts.gov

bernard herrmann - the egyptian

genossenie schaft feiert 111jähriges jubiläu...günter...

herrmann ...

thuner stadtschütz - stadtschuetzen-thun.ch · 31...

!!programmänderungen vorbehalten!! einlass und … · von...

ulrike herrmann

guido f. herrmann:

martin herrmann

ulf herrmann, flagsol

capillary forces in assemblies of three spherical particles...

dr. karl-josef schmidt wird neues mitglied der ... ·...