FRACTALS

LINDENMAYER SYSTEMS

November 22, 2013

Rolf Pfeifer

Rudolf M. Füchslin

RECAP

HIDDEN MARKOV MODELS

What Letter Is Written Here?

What Letter Is Written Here?

What Letter Is Written Here?

The Idea Behind Hidden Markov Models

First letter: Maybe „a“, maybe „q“

Second letter: Maybe „r“ or „v“ or „u“

Take the most probable combination as a guess!

Hidden Markov Models

Sometimes, you

don‘t see the states,

but only a mapping

of the states.

A main task is then

to derive, from the

visible mapped

sequence of states,

the actual underlying

sequence of

„hidden“ states.

HMM: A Fundamental Question

What you see are

the observables.

But what are the

actual states behind

the observables?

What is the most

probable sequence

of states leading to

a given sequence

of observations?

The Viterbi-Algorithm

We are looking for indices M1,M2,...MT, such that

P(qM1,...qMT) = Pmax,T is maximal.

1. Initialization

2. Recursion (1 t T-1)

3. Termination

4. Backtracking

11

1

( )

( ) 0

i i ki b

i

11

1

( ) max( ( ) )

( ) : ( ) max.

tt t i j j ki

t t i j

j i a b

j i i a

max,

max,

max( ( ))

: ( ) max.

T T

T i T

P i

q q i

1 1( )t t tM M

Efficiency of the Viterbi Algorithm

• The brute force approach takes O(TNT) steps. This is

even for N = 2 and T = 100 difficult to do.

• The Viterbi – algorithm in contrast takes only O(TN2)

which is easy to do with todays computational means.

Applications of HMM

• Analysis of handwriting.

• Speech analysis.

• Construction of models for prediction.

Only few processes are really Markov processes (neither

writing nor speech is), but often, models based on Markov

processes are good approximations.

END RECAP

FRACTALS

Natural Geometry

Geometry in

text books Geometry in

nature

Fractals: Informal Definition

• Termed coined by Benoit

Mandelbrot

• Geometry without

smoothness Structure on

all scales (detail persists

when zoomed arbitrarily)

• Geometrical objects

generally with non-integer

dimension

• Self-similarity (contains

infinite copies of itself)

Fractals in the Human Body

Lung

Kidney

Cortical surface (?)

The Length of Borders

• Lewis Fry Richardson: Probability of war between two

adjacent countries proportional to length of border?

• Checking the theory required gathering data about

border lengths.

• Surprising finding: There are strongly varying numbers in

the literature.

The Border of Great Britain

The Border of Great Britain

The closer you look, the longer the border.

And the growth doesn‘t stop!

A Slightly Different View on Dimension

One-Dimensional Objects

1

1

1

: Diameter of disk

: Number of disks

: Constant

N c

N

c

Two-dimensional Objects

2

?

: Diameter of disk

: Number of disks

: Constant

N

N

c

Two-dimensional Objects

2

2

2

1

: Diameter of disk

: Number of disks

: Constant

N c

N

c

Definition of the Fractal Dimension

1

: Diameter of disk

: Number of disks

: Constant

D: Hausdorff Dimension

D

N c

N

c

0 0 0

log( ( )) log( ) log( ( )) log( ( ))lim lim lim

1 1 log( )log( ) log( )

N c N ND

The number of disks

necessary for covering

a structure grows with

shrinking λ.

Example: The Sierpinski Triangle

Construction of the

Sierpinski-triangle

A Sierpinski triangle

contains a whole copy of

itself in its parts.

Example: The Sierpinski Triangle

Construction of the

Sierpinski-triangle

0

log( ( ))lim

log( )

ND

Example: The Sierpinski Triangle

log(3 ) log(3)lim 1.585

1 log(2)log( )

2

n

n

n

D

2

2 1lim3 02

n

nnA N

Hausdorff Dimension:

Area of Sierpinski triangle

13 lim3 3

2

n

nnL KN K

Boundary length of ST:

Example: Cantor Dust

Take out the middle third!

0

log( ( ))lim

log( )

ND

Example: Cantor Dust

Take out the middle third!

1lim(no elements = 2 ) (length element = )

3

n

nnL

log(2 ) log(2)lim

1 log(3)log( )

3

n

n

n

D

Example: The Mandelbrot Set

Example: The Mandelbrot Set

2

1

0 0,

n nz z c

z c C

2 2

1

1 2

,

n n n

n n n

x x y a

y x y b

z x iy c a ib

The Mandelbrot Set

Three-Dimensional L-Systems

Compressibility

• The Mandelbrot looks complex.

• The algorithm describing the Mandelbrot-set is very

short.

• Procedures generating fractal structures give a very

compressed form of storing complex-looking shapes.

• Directly storing these pictures is actually impossible.

2

1

0 0,

n nz z c

z c C

Self-Similarity and Scale-Invariance

• “When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar.” (B. Mandelbrot)

• Contains infinite copies of itself

• Scaling/Scale = The value measured for a property does not depend on the resolution at which it is measured

• Two types of invariance:

- Geometrical

- Statistical

Fractals in Reality

• Strict self-similarity is mostly found in mathematical

examples.

• Statistical interpretation of self-similarity: If a part of

system can be zoomed up, and this part shows the

same statistical properties as the whole system, one

speaks of self-similarity.

• In reverse (and more important), if coarse-graining

does not change the statistical properties of a system,

one speaks of self-similarity.

Non-Fractal

Random distribution of spheres with uniform radius.

Fractal

Random distribution of spheres with varying random

radius (power law distribution).

Self-Similarity and Coarse-Graining

or Coarse graining =

formation of blocks with

averaged properties

Self-Similarity in Reality

ρ= 0.7

ρ= 0.6

ρ= 0.55

ρ= 0.5

SELF SIMILARITY

Physically important in the description of phase transition.

Fractal Dimension of Time Series

EEG EEG during epileptic seizure

Univ. Zürich:

G. Wieser, P. F. Meier, Y. Shen, HR. Moser. R. Füchslin

Some statistical measures such as the

fractal correlation dimension D2 decrease

shortly before and during an epiliptic

seizure. D2 can be used as a diagnostic

measure.

Random Numbers

• Question: Is a random number self similar?

LINDENMAYER-SYSTEMS:

STUDYING DEVELOPMENT USING

FORMAL LANGUAGES

A Real Puzzle

• Nature is full of well-structured objects.

• These objects are not assembled using global control

and a blue-print, but emerge from local behavior.

External control Self-organization

Self-Assembly is Powerful, but ….

Even if self-assembly

processes may lead to

non-trivial and finite

structures with global

shape and mesoscopic

pattern induced by

microscopic

interactions, it is not

the way how nature

works.

Paul. W. Rothemund

Developmental Representation vs. Blueprint

• All higher living organism develop from a fertilized egg (a

zygote) into their adult form. This process is, at least to

a large degree, controlled by their respective genome.

• Is it that the genome contains a sort of "blueprint" of the

organism?

Developmental Representation vs. Blueprint

• All higher living organism develop from a fertilized egg (a

zygote) into their adult form. This process is, at least to

a large degree, controlled by their respective genome.

• It is NOT TRUE that the genome contains a sort of

"blueprint" of the organism. Rather, the genome

contains instructions which lead to molecules that in

the interaction with the environment lead to

organisms.

Developmental Representation vs. Blueprint

Developmental process is influence by:

• An initial seed.

• The (probably time-dependent) interactions of the

building blocks of a body

• The environment and the physical and chemical laws

ruling this environment.

Developmental representations are iterative in the

sense that they tell you how to proceed if there is

already something there.

Developmental Representation vs. Blueprint

The genome does not contain all the

information it needs to build your body.

Development requires embodiment!

Instructions for construction

=

Developmental representation

+

Laws of the environment

Formal Languages Are Not Enough

• How to describe development by a formal system?

• Problem: The languages we know do not necessarily

lead to globally structured outcomes with repetitive

patterns.

• Reason: External decision of location where a

replacement rule is applied.

On Growth and Form: L-Systems

• The patterns observed in multicellular algae are the

result of developmental processes

• Mathematical formalism introduced in 1968 by Aristid

Lindenmayer.

• Productions are rewriting rules which state how new

symbols (or cells) can be produced from old symbols (or

cells)

L-Systems / Rewriting Systems

• Lindenmayer systems belong to the general class of

parallel grammars or parallel rewriting systems.

• Difference to grammars as we know them: In a parallel

rewriting system, rules are applied to all possible

instances simultaneously. L-systems are subsets of

languages.

• Most practical L-systems are related to context-free

languages.

• Context-free grammars suit the emulation of maturation

and division.

, ( )

A

A V V

Context-free language

The Cantor Set as an L-System

Non-Terminals (variables): ,

Terminals (constants) : none

Start :

Rules :

A B

A

A ABA

B BBB

The Cantor Set as an L-System

Non-Terminals (variables): ,

Terminals (constants) : none

Start :

Rules :

A B

A

A ABA

B BBB

1. A

2. ABA

3. ABABBBABA

4. ABABBBABABBBBBBBBBABABBBABA

5. .....

Anabaena Catenula

A Bio-Inspired L-System

Anabeana catenula: Two types of polar cells,

photosynthesis and nitorgen fixation

Variables : , , ,

Constants :

Start :

Rules :

A A B B

none

A

A AB

A BA

B A

B A

Visualization: Turtle Graphics

F

Move forward by

distance F

+

Rotate by angle δ

-

Bracket notation:

[ : Store position and direction

] : Go back to position and direction of matching [

Rotate by angle -δ

Visualization; Turtle Graphics

Simple system:

Axiom (Start): F

Rule: F→F[-F][+F]

Angle: 30°

More Plants

Professional Visualization

Generalizations

?

Generalizations

• Stochasticity

• Contex sensitive rules

• Delay times

• Reaction-diffusion systems

Homology of Structure

Potential Advantages of Development

• Compact description

• Supports symmetry

• Supports modularity

• Supports reuse of mechanisms in different contexts.

• Supports scalability

• Decentralized control by self-orgnaization and parallelism.

• Turns out to be robust

• Enables adaptivity

• Change of structure can be achieved by small changes.

• Change of structure by change of timing (heterochrony).

• Evolvability