Diapositiva erronea, Ded

1. 11
Artificial Life & Complex Systems
Lecture 3Lecture 3
PatternsPatterns
May 25, 2007May 25, 2007
MaxMax LungarellaLungarella

2. 2
Contents
•• PatternsPatterns
•• Dynamical SystemsDynamical Systems
•• FractalsFractals
•• LL--SystemsSystems
•• Pattern formationPattern formation
•• MorphogenesisMorphogenesis
[Paper:[Paper: A.M.TuringA.M.Turing (1952) CLASSIC!](1952) CLASSIC!]

3. 3
What is a Pattern?
Pattern = organized arrangement of objects in space andPattern = organized arrangement of objects in space and
timetime
Examples of biological patterns:Examples of biological patterns:
-- Schools of fishSchools of fish
-- Raiding column of army antsRaiding column of army ants
-- Ocular dominance stripes in visual cortexOcular dominance stripes in visual cortex
-- Synchronous flashing of firefliesSynchronous flashing of fireflies
-- Pigmentation patterns on shellsPigmentation patterns on shells
inputs from left eye (black) andinputs from left eye (black) and
right eye (white)right eye (white)firefliesfireflies
school of fishschool of fish

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More Patterns

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More Patterns

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Pattern Formation
If Nature is at all economical (probably, yes), weIf Nature is at all economical (probably, yes), we
can expect that she will choose to createcan expect that she will choose to create
complex form not by piececomplex form not by piece--toto--piece constructionpiece construction
by organizational andby organizational and patternpattern--forming phenomenaforming phenomena
we see also in the nonwe see also in the non--living world.living world.
If this is true, then there must be similarities in theIf this is true, then there must be similarities in the
form and patterns of living and purely inorganicform and patterns of living and purely inorganic
or physical systems.or physical systems.

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How Does Nature Build Patterns?
Buildings blocks are:Buildings blocks are:
-- Living units (fish, ants, nerve cells,Living units (fish, ants, nerve cells, ……))
biological systemsbiological systems
-- Inanimate objects (bits of dirt, sand,Inanimate objects (bits of dirt, sand, ……))
physical/physical/nonbiologicalnonbiological systemssystems
Differences between physical and biologicalDifferences between physical and biological
systems:systems:
-- In biological systems the subunits are moreIn biological systems the subunits are more
complexcomplex
-- The interactions among the subunits areThe interactions among the subunits are
influenced also by genetically controlledinfluenced also by genetically controlled
properties (in physical systems interactions areproperties (in physical systems interactions are
purely physical)purely physical)

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How Are Patterns Formed? Any Ideas?

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How Are Patterns Formed?
Simple answer:Simple answer: ““Form follows functionForm follows function””
That is, the shape and structure of aThat is, the shape and structure of a
biological/technological systembiological/technological system –– a proteina protein
molecule, a limb, an organism, a colony, themolecule, a limb, an organism, a colony, the
InternetInternet –– is that which best equips the system foris that which best equips the system for
survivalsurvival
A form which gives organism (system) anA form which gives organism (system) an
evolutionary advantage tends to stickevolutionary advantage tends to stick
Power of cumulative selectionPower of cumulative selection

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Natural selection is not entirely satisfyingNatural selection is not entirely satisfying …… Why?Why?
How Are Patterns Formed?

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How Are Patterns Formed?
1)1) It says nothing about theIt says nothing about the
underlyingunderlying mechanismsmechanisms!!
2)2) Why should the pattern on a shellWhy should the pattern on a shell
be of advantage?be of advantage?
Once you start to ask theOnce you start to ask the ‘‘how?how?’’ ofof
mechanism, you are up against themechanism, you are up against the
rules of chemistry, physics, andrules of chemistry, physics, and
mechanism, and the questionmechanism, and the question
becomes not justbecomes not just ‘‘is the formis the form
successful?successful?’’ butbut ‘‘is it physicallyis it physically
possible?possible?’’ (1917)(1917)

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Another simple answer: Emergence!Another simple answer: Emergence!
Patterns emerge from large populations ofPatterns emerge from large populations of
interactinginteracting ‘‘unitsunits’’ (living organisms and non(living organisms and non--livingliving
entities)entities)
Emergence implies that the largeEmergence implies that the large--scale organizationscale organization
could never be deduced by close inspection of thecould never be deduced by close inspection of the
individual units (holistic perspective, e.g. Hindividual units (holistic perspective, e.g. H220)0)
How Are Patterns Formed?

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How Are Patterns Formed?
Can simple rules give rise to patterns and forms?Can simple rules give rise to patterns and forms?

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Mechanisms
-- IterationIteration
-- RecursionRecursion
-- FeedbackFeedback
Can give rise to selfCan give rise to self--similar structures such as trees,similar structures such as trees,
plants, clouds, mountains,plants, clouds, mountains, ……

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Characterization:Characterization:
•• AA systemsystem thatthat evolves in time according to a set of rulesevolves in time according to a set of rules
•• What changes in time is a variable (e.g. position,What changes in time is a variable (e.g. position,
concentration, temperature, viscosity,concentration, temperature, viscosity, ……))
•• PPresent conditions determine the future.resent conditions determine the future.
•• The rules are usually nonlinearThe rules are usually nonlinear (e.g. set of diff.(e.g. set of diff. eqeq. defining. defining
rates of change)rates of change)
•• There may be many interacting variablesThere may be many interacting variables
What is a Dynamical System?

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Dynamical Systems at a Glance

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Other Examples of (Nonlinear) Dynamical Systems
•• The Solar SystemThe Solar System
•• The atmosphere (the weather)The atmosphere (the weather)
•• The economy (stock market)The economy (stock market)
•• The human body (heart, brain, lungs, ...)The human body (heart, brain, lungs, ...)
•• Ecology (plant and animal populations)Ecology (plant and animal populations)
•• Cancer growthCancer growth
•• Spread of epidemicsSpread of epidemics
•• Chemical reactionsChemical reactions
•• The electrical power gridThe electrical power grid
•• The InternetThe Internet

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Def.: Map of variables in timeDef.: Map of variables in time
-- Time is parameterTime is parameter
-- Trajectory (orbit) in stateTrajectory (orbit) in state
spacespace
E.g.E.g. Z(tZ(t) = [x(t),x(t+1),x(t+2)]) = [x(t),x(t+1),x(t+2)]
Interestingly, forInterestingly, for continuouscontinuous
(reversible) systems(reversible) systems: Only: Only
one trajectory passesone trajectory passes
through each point of athrough each point of a
statestate--space (not true inspace (not true in
discrete systems)discrete systems)
Phase or State-Space

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Vector Fields
DEMO:DEMO:

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Attractors
PhasePhase--space volume to where dynamical system convergesspace volume to where dynamical system converges
asymptotically over timeasymptotically over time

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Why Attractor Behavior?
Energy dissipation (thermodynamic systems)Energy dissipation (thermodynamic systems)
-- Friction, thermodynamic loss, loss of material, etc.Friction, thermodynamic loss, loss of material, etc.
-- Volume contraction in phaseVolume contraction in phase--space (system tends tospace (system tends to
restrict itself to small basins of attraction)restrict itself to small basins of attraction)
-- SelfSelf--organization (dissipative systems)organization (dissipative systems)
Compare withCompare with ……
Hamiltonian systemsHamiltonian systems
-- frictionless, no attractorsfrictionless, no attractors
-- Conservation of energyConservation of energy
-- ergodicityergodicity

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Types of Attractors
Fixed pointFixed point
Limit cycleLimit cycle
QuasiQuasi--periodicperiodic
StrangeStrange

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Types of Attractors

24. 24
Fix-Point Behavior
•• 00--dimensional attractordimensional attractor
•• Volume of phase space to which the systemVolume of phase space to which the system
converges after a long enough time = basin ofconverges after a long enough time = basin of
attraction (defined by all trajectories leading intoattraction (defined by all trajectories leading into
the attractor)the attractor)
•• Example: Pendulum when friction and gravityExample: Pendulum when friction and gravity
bring its motion to a haltbring its motion to a halt

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Limit Cycle Behavior
•• Periodic motionPeriodic motion
•• Repetitive oscillation among a number of statesRepetitive oscillation among a number of states
(e.g. loop)(e.g. loop)
•• Example: lone planet orbiting around a star in anExample: lone planet orbiting around a star in an
elliptical orbitelliptical orbit

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Quasi-Periodic Attractor
•• Several independent cyclic motionsSeveral independent cyclic motions
•• Never quite repeat themselvesNever quite repeat themselves
•• ToroidalToroidal attractorsattractors
•• Example: the moon orbits Earth, which orbits theExample: the moon orbits Earth, which orbits the
sun, which, in turn, orbits the galactic centersun, which, in turn, orbits the galactic center

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Strange or Chaotic Attractors
-- Pervasive (chaos is everywhere: weather patterns, humanPervasive (chaos is everywhere: weather patterns, human
brainbrain’’s electrochemical activity,s electrochemical activity, ……))
-- Sensitivity to initial conditions (Butterfly effect)Sensitivity to initial conditions (Butterfly effect)
-- Deterministic chaos (if we could know the exact initialDeterministic chaos (if we could know the exact initial
conditions, trajectories would be determined)conditions, trajectories would be determined)
-- LowLow--dimensional chaos (strange attractors are restricteddimensional chaos (strange attractors are restricted
to small volumes of phaseto small volumes of phase--space)space)
-- Weak causalityWeak causality
- 3-body problem (any slight measurement difference results
in very different predictions)
- Butterfly effect (Lorenz attractor)

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Simple Chaotic Systems in Physics

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Strange Attractors
Limit set asLimit set as tt →→ ∞∞
Set of measure zeroSet of measure zero
Basin of attractionBasin of attraction
Fractal structureFractal structure
• non-integer dimension
• self-similarity
• infinite detail
Chaotic dynamicsChaotic dynamics
• sensitivity to initial conditions
• topological transitivity
• dense periodic orbits
Aesthetic appealAesthetic appeal

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Characteristics of Chaos
•• Never repeatsNever repeats
•• Depends sensitively on initial conditions (Butterfly effect)Depends sensitively on initial conditions (Butterfly effect)
•• Allows shortAllows short--term prediction but not longterm prediction but not long--term predictionterm prediction
•• Comes and goes with a small change in some control knobComes and goes with a small change in some control knob
•• Usually produces a fractal patternUsually produces a fractal pattern
•• Not random (or stochastic); unpredictable in the shortNot random (or stochastic); unpredictable in the short--
term butterm but ““predictablepredictable”” in the longin the long--termterm ……

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Example 1: A Planet Orbiting a Star
Elliptical OrbitElliptical Orbit Chaotic OrbitChaotic Orbit

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Example 2: The Logistic Map
Also known as the quadratic map or the Feigenbaum map
Simple population growth model defined by the iterative
map: x(t+1) = rx(t)*(1-x(t)), 0 < r < 4, 0 < x < 1 (r =
reproduction rate or bifurcation parameter)
if x = 0: population is extinct
if x = 1: population will be extinct at the next time step
rx(t) positive feedback term
1-x(t) negative feedback term (decrease to
overpopulation)

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Example 2: The Logistic Map
Question: What happens to the longQuestion: What happens to the long--term behaviorterm behavior
ofof x(tx(t) for different values of) for different values of ‘‘rr’’??
Special cases:Special cases:
0<r<1: all terms <10<r<1: all terms <1 extinctionextinction
1<r<3: fix point behavior1<r<3: fix point behavior
r>3: period doubling (bifurcation)r>3: period doubling (bifurcation)

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Logistic Map – Fix Point

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Logistic Map – Chaos
Cobweb plot at:Cobweb plot at: http://www.emporia.edu/mathhttp://www.emporia.edu/math--cs/yanikjoe/Chaos/CobwebPlot.htmcs/yanikjoe/Chaos/CobwebPlot.htm

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The Logistic Map – Four Types of Attractors

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The Logistic Map – Bifurcation Diagram
DEMO:DEMO: http://www.emporia.edu/mathhttp://www.emporia.edu/math--cs/yanikjoe/Chaos/Bifurcation.htmcs/yanikjoe/Chaos/Bifurcation.htm

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Logistic Map – Bifurcations and Self-Similarity

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Discovered sensitivity to initial conditions inDiscovered sensitivity to initial conditions in
a simple 3a simple 3--variable dynamical system;variable dynamical system;
simplified model of weather; convectionsimplified model of weather; convection
flows in the atmosphere (x: convection; y:flows in the atmosphere (x: convection; y:
temperature difference; z: distortion of thetemperature difference; z: distortion of the
temperature profile)temperature profile)
DEMO:DEMO: http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demo8.htmlhttp://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demo8.html
DEMO:DEMO: http://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.htmlhttp://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.html
Example 3: Lorenz Attractor

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Example 3: Lorenz Attractor

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Example 3: The Hénon Attractor
xn+1 = a - xn
2 + bxn-1
Example ofExample of
infinite detailinfinite detail
……

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Example 3: The Hénon Attractor

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Example 4: General 2-D Quadratic Map
xn+1 = a1 + a2xn + a3xn
2 + a4xnyn + a5yn + a6yn
2
yn+1 = a7 + a8xn + a9xn
2 + a10xnyn + a11yn + a12yn
2
xn+1 = xn
2 - yn
2 + a
yn+1 = 2xnyn + b
a
b

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Take-Home-Message
•• Simple equations can give rise to complexSimple equations can give rise to complex
behaviorbehavior
•• Chaotic systems are deterministic and not randomChaotic systems are deterministic and not random
(but also unpredictable in the long(but also unpredictable in the long--term;term;
prediction in the shortprediction in the short--term fairly accurate)term fairly accurate)
•• Chaotic systems are extremely sensitive to theChaotic systems are extremely sensitive to the
initial conditionsinitial conditions
•• Presence of iterationsPresence of iterations
•• The devil is in the detailsThe devil is in the details

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Fractals
•• Termed coined by Benoit Mandelbrot toTermed coined by Benoit Mandelbrot to
differentiate pure geometric figuresdifferentiate pure geometric figures
(circle, square,(circle, square, ……) from other types of) from other types of
figures that defy such simplefigures that defy such simple
classificationclassification
•• Structure on all scales (detail persistsStructure on all scales (detail persists
when zoomed arbitrarily)when zoomed arbitrarily)
•• Geometrical objects generally with nonGeometrical objects generally with non--
integer dimensioninteger dimension
•• SelfSelf--similarity (contains infinite copies ofsimilarity (contains infinite copies of
itself)itself)

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Fractals in Nature

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Which One is Digital?

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Fractals in the Human Body

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Properties: Anyone?

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Properties
•• They are fractal!They are fractal! ☺☺
•• Display selfDisplay self--similaritysimilarity
•• Display scaleDisplay scale--invarianceinvariance

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Non-Fractal
Size of features
1 cm
1 characteristic
scale

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Fractal
2 cm
1 cm
1/2 cm
1/4 cm
manymany characteristic scalecharacteristic scaless
Size ofSize of ffeatureseatures
Structure on all scales (detail persists when zoomedStructure on all scales (detail persists when zoomed
arbitrarily)arbitrarily)

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Self-Similarity and Scale-Invariance
•• ““When each piece of a shape is geometricallyWhen each piece of a shape is geometrically
similar to the whole, both the shape and thesimilar to the whole, both the shape and the
cascade that generate it are called selfcascade that generate it are called self--
similar.similar.”” (B. Mandelbrot)(B. Mandelbrot)
•• CContains infinite copies of itselfontains infinite copies of itself
•• Scaling/Scale =Scaling/Scale = The value measured for aThe value measured for a
propertyproperty does notdoes not depend on the resolution atdepend on the resolution at
which it is measuredwhich it is measured
•• Two types of invariance:Two types of invariance:
- Geometrical
- Statistical ImportantImportant
ConceptConcept

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Self-Similarity and Scale-Invariance

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Self-Similarity and Scale Invariance

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Scale-Invariance

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Scaling
one measurement:one measurement: not sonot so
interestinginteresting
slope
Logarithmof
themeasurement
Logarithmof
themeasurement
one
value
Logarithm of the
resolution used to make
the measurement
Logarithm of the
resolution used to make
the measurement
scaling relationship:scaling relationship: muchmuch
more interestingmore interesting
Fractal?

58. 58
Why Fractals?
-- Packing efficiency: fractals are very good atPacking efficiency: fractals are very good at
squeezing a great deal length (or surface)squeezing a great deal length (or surface) ––
perhaps infiniteperhaps infinite –– into a small finite areainto a small finite area
-- Reason: minimize material requirements whileReason: minimize material requirements while
maximizing functionalitymaximizing functionality

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Branching Patterns
nerve cells
in the retina, and in culture

60. 60
Branching Patterns
blood vessels
Family, Masters, and Platt 1989
Physica D38:98-103
Mainster 1990 Eye 4:235-241
in the retina
air ways
in the lungs
West and Goldberger 1987
Am. Sci. 75:354-365

61. 61
Branching Patterns

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Currents Through Ion Channels
ATP sensitive potassium channel in
cell from the pancreas
Gilles, Falke, and Misler (Liebovitch 1990 Ann. N.Y. Acad. Sci. 591:375-391)
5 sec
5 msec
5 pA
FC = 10 Hz
FC = 1k Hz

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How Are Fractal Patterns Generated?
How are fractals grown?How are fractals grown?
Three known mechanismsThree known mechanisms
-- IterationIteration
-- DiffusionDiffusion
-- RecursionRecursion

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Example: The Cantor Set (Mathematical Monster)
Requires: iteration and recursionRequires: iteration and recursion
Note: points in Cantor set areNote: points in Cantor set are uncountablyuncountably infinite but width of allinfinite but width of all
the points is (2/3)^nthe points is (2/3)^n 0, for n0, for n 00

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Example: The Koch Curve (Math. Monster)
Question: what is the length ofQuestion: what is the length of
the Koch curve after n=100the Koch curve after n=100
iterations if the length of oneiterations if the length of one
segment is 1m?segment is 1m?

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Example: Sierpinski Triangle

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Example: The Mandelbrot Set

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Example: The Mandelbrot Set
Technically speaking:Technically speaking: ““iterative dynamical systemiterative dynamical system””
PseudoPseudo--code:code:
For each number, c, in a subset of the complex planeFor each number, c, in a subset of the complex plane
SetSet xtxt=0=0
For t=1 toFor t=1 to tmaxtmax
ComputeCompute xtxt = xt^2 + c= xt^2 + c
If |If |xtxt|>2, then break out of loop|>2, then break out of loop
EndForEndFor
If t<If t<tmaxtmax, then color point c white, then color point c white
If t=If t=tmaxtmax, then color point c black, then color point c black
EndForEndFor
Definition: The Mandelbrot set is defined as the set of pointsDefinition: The Mandelbrot set is defined as the set of points
thatthat never divergesnever diverges!!

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Example: The Mandelbrot Set
Special cases:Special cases:
1.1. c = 0c = 0 (origin of complex plane):(origin of complex plane): xtxt = xt^2 = 0= xt^2 = 0
(stays bounded)(stays bounded)
2.2. c = ic = i: x[0] = 0; x[1] = i; x[2] =: x[0] = 0; x[1] = i; x[2] = --1 + i;1 + i; ……
3.3. c = 1+ic = 1+i:: x[tx[t]] unboudedunbouded (sequence diverges before(sequence diverges before
tmaxtmax iterations)iterations)
4.4. c = 0.01+ic = 0.01+i (compare with 2): sequence diverges(compare with 2): sequence diverges
beforebefore tmaxtmax iterationsiterations
Butterfly effect!Butterfly effect!

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Example: The Mandelbrot Set
For each pixel on the screen do:For each pixel on the screen do:
{{
x =x = aa = x co= x co--ordinate of pixelordinate of pixel
y =y = bb = y co= y co--ordinate of pixelordinate of pixel
x2 = x*xx2 = x*x
y2 = y*yy2 = y*y
iteration = 0iteration = 0
maxitermaxiter = 1000= 1000
while ( x2 + y2 < (2*2) AND iteration <while ( x2 + y2 < (2*2) AND iteration < maxitermaxiter ))
{{
x = x2x = x2 -- y2 +y2 + aa
y = 2*x*y +y = 2*x*y + bb
x2 = x*xx2 = x*x
y2 = y*yy2 = y*y
iteration =iteration = iterationiteration + 1+ 1
}}
if ( iteration ==if ( iteration == maxitermaxiter ))
color = blackcolor = black
elseelse
color = iterationcolor = iteration
}}
xn+1 = xn
2 - yn
2 + a
yn+1 = 2xnyn + b

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On Compressibility
The Mandelbrot set despite itThe Mandelbrot set despite it’’s complex nature iss complex nature is
actually extremely compressible in the sense thatactually extremely compressible in the sense that
an algorithm for the image is far simpler to storean algorithm for the image is far simpler to store
than the image itself!than the image itself!

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The Mandelbrot Set

73. 73
The Mandelbrot Set

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Self-Similarity and Scale-Invariance

75. 75
Diffusion-Limited Aggregation
Is this a living organism?Is this a living organism?

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Diffusion-Limited Aggregation
What about this one?What about this one?

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Diffusion-Limited Aggregation
How was this pattern generated?How was this pattern generated?

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Diffusion-Limited Aggregation
Two processes:Two processes:
-- DiffusionDiffusion
-- AggregationAggregation

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Diffusion (Random Walk)
•• Diffusion is a random motionDiffusion is a random motion
•• Although the motion of individual particles is totallyAlthough the motion of individual particles is totally
random with respect to the direction, it may happen thatrandom with respect to the direction, it may happen that
particles walk somewhat far relative to a starting pointparticles walk somewhat far relative to a starting point
•• But, in contrary to a normal flow, where all particlesBut, in contrary to a normal flow, where all particles
under investigation move more or less into the sameunder investigation move more or less into the same
direction,direction, the average of walked distance of all particlesthe average of walked distance of all particles
within a random walk is zerowithin a random walk is zero

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•• When particles have the possibility to attract each otherWhen particles have the possibility to attract each other
and stick together, they may formand stick together, they may form aggregatesaggregates
•• The forces between the particles may be weak or strongThe forces between the particles may be weak or strong
•• For particles which carry some electrical charge (ions) theFor particles which carry some electrical charge (ions) the
forces are typically very strong and thus there is a gain inforces are typically very strong and thus there is a gain in
energy when they build aggregatesenergy when they build aggregates
Aggregation

81. 81
Diffusion-Limited Aggregation
In diffusionIn diffusion--limited aggregateslimited aggregates the shape of thethe shape of the structurestructure
is controlled by the possibility of particles to reach theis controlled by the possibility of particles to reach the
aggregateaggregate
Starting with aStarting with a uniform distribution,uniform distribution, and someand some ‘‘seedseed’’
particles which start the growth processparticles which start the growth process
The rate of growth is governed/limited by the diffusionThe rate of growth is governed/limited by the diffusion
SSome particles might meetome particles might meet, and t, and the aggregate may growhe aggregate may grow
as long there are particles moving aroundas long there are particles moving around
Preferential attachment at tip of branchesPreferential attachment at tip of branches
The volume is not filled in its entirety, but there areThe volume is not filled in its entirety, but there are
many gapsmany gaps FractalityFractality!!
Found in crystal growth, coral reefs, bacterial colonies,Found in crystal growth, coral reefs, bacterial colonies,
and other natural systemsand other natural systems

82. 82
Diffusion-Limited Aggregation
http://apricot.ap.polyu.edu.hk/~lam/dla/dla.htmlhttp://apricot.ap.polyu.edu.hk/~lam/dla/dla.html

83. 83
How Do We Measure Self-Similarity and Scaling?

84. 84
Dimension
•• A quantitative measure of selfA quantitative measure of self--similarity andsimilarity and
scalingscaling
•• Intuitively: The dimension tells us how manyIntuitively: The dimension tells us how many
pieces we see when we look at a finer resolution.pieces we see when we look at a finer resolution.
•• Definitions:Definitions:
- Topological dimension (always integer)
- Self-similarity dimension (fractal)
- Capacity dimension (fractal)
- Hausdorff dimension
- Embedding dimension

85. 85
Topological Dimension
-- Always an integer (not fractal!)Always an integer (not fractal!)
- in a minimal covering, each point of the object is covered
by no more than G sets
- Topological dimension d = G-1
- For a plane: G = 3 d = 3-1 = 2

86. 86
Self-Similarity Dimension
Simplest fractals are selfSimplest fractals are self--similar!similar!
How many times does scaledHow many times does scaled--downdown
copy fit into the whole object?copy fit into the whole object?
N = number of copiesN = number of copies
rr = scale factor= scale factor
SelfSelf--ssimilarity dimension =imilarity dimension = ln(N)/lnln(N)/ln(r(r))
E.g. square: N = 9, r = 3E.g. square: N = 9, r = 3 SsdSsd ==
ln(9)/ln(3) = 2ln(9)/ln(3) = 2
What is the selfWhat is the self--similarity dimension ofsimilarity dimension of
the Koch curve?the Koch curve?

87. 87
Fractal Dimension: Koch Curve
Log (number of new pieces)
Log (factor of finer resolution)
D =
Log 4
= 1.2619=
Log 3Log 3

88. 88
Box Dimension
•• Used for fractals that are not selfUsed for fractals that are not self--similarsimilar
•• NN (r)(r) balls or squaresballs or squares of radiusof radius 1/1/r needed to cover ther needed to cover the
objectobject
•• Box dimensionBox dimension
•• D = 1.65 (?)D = 1.65 (?)
r
rN
d
r ln
)(ln
lim
0→
=

89. 89
Definition of a Fractal
Mathematical:Mathematical:
d(fractald(fractal) >) > d(topologicald(topological))
Origin:Origin:
Fractal = fragmented, many pieces, one alsoFractal = fragmented, many pieces, one also
speaks of fractal dimensionspeaks of fractal dimension

90. 90
Example: Koch Curve
Perimeter:Perimeter:
d(fractald(fractal) = 1.2619) = 1.2619
d(topologicald(topological) = 1.) = 1.
In other words: Perimeter covers more space than aIn other words: Perimeter covers more space than a
11--D line; covers less space than a 2D line; covers less space than a 2--D area.D area.

91. 91
Fractal Dimensions in Biology: Measured
•• surfaces of proteinssurfaces of proteins
•• surface of cell membranessurface of cell membranes
•• growth of bacterial coloniesgrowth of bacterial colonies
•• islands of types of lipids inislands of types of lipids in cell membranescell membranes
•• dendrites of neuronsdendrites of neurons
•• blood flow in the heartblood flow in the heart
•• blood vessels in the eye,blood vessels in the eye, heart, and lungheart, and lung
•• shape of herpes simplexshape of herpes simplex
•• ulcers in the corneaulcers in the cornea
•• textures of Xtextures of X--rays of bonerays of bone and teethand teeth
•• texture of radioisotopetexture of radioisotope tracer in the livertracer in the liver
•• MANY MORE!MANY MORE!

92. 92
Pulmonary Hypertension
HIGH BLOOD PRESSURE IN THE LUNGS
Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser
1990 Circ. Suppl, lll, 82: 100
D = 1.65
normal
20% 02

93. 93
D = 1.53
hypoxic
10% 02
HIGH BLOOD PRESSURE IN THE LUNGS
Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser
1990 Circ. Suppl, lll, 82: 100
Pulmonary Hypertension

94. 94
Pulmonary Hypertension
D = 1.43
hyperoxic
90% 02
HIGH BLOOD PRESSURE IN THE LUNGS
Boxt, Katz, Czegledy, Liebovitch, Jones, Reid, & Esser
1990 Circ. Suppl, lll, 82: 100

95. 95
Statistical Indicators of Fractal StructureMean
More Data Mean
More Data
0
More Data
Mean
∞
areaarea 00 lengthlength infinf

96. 96
Statistical Indicators of Fractal Structure
Log avg
density within
radius r
Log radius r
r
rN
d
r ln
)(ln
lim
0→
=
Remember!

97. 97
Statistical Indicators of Fractal Structure
Log avg
density
within
radius r
Log radius r
.5
-1.0
-2.0
-1.5
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00
-2.5
0
Meakin 1986 In On Growthand Form: Fractal and Non-Fractal Patterns in
Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp. 111-135

98. 98
Natural Design Principles

99. 99
Mathematics: the Language of Nature

100. 100
On Growth and Form
On Growth and Form (1917) laid the foundations ofOn Growth and Form (1917) laid the foundations of
biobio--mathematicsmathematics
““Organic form itself is found, mathematicallyOrganic form itself is found, mathematically
speaking, to be a function of time [speaking, to be a function of time [……] We might call] We might call
the form of an organism an event in spacethe form of an organism an event in space--time, andtime, and
not merely a configuration in space.not merely a configuration in space.””
EqsEqs. to describe static patterns of living organisms. to describe static patterns of living organisms
Transforming form changing a few parametersTransforming form changing a few parameters
““In many growth processes of living organism,In many growth processes of living organism,
especially of plants, regularly repeated appearancesespecially of plants, regularly repeated appearances
of certainof certain multicellularmulticellular structures are readilystructures are readily
noticeable [noticeable [……] In the case of a compound lead, for] In the case of a compound lead, for
instance, some of the leaflets which are parts of ainstance, some of the leaflets which are parts of a
leaf at an advanced stage, have the same shape asleaf at an advanced stage, have the same shape as
the whole leave has at an earlier stage.the whole leave has at an earlier stage.”” (Herman et(Herman et
al.)al.)

101. 101
On Growth and Form

102. 102
Affine Transformations
Theory of transformation:Theory of transformation:
Comparison of relatedComparison of related
morphological forms (Thompson,morphological forms (Thompson,
1917)1917)

103. 103
Homology of Structure

104. 104
On Growth
Three mechanisms:Three mechanisms:
-- IterationIteration
-- RecursionRecursion
-- DiffusionDiffusion

105. 105
On Growth and Form: L-Systems
•• The selfThe self--similarity of plants is the result ofsimilarity of plants is the result of
developmental processesdevelopmental processes
•• Mathematical formalism introduced in 1968Mathematical formalism introduced in 1968
byby AristidAristid LindenmeyerLindenmeyer (1925(1925--1989)1989)
•• Essentially, production systemEssentially, production system
•• Productions are rewriting rules which stateProductions are rewriting rules which state
how new symbols (or cells) can be producedhow new symbols (or cells) can be produced
from old symbols (or cells)from old symbols (or cells)

106. 106
Example
•• Axiom = seedAxiom = seed ‘‘cellcell’’
•• Rule = description of how to grow newRule = description of how to grow new
‘‘cellcell’’
F = draw forwardF = draw forward
G = go forward (but do not draw)G = go forward (but do not draw)
+ = right turn+ = right turn
-- = left turn= left turn
[ = save turtle[ = save turtle’’s current poss current pos
] = remove last saved pos] = remove last saved pos

107. 107
L-Systems

108. 108
More sophisticated L-Systems

109. 109
Three-Dimensional L-Systems

110. 110
Movies
……

111. 111
Book
Available for free online!Available for free online!

112. 112
Project Idea
http://http://algorithmicbotany.org/virtual_laboratoryalgorithmicbotany.org/virtual_laboratory//

113. 113
L-Systems
Needless to sayNeedless to say …… LL--Systems display fractalSystems display fractal
structurestructure ☺☺

114. 114
On Growth and Form: Morphogenesis
Morphogenesis = morphology (shape, form) + genesisMorphogenesis = morphology (shape, form) + genesis
(development)(development)
Technically speaking: stage in embryonic developmentTechnically speaking: stage in embryonic development
Cells begin to cluster and form patterns:Cells begin to cluster and form patterns:
-- Cell differentiationCell differentiation
-- Cell growthCell growth
-- Cell divisionCell division
-- Cell migrationCell migration
-- Chemical secretion/diffusionChemical secretion/diffusion ……

115. 115
Cell Differentiation: Vertebrates
•• Three stagesThree stages
•• Four cell typesFour cell types

116. 116
On Growth and Form: Morphogenesis
•• How does body structure arise?How does body structure arise?
•• If arm cells had an extra cycle of cell division, our armsIf arm cells had an extra cycle of cell division, our arms
would be 1.8m longwould be 1.8m long …… how is growth controlled?how is growth controlled?
•• How is cell division turned on or off? The early embryoHow is cell division turned on or off? The early embryo
is like a cancerous tumoris like a cancerous tumor –– growing very rapidlygrowing very rapidly –– butbut
the embryothe embryo ““knowsknows”” when to stop growing and diving.when to stop growing and diving.
•• What are theWhat are the mechanisms that determine themechanisms that determine the
anatomical structure of an organismanatomical structure of an organism??

117. 117
Striped Patterns and Spotted Bodies

118. 118
Morphogenetic Model
[Classic][Classic] ““The Chemical Basis of Morphogenesis,The Chemical Basis of Morphogenesis,”” 1952,1952, Phil. Trans.Phil. Trans.
Roy. Soc. of LondonRoy. Soc. of London, Series B: Biological Sciences,, Series B: Biological Sciences, 237237:37:37——72.72.

119. 119
Reaction-Diffusion Systems
Key idea: Morphology is the result of two substancesKey idea: Morphology is the result of two substances ‘‘aa’’ andand
‘‘bb’’ ((morphogensmorphogens) diffusing at different rates) diffusing at different rates
MorphogenMorphogen:: ““is simply the kind of substance concerned in thisis simply the kind of substance concerned in this
theorytheory…”…” in fact, anything thatin fact, anything that diffusesdiffuses into the tissue andinto the tissue and
““somehow persuades it to develop along different lines fromsomehow persuades it to develop along different lines from
those which would have been followed in its absencethose which would have been followed in its absence””
qualifies.qualifies. (Generator of form)(Generator of form)
Investigation of pattern due to breakdown of symmetry andInvestigation of pattern due to breakdown of symmetry and
homogeneity in initially homogeneous continuous mediahomogeneity in initially homogeneous continuous media
Breakdown captured by set of differential equationsBreakdown captured by set of differential equations
Equations capture dynamics of autocatalytic andEquations capture dynamics of autocatalytic and
antagonistic chemical reactions with diffusionantagonistic chemical reactions with diffusion
Result = Turing patternsResult = Turing patterns

120. 120
Pattern Formation from Homogenous I.C.
ActivatorActivator--Inhibitor ModelInhibitor Model
Requirements:Requirements:
-- Activator:Activator: local selflocal self--enhancement (positive feedback)enhancement (positive feedback)
-- Inhibitor:Inhibitor: LongLong--range antagonistic effect (negative feedback)range antagonistic effect (negative feedback)
-- Diffusion:Diffusion: Diffusion of inhibitor >> diffusion of activator (>7 timesDiffusion of inhibitor >> diffusion of activator (>7 times
faster)faster) inhibitor must act rapidly (rapid adaptation of inhibitorinhibitor must act rapidly (rapid adaptation of inhibitor
concentration)concentration) –– otherwise oscillations will occurotherwise oscillations will occur
-- Decay (removal):Decay (removal): Decay inhibitor > decay activatorDecay inhibitor > decay activator
Pattern formation:Pattern formation:
-- Small deviations from a homogeneous distribution create a strongSmall deviations from a homogeneous distribution create a strong
positive feedbackpositive feedback which causes deviations to grow even more.which causes deviations to grow even more.
-- AA longlong--range antagonistic effectrange antagonistic effect restricts the selfrestricts the self--enhancing reactionenhancing reaction
and causes a localization.and causes a localization.

121. 121
Activator-Inhibitor Scheme

122. 122
Diffusion: 1-D Case
x
a
D
t
a
txa
a 2
2
),(
∂
∂
=
∂
∂
•• Constant concentration difference betweenConstant concentration difference between
neighboring cellsneighboring cells a(x,ta(x,t) and a(x+1,t): net) and a(x+1,t): net
exchange of molecules by diffusion is 0exchange of molecules by diffusion is 0
•• Linear concentration gradient: net exchangeLinear concentration gradient: net exchange
of molecules by diffusion is also 0 (whatof molecules by diffusion is also 0 (what
goes out come in)goes out come in)
•• Hence, second derivate!Hence, second derivate!
first
temporal
derivative:
rate
second
spatial
derivative:
flux

123. 123
Activator-Inhibitor System: Sea Shells

124. 124
Activator-Inhibitor System: Sea Shells
2
2
2
2
22
x
b
Dbbrsa
t
b
x
a
Darb
b
a
s
t
a
bba
aaa
∂
∂
++−=
∂
∂
∂
∂
+−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+=
∂
∂
a(x,ta(x,t),), b(x,tb(x,t): concentration of activator and inhibitor): concentration of activator and inhibitor
s: ability of cells to perform autos: ability of cells to perform auto--catalysiscatalysis
DaDa, Db: diffusion constants, Db: diffusion constants
rara,, rbrb: decay rates: decay rates
sasa22
/b : non/b : non--linear autocatalytic influencelinear autocatalytic influence
--rraaaa : rate of removal (proportional to concentration): rate of removal (proportional to concentration)
bbaa, b, bbb: basic activator/inhibitor production: basic activator/inhibitor production

125. 125
Turing Patterns in a Chemical Medium
Patterns arise spontaneously through competition between localizPatterns arise spontaneously through competition between localizeded
autocatalytic chemical reaction and longautocatalytic chemical reaction and long--ranged diffusion of aranged diffusion of a
substance inhibiting the reactionsubstance inhibiting the reaction

126. 126
Results: Often Surprising!

127. 127
Activator-Inhibitor System
•• ShortShort--range autocatalysis / longrange autocatalysis / long--range inhibitionrange inhibition
•• Local instability but global stabilizationLocal instability but global stabilization
•• Behavior of system is often difficult to predictBehavior of system is often difficult to predict
(fluctuations in initial conditions render accurate(fluctuations in initial conditions render accurate
predictions impossible)predictions impossible)