2006 RET.bujak.fractals

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2006 Penn RET - Bujak Page 1 of 22

Simulating the Natural Process of Growth through Fractals

and

Measuring Its Complexity Through Fractal Dimension

Edward Bujak

2006 RET University of Pennsylvania

Preface

During the 2006 Summer I was an NSF Fellow in a NSF-sponsored Research Experience

for Teachers (RET) program at the University of Pennsylvania Laboratory for Researchon Structure of Matter (LRSM) working with Prof. Ritesh Agarwal, MSE on 'Synthesis

and Characterization of Cadmium Sulfide (CdS) Nanowires.

(www.lrsm.upenn.edu/outreach/#ret ). We developed processes to grow CdS nanowiresto control the morphology and other characteristics such as scaling, uniformity,

composition, purity, and distribution. CdS has phenomenal luminescent properties and

emits visible green light and hence is used for Light-Emitting Diodes (LEDs). With

nanowires we desire a nanostructure that is crystalline (aligned structure of similar atoms)and very thin with an aspect ratio of 1000:1. The best wire is a line segment; the

straighter the better. For your curiosity there are many nano-scale shapes that exhibit

desirable properties: nano-rods (like capsules), nano-tubes (single walled and multi-walled), nano-checkerboards, and many others.

In the lab we also grew other nanostructures where our desired shape was more fibrous or

dendritic like a branching tree. In this case we wanted maximum surface area in a fixed

volume (mathematically this is more important than it appears!).

This paper/presentation is an introductory mathematical study into growth simulation

using mathematical models; particularly fractals with simple branching and iteration rulesusing computer software to automate the repetitive process. This will be taught as a few

lessons within a unit.

This should probably be the first lesson using fractals and might be prefaced by somevisually appealing youtube.com, video.google.com, or similar videos which will catch the

students attention and interest.

This unit will investigate patterns, symmetry, beauty (order in chaos) with a very simple

iterative process to created structures. Some fractal terms; such as dimension, self-

similarity, iteration (recursion), need to be understood.


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Brief History (taken from http://www.math.umass.edu/~mconnors/fractal/hist.html )

Historically, the revolution was forced by the discovery of mathematical structures that

did not fit the patterns of Euclid and Newton. Many of the fractals go back to classicalmathematics and mathematicians of the past like Georg Cantor, Guisseppi Peano, Helge

Von Koch, Waclaw Sierpinski and many others. However, reputable scientists andmathematicians called these structures a "gallery of monsters." Jules Henri Poincardeemed many of Cantor's creations "pathological." Cantor's Dust, for example,

constructed in 1877, seemed to jump dimensions. It is constructed by chopping up one

dimensional line segments - but in the end it contains only zero dimensional points

without length or width.

Peano's curve is a space filling curve

These monster curves were seen as shapes intended to show deviation from the norm

rather than to typify the normal. Their profound meaning had been somewhat forgotten

until Benoit Mandelbrot took them out in the 1960's, dusted them off and saw thesepathological shapes as an important clue to a new mathematics of natural forms such as

clouds, trees, and mountain ranges. Mandelbrot is often called the father of fractal

geometry. He coined to word fractal from the Latin frangere meaning to break into

irregular fragments and published his book, The Fractal Geometry of Nature.

Introduction

We are mesmerized by the regularity of shapes; the predictability of patterns. There ismath, an equation, behind much of nature. We try to set out to create the pattern in nature.

Patterns can be explained.

Some natural fractal objects are: mountain ranges, snowflakes, clouds, crystals, leaves,

leaf veins, trees, tree branching, blood vessels, lungs, sea shells, and many others.

Fractals can be characterized by just a few properties:

Scale invariance Self-similarity part looks like the whole (self-organizing) Dimension

Fractals belong to a non-Euclidean way of looking at the universe which may have a non-

integer dimension. They are geometric shapes or patterns that help describe the forces ofgrowth.

In 1975 the French mathematician Benoit Madelbrot defined fractals as objects that donot lose their detail or their proportions when they are magnified or shrunk.

This property is reminiscent ofphi ( , , the Golden Mean, the Golden Section, or

1.618...) where the same essential proportion is retained every time you cut the line, the


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rectangle, or other shape. In fact, the qualities of both fractals andphi are concerned withgrowth.

Features of Fractals

There are two different types of fractal, the geometric fractal and the random fractal.The snowflake is an example of a geometric fractal that grows (in the simplest terms) by

the addition of equilateral triangles in specific patterns. Random fractals are computer

generated, in both modeling and games.

Fractal geometry can lead to convincing images of natural growth phenomena, such as

coastlines, ferns, and tree bark. They can also emerge from climate and manmade

phenomena, such as stock price graphs or economic predictions, which show self-similarity.

Some ferns are classic natural examples of a fractal, with each section (pinna) of leafbeing miniature replication of the whole leaf. A single pinna if magnified looks like a

whole leaf. In addition, in some species, their buds unfold in the shape of a logarithmic

spiral (Fibonacci number Golden Mean). This means that nature does not have to

redesign the leaf at every stage of its growth, but the initial design keeps on replicating.This is efficiency.

The other important feature of fractals is scaling. In a fractal the degree of irregularity orfragmentation is identical on all scales. Fractals do not get smoother as the magnification

brings you closer; they simply continue to generate new irregularities that arecommensurate with the speed at which you zoom in on them.

Order not chaos

Fractals are associated with the mathematics of chaos, but they are in fact, very ordered just millions of interlocking self-replicating, interlocking objects. They only look

chaotic yet are governed by a definite geometry.

Because fractals contain repeating elements, they can be created using mathematics.

Usually simple growth specification is repeated iteratively by a generator (rule) with an

initiator and is called an IFS Iterated Function System. You can do IFS withGeometers Sketchpad with the Transform then Iterate menu choices.

IF time permits or students want to investigate (or differentiate) them have some play thechaos game.


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Creating Fractals by hand

Cauliflower Fractal separate handout, need ruler, compass, paper, and pencil.

Dimension - background

Fractals are called fractals because they may a non-integer dimension. We know 1-D,

2-D, 3-D, and maybe even 4-D, but there are objects below dimension one, in between 1

and 3, and beyond.

Mention methods to measure dimension such as simple box counting dimension,

similarity dimension, or mass dimension (these are very labor intensive, but time well

spent especially if cooperative and/or inquiry-based exercises are desired). Derive andexplain the dimension formula and validate that it does work for geometric objects that

you know: a line should have D=1, a planar shape (such as a rectangle or a square) should

have D=2, a prism (such as a cube) should have D=3, etc. Now measure the dimensionof a Koch curve with a simple box counting algorithm by using successively smaller

grids (on transparencies laid over the image).

The following table is very useful in trying to grasp the notion of a fractal (fractional)dimension. Make sure students understand the native dimension for a 0-D object (a

point) is simply counting the points, that the native dimension for a 1-D object is a line,

and so forth. Be sure to explain that a 1-D object has no area or volume, and so forth. Besure to explain that a 3-D object is measured by volume and that it has an infinite number

of cross-sectional slices that make up this volume, and so forth. Some of these are hardto explain or conceptualize, like dimensions above 3 which are not outlined here, but note

that the chart can be extended to any integer dimension.

Ways to measure dimension-n compact objects:

Dimension

0 1 2 3

Count finite

Length 0 finite

Area 0 0 finite Measure

Volume 0 0 0 finite

So why is a line one-dimensional and the plane two-dimensional? Note that both of these

objects are self-similar. We may break a line segment into 4 self-similar intervals, each

with the same length, and each of which can be magnified by a factor of 4 to yield theoriginal segment. We can also break a line segment into 7 self-similar pieces, each with


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magnification factor 7, or 20 self-similar pieces with magnification factor 20. In general,we can break a line segment into N self-similar pieces, each with magnification factor N.

Note: It is important to realize that similar is in the geometric sense of the word; the

pieces must have the save proportional shape, the pieces need to look alike, but can bescaled (in the tree branching lab, the left and right branches are scaled differently)

A square is different. We can decompose a square into 4 self-similar sub-squares, andthe magnification factor here is 2. Alternatively, we can break the square into 9 self-

similar pieces with magnification factor 3, or 25 self-similar pieces with magnification

factor 5. Clearly, the square may be broken into N2 self-similar copies of itself, each of

which must be magnified by a factor ofN to yield the original figure. See figure below.Finally, we can decompose a cube into N

3 self-similar pieces, each of which has

magnification factor N.

A square may be broken into N2 self-similar pieces,

each with magnification factor N

Now we see an alternative way to specify the dimension of a self-similar object: Thedimension is simply the exponent of the number of self-similar pieces with magnification

factor N into which the figure may be broken.

Dimension mathematically

Fractal dimension is a measure of how "complicated" a self-similar figure is. In a roughsense, it measures "how many points" lie in a given set. A line is larger than a point. A

planar is "larger" than a line. A solid is larger than a planar shape.

The simplest dimension is the box counting dimension. If time permits, do a classic

coastline counting exercise with successively smaller unit squares. Use transparent graph

paper to overlay the map. It can be empirically shown that:

( ) 1d

N rr

=

where:

r= radius of side length of unit box used to count along the edge

N(r) = number of boxes with side length r along edge your counted boxes along

d= dimension


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To isolate the dimension, logarithms are necessary:

( )

( )

1log log

log

1log

N r d r

N rd

r

=

=

This is like y = mx + b for a log-log plot, where the independentx = log (1/r) anddependenty = log N(r). The linear plots slope will be the fractal dimension d. To

convince you that this works, apply it to a line (d=1), a concave planar shape (d=2), and a

concave solid (d=3). Is a hypercube really 4 dimensions? Who said the 4th

dimension hasto be time?

When shapes are self-similar (the basic pattern iterates), then a similarity dimension canbe used which is a special case of the box counting dimension, but is much easier.

( )log1

log

N rd

r

=

where:

r= scaling factor (or magnification factor)

N(r) = number of copies (or # of self-similar pieces)

There are other measures of fractal dimensionality; such as the mass dimension, which is

a power law relationship between mass and radius:M = k rd

log M = d log r + log k

like y = mx + bWhat does the y-intercept (log k) mean is it the (linear) density of the material?

Should it always be the Origin (I would hope not)? Can it be non-zero?

This is skipped here, but if time permits, a great hands-on would be (jelly) bean counting,

to calculate dimensionality. Is packing an issue? Assuming the shape could be scaled

would the size of the similar objects affect the dimension? Packing of atoms is extremely

important in materials science the structure of the atoms affect the materials properties.

Examples of standard crystal lattice structures are SC (simple cubic) unit cell, FCC (face-centered cubic) unit cell, BCC (body-centered cubic) unit cell, and hexagonal closest

packing unit cell, and others. Note that the packing efficiency is not the dimension!


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Now lets look specifically at the Koch Curve in detail.

n = Stage =

IterationPicture

N(r) = #

Line

Segments

S = 1/r =

Segment

Length

Ln =

Length

0 1 = 40

01

13

=

04

13

=

1 4 = 41

11 1

3 3

=

14 4

3 3

=

2 16 = 42

21 1

9 3

=

216 4

9 3

=

3 64 = 43

31 1

27 3

=

364 4

27 3

=

4 256 = 44

41 1

81 3

=

n 4n

1

3

n

Using the similarity dimension formula:

( )log log4

1 log3log

log 4 log 41.26

log3 log3

n

n

N rd

r

nd

n

= =

= =


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The length of the Koch curve is greater than each Ln, so greater than every number. Thatis, the Koch curve has infinite length.

Here's an interesting corollary to the infinite length of the Koch curve.

Not only does the Koch curve itself have infinite length, but measured along the curve,the length of any pair of points in the Koch curve is infinite.

To make these computations concrete, note that if the original L0 is 1 meter, then

n = Stage =

IterationPicture N(r) = # Line Segments

S = 1/r = Segment

Length

Ln =

Length

24 81,474,976,710,656 = 424

24

12 13.5 x 10 m3

1 km

128

115,792,089,237,316,195,423,

570, 985,008,687,907,853,269,984,665,640,564,039,457,584,

007,913,129,639,936

= 4128

128

62 18.5 x 10 m3

1 lightyear

In summary the Koch curve has:

length d > 1and

0 area d < 2

which agrees with our integer dimension chart (first figure). The Koch curve dimension

is between 1 and 2. It has a fractal dimension d 1.26

Advanced students might want to study a Koch curve with non-equal segment lengths,

not all one-third. How and when does this affect the dimensionality? Does the shape

fold back upon itself and when? The regular Koch curve will not fold upon itself?Why? Do you think that not folding upon oneself is desirable and when/why?


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So what is the dimension of the Sierpinski triangle (or Sierpinski gasket)?

n = Stage =

(Iteration)Picture

N(r) = # dark

Triangles

S = 1/r = Segment

Length

0 1 = 30

01

12

=

1 3 = 31

11 1

2 2

=

2 9 = 32

21 1

4 2

=

3 27 = 33

31 1

8 2

=

4 81 = 34

41 1

16 2

=

n 3n

1

2

n

Using the similarity dimension formula:

( )log

log31 log2

log

log3 log31.58

log 2 log 2

n

n

N rd

r

nd

n

= =

= =


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More about Dimension

More examples can be found athttp://www.math.umass.edu/~mconnors/fractal/sierp/sierp.html where there is also a

great collection of worksheets for calculating dimension

http://www.math.umass.edu/~mconnors/fractal/fractal.html .

Growing Fractals - Hands-On Virtual Manipulative

Use Geometers Sketchpad (computer software application) to create the classicSierpinski triangle (play the chaos game-very time consuming, but demonstrates order

from chaos/randomness), fractal trees, and the classic Koch curve (or snowflake) fractal -separate handout with detailed instructions. Use an initial pattern (rule) and iterate. Note

that the Koch snowflake is simply the Koch curve placed on each of the three edges of a

triangle. Also note and experiment with segmenting the Koch curve in non equal 3-

segments. Experiment with the Koch snowflake not based on an equilateral triangle does not look like a snowflake, why are snowflakes symmetric and regular?

Explore starting with a square, dividing it into 9 equal smaller sub-squares and cutting

out the middle. If you iterate this pattern you will end up with a Sierpinski carpet.

Extension on fractal trees

Dynamic broccoli tree or forest showing iteration with a simple branching pattern.


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In The Essence of Chaos, pages171-173, a fractal branching is outlined. We can use the

pattern of the Geometers Sketchpad exercise A Tree Fractal

Begin with a vertical line segment; this is the trunk. From the top of the trunk,draw line segments extending horizontally to either side, each six-tenths as long

as the trunk; these are the limbs. From the end of each limb , draw a segmentextending upward and one extending downward, each six-tenths as long as a limb.Continue, alternating between horizontal and vertical segments, until lines are as

close together as the width of pencil marks. You will obtain something like

Figure 59. If you carry out the process for an infinite number of steps, you would

have a fractal, with dimension 1.356. The leaves, that is, the point to which thebranches eventually converge, also form a fractal with dimension 1.356.


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Extensions

A great hands-on (time-intensive) is to create a Sierpinski tetrahedron which to all

appearances is 3-D, but in fact is exactly 2-D ! This can be done with cutting standardsmall office envelopes and folding and joining the tetrahedrons in a Sierpinski pattern.

This can also be used to explain geometric sequences; here 1,4,16,64,256,

An even more intense object is the Menger Sponge

which has d 2.73


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Folding and cutting construction paper to make fractal lines, stairs, and other shapes

Advanced IFS: Scale, Reflect, Rotation, Translate use specific notation used with

fractal geometry

Iterative representations of sequences and series

Explore how complex (imaginary) numbers are utilized in the Julia Sets and the more

specific Mandelbrot set. Need technology tools for this mathematically intensive and

repetitive computations.

Explore the synthetic creation and calculation of dimension for other objects:

Is a Moebius strip one dimensional and when you cut it did the dimension change? Whyor why not? Investigate wallpaper groups that tessellate with only 17 known patterns

its about finding the basic shape that iterates and how it iterates through scaling(s),

rotation(s), translation(s), reflecting/mirroring. Investigate topology try to explain the

Klein bottle.

Cantor Set (the Cantor middle-thirds set has d = 0.63)Julia Sets (plural)Mandelbrot Set (singular)

Mapping of set (particularly Madelbrot set of is mapping of a very special set of Julia

Sets)

Use math software which becomes necessary; especially when complex numbers is

introduced with the numerous iterative calculations in Julia Sets and Mandelbrot Set.

Maple computer

Mathematica - computer Excel computer Graphing calculator - handheld

Cellular automata (fractal grids) especially with software like Mathematica

Space filling curves, such as the Hilbert curve (use IFS, actually easy with GeometersSketchpad)


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Fractal Toys

1. http://www.constructiontoys.com/store/fractiles.php 2. www.zometool.com

Websites:

1. http://classes.yale.edu/fractals/ - Fractal Geometry by Michael Frame, BenoitMandelbrot, and Nial Neger really phenomenal site! many links and resources.

2. http://www.dynamicgeometry.com/ - Key Curriculum Press Resource Center forDynamic Geometry. Search on fractal for phenomenal Geometers Sketchpad

examples and files.

3.

http://www.geom.uiuc.edu/- The Geometry Center search on fractals.

4. http://id.mind.net/~zona/mmts/geometrySection/fractals/tree/treeFractal.html -Java-based tutorial on Fractal Tree nice

5. http://math.bu.edu/people/bob/ - Bob Devaneys site, author of 2 books inReferences amazing Java Applets and other dynamic systems resources

6. http://www.charlottemathtutor.com/htmlfile/fractal.htm - fractal linksWebsites with dynamic images or virtual manipulatives

1. http://math.bu.edu/DYSYS/ - portal to The Dynamical Systems and TechnologyProject at Boston University

2. http://www.shodor.org/master/fractal/ fractal modeling tools3. http://id.mind.net/~zona/mmts/geometrySection/fractals/tree/treeFractal.html -

fractal trees

4. http://math.bu.edu/DYSYS/chaos-game/chaos-game.html - the chaos game5. http://didio.wikidot.com/ - Dr. Richard A. DiDio, from La Salle University,

presented the chaos game at a colloquium at Arcadia University on November 14,

2006.

6. http://www.scienceu.com/geometry/fractals/ - fractal color scheme chooser(actually only the Mandebrot set)


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Websites with fractal movies and presentations

1. http://www.unitedstreaming.com/ video on Patterns, Symmetry, and Beauty(26:00 minutes)

2.

http://www.youtube.com/results?search_query=fractal many great videos onfractals. Note: the images are better appreciated when the viewer has a littleunderstanding of fractals.

3. http://www.veoh.com/many videos on fractals.4. http://video.google.com/many videos on fractals.


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References

1. An Eye for Fractals - A Graphic & Photographic Essay, Michael McGuire,Addison-Wesley Publishing Company, Inc., 1991

- Easy book to read with many photographs and illustrations, introductorymath

2. Introducing Fractal Geometry, Nigel Lesmoir-Gordon, Will Rood, and RalphEdney, Totem Books, ISBN 1840461233, www.iconbooks.co.uk

- Easy book to read with many illustrations, introductory math3. Fractals A Tool Kit of Dynamics Activities, Jonathan Choate, Robert L.

Devaney, Alice Foster, Key Curriculum Press, 1999, ISBN 1-55953-355-2,www.keypress.com, http://www.keypress.com/x8192.xml

- Blackline Activity Masters, great for classroom activities, part of a seriesthat includes 4 activities books: Iteration, Fractals, Chaos, and The

Mandelbrot and Julia Sets.

4. Iteration A Tool Kit of Dynamics Activities, Jonathan Choate, Robert L.Devaney, Alice Foster, Key Curriculum Press, 1999, ISBN 1-55953-354-4,www.keypress.com, http://www.keypress.com/x8192.xml

- Blackline Activity Masters, great for classroom activities, part of a seriesthat includes 4 activities books: Iteration, Fractals, Chaos, and TheMandelbrot and Julia Sets.

5. Sacred Geometry A Tool Kit of Dynamics Activities, Stephen Skinner, SterlingPublishing Co., New York, 20006, ISBN 1-4027-4129-4, www.sterlingpub.com

- Colorful coffee table book, examines math, beauty, symmetry in nature6. The Essence of Chaos, Edward N. Lorenz, Seattle: University of Washington

Press, 1993, ISBN 0-295-97514-8

- A great book on chaos from the discoverer of Lorenz curves, includes anice discussion about fractality and dimensions.

7. Chaos Making a New Science, James Gleick, New York: Penguin Books, 1987,ISBN 0-14-009250-1

- national bestseller, probably the popular book that started a great deal ofinterest in chaos, very readable, mostly text, hardly any math, butworthwhile

Appendices

Worksheets and handouts for labs follow.


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