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Fractal (Fractile)

Introduction

1. Derived from the Latin wordfractus, meaning fragmented, or broken. It was coined by the Polish-born mathematician Benoit B. Mandelbrot in 1975 when he asked a question about the length of the

coastline of Britain. If one takes aerial pictures of the coastline at different heights (or use ruler of

different divisions), one sees different amount of details. Since the coastline is irregular with bays, inlets,

etc., the estimates of the coastline become arbitrarily large. Does it approach infinity?

2. This idea is different from estimating the circumference of a circle (which is regular) using regular

polygons. The estimates converge.

3. A fractal curve is one whose length becomes arbitrarily large as the measuring scale becomes smaller and

smaller. A fractal may have one or more of the following characteristics:

(a) complex structure at all scales

(b) self-similarity

(c) fractional dimension

(d) infinite branching

(e) chaotic dynamics

4. There are numerous applications to diverse fields dealing with irregularly shaped objects or spatially non-

uniform and random phenomena: the rugged terrain of mountains, coastlines, Brownian movement,

vascular networks, population growth, the shapes of polymer molecules, distribution of galaxies, and the

intricate branch systems of trees and ferns in nature. Computer graphics are used to generate lifelike

images of these complicated, highly irregular natural objects and processes.

5. Introduction of fractals in school mathematics has the following benefits:

(a) as a recent topic in mathematics, it shows students that mathematics is a constantly developing

discipline (unlike most school mathematics which has been fossilised for thousands of years);(b) they provide visual illustrations of many basic mathematical results, such as sequences, indices,

limit, complex numbers, and symmetry (a multi-modal approach);(c) challenge students to find and explore visual patterns, linking these activities to computer graphics

(use ofExcel, Logo, and special fractal-generating programs); the planet in Star Trekwas computer-generated fractal landscape;

(d) it is fun to discover new fractals;

(e) develop more positive attitudes towards learning mathematics.

Self-Similar Objects

1. A self-similar object: its parts resemble the whole; each part, when magnified, will look basically like the

object as a whole; it remains invariant under changes of scale, i.e., it has scaling symmetry (self-symmetry

under magnification).

2. Koch curve (by Helga von Koch,1870 1924, a Swedish mathematician, introduced this curve in1904).

On a piece of A4 paper, draw a line segment of length 18 cm. Take this as a unit length. Trisect it and

replace the middle segment by the other two sides of an equilateral triangle of side31

the original length.

Repeat the process up to stage 3. Use pen of different colour for each stage.

3. (Logo) Load fractal.lgo. Type side1 50 0 and change the side (first input) and stage (second input).

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Stage

nLength of each segment

sn

Number of segments

pn

Total length

Ln0 1 1 1

131 4

34

2

n ( )n31 4

n

( )n34

4. The Koch curve is self-similar. As n , the total length . Imagine this curve is used to estimate thelength of a coastline. As the scale becomes smaller, the total length becomes increasingly larger.

5. Repeat the above using the following division (quadric Koch curve). (Logo) Type side2 50 0 and so on.

6. These fractal curves are highly complex and contain a great deal of detail. The degree of complexity of

self-similar fractals is described by thesimilarity fractal dimension. These curves have fractional

dimensions, so FRACTAL also stands for FRACTionAL dimension.

7. LetNbe the ratio of the number of final pieces to the number of original pieces (replacement ratio) and rthe ratio of the length of the originalpiece to the length of each final piece (scaling ratio). Then the

dimensionD is defined by NrD = or rN

Dlog

log= . This is an extension of the dimensions of classical

geometric objects, and is based on the notion of Hausdorff dimension for topological spaces.

Figure r N Dimension,D

Line 2 2 = 21 1

Square 2 4 = 22 2

Cube 2 8 = 23 3

Koch curve 3 4 = 3D 1.26

Quadric Koch curve 3 5 1.46

8. These curves form the L-system fractals, where L stands for Lindenmayer, a mathematician who wrote

about the fractal nature of plant development.

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8. Curves like the Sierpinski curve are closed paths that pass through every point in the unit square. Space-

filling curve will do the same for a cube.

9. Sierpinski triangle (or gasket). Construct an equilateral triangle of side 12 units on an isometric dot paper.

At step 1, bisect each side and shade the middle triangle. This leaves three unshadedtriangles. See the first

two stages below.

Stage

n

Number ofunshadedtriangles

pn

Total area ofunshadedtriangles

An0 1 1

1 34

3

n 3n ( )n4

3

Show that its dimension is2log

3log 1.58.

10. Sierpinski triangle and the chaos game. This activity demonstrates that a non-deterministic process may

lead to regularity.

Draw an equilateral triangle with top vertex T and left and right vertices L and R. Take any point inside

the triangle. Throw a die and mark a point accordingly:

For 1 or 2, mark the mid-point between the current point and L.For 3 or 4, mark the mid-point between the current point and T.

For 5 or 6, mark the mid-point between the current point and R.

Repeat this process for a number of points.To collect class result, give each member a transparency of the same triangle and ask them to mark using

different colours. Put them together later.

11. Let L = (-1, 0), R = (1, 0), and T = (0, 3). The chaos game is generated by three linear mappings:

L =

221 ,

yx, R =

+

221 ,

yx, T =

+

2

3

2,yx

.

12. Instead of taking the mid-point between a point and a vertex, choose a line that divides the segment from

the point to a vertex in the ratio k: (1 k). Ifk< 0.5, we have an overlaid Sierpinski triangle; ifk> 0.5,we have a strict Sierpinski triangle where the triangles have no common point. (Logo) Type sierst andenter the value ofkto use.

13. Repeat the above using a square (or any regular polygon!). (Logo) Type siersqst.

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Chaos

Many physical phenomena like a swinging pendulum are deterministic and regular. However, some aredeterministic but irregular (atmospheric conditions); a minor change in one parameter may give rise to drastic

changes; this is a state of chaos.

A Logistic Example

1. Letpn + 1 be the population at year (n + 1). A simple model may be:

pn + 1 =pn + b pn= (1 + b) pn, where b is the birth rate (> 0).This model will show that the population will grow without bound!

2. Letpn + 1 = (1 + b)s pn, wheres is the survival rate. Here the overall growth rate is (1 + b)s.

If 0 < (1 + b)s < 1, the population will eventually die out.If (1 + b)s = 1, the population will remain constant.If (1 + b)s > 1, the population will grow without bound.

3. These linear models are not realistic. May and Feigenbaum examine a logistic function. Suppose there is

a maximum populationPthat the resources can support. Multiple the above equation by (P pn); notice

that aspn increases, this term becomes smaller. We have

pn + 1 = (1 + b)s(P pn)pn orP

p

P

p

P

p nnn

=+ 11 , or

xn+1 = (1 xn) xn,where 0

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4. The boundary of the prisoner set (consisting of thez0 values) is called a Julia set for the specific value ofc.

5. (Logo) Type juliaset to plot all the points defined by -1.5 x 1.5 and -1.5 y 1.5. Examples to try:c = 0.3; c = -0.05 + 0.745i, c = -0.5; c = -0.71 + 0.36i; c = 0.27 + 0.49i. The cross shows the position of thepoint c; in the last two cases, the regions cover the point c. Each diagram takes about 15 minutes.

c = 0.3 c = -0.05 + 0.745i c = -0.6

Mandelbrot Set

1. These sets are named after the mathematician, Benoit Mandelbrot, who first produced them in 1979. The

Mandelbrot set is shown below. It consists of a main bay (the lake), three baylets, and fractal shorelines.

2. Each value of the Mandelbrot set corresponds to one Julia set c. Ifc is inside the lake, the correspondingJulia set is connected. As c moves towards the shore, the Julia set becomes convoluted and eventually

becomes disconnected.

3. (Logo) The mandelst procedure useszz2 + 1.5c,and points are plotted for 1.5 x 0.5 and 1 y 1.Using the default parameters, the following plot is obtained; time taken is about 12 minutes. In the

construction, take a value ofc and check whether the origin is a prisoner. If so, the value ofc is included(plotted).

4. In colouring Mandelbrot set, points outside the set are coloured according to the number of iterations to

test a point. For example, if a test point escapes in a few iterations, it might be coloured red; if it escapes

in many iterations, it might be coloured blue, and so on.

References

Bills, C., & Bills, L. (1997). A little bit of chaos. In M. Sewell (Ed.), Mathematics masterclass: Stretching the

imagination (pp. 1-15). Oxford: Oxford University Press.Field, M., & Golubitsky, M. (1992). Symmetry in chaos: A search for pattern in mathematics, art and nature.

Oxford: Oxford University Press.Muller, J. (1997). The great Logo adventure: Discovering Logo on and off the computer. Madison, AL: Doone

Publications.

Peitgen, H.O., Jrgens, H., Saupe, D., Maletsky, E., Perciante, T., & Yunker, L. (1991).Fractals for theclassroom: Strategic activities (Vol. 1). New York: Springer-Verlag & NCTM .

Fractals: A Fractal Unit for Elementary and Middle School Students (http://math.rice.edu/~lanius/frac/)

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http://math.rice.edu/~lanius/frac/http://math.rice.edu/~lanius/frac/

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The Spanky Fractal Database of the Stone Soup Group (http://spanky.triumf.ca/www/fractint/fractint.html) for

the freeware,Fractint, used to generate various types of fractals under DOS. Instructions and examplesgiven in Wegner, T. & Tyler, B. (1993).Fractal creations (2nd ed.).

Fractal Pictures and Animations (http://www.cnam.fr/fractals.html)

Logo Procedures

TO SIDE1 :SIZE :LEVEL

IF :LEVEL = 0 [FD :SIZE STOP]SIDE1 :SIZE / 3 :LEVEL - 1 LT 60

SIDE1 :SIZE / 3 :LEVEL - 1 RT 120

SIDE1 :SIZE / 3 :LEVEL - 1 LT 60

SIDE1 :SIZE / 3 :LEVEL - 1

END

TO KOCH :SIZE :LEVEL

REPEAT 3 [RT 120 SIDE1 :SIZE :LEVEL]

END

TO KOCHST

CS WINDOW PU SETXY [-200 80] PD

SETPC 0 KOCH 200 1 WAIT 30SETPC 9 KOCH 200 2 WAIT 30

SETPC 5 KOCH 200 3 WAIT 30

SETPC 12 KOCH 200 4

END

TO SIDE2 :SIZE :LEVEL

IF :LEVEL = 0 [FD :SIZE STOP]






END

TO QKOCH:SIZE :LEVEL

REPEAT 4 [RT 90 SIDE2 :SIZE :LEVEL]

END

TO QKOCHST

CS WINDOW PU SETXY [-200 80] PD

SETPC 0 QKOCH 100 1 WAIT 30



SETPC 12 QKOCH 100 4

END

TO SIDE3 :SIZE :LEVELIF :LEVEL = 0 [FD :SIZE STOP]










END

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TO SIER

PU

MAKE "V RANDOM 3

IF :V=1 [SETPC 1 SETXY LIST (1-:C)*XCOR (1-:C)*YCOR+119*:C]

IF :V=2 [SETPC 2 SETXY LIST (1-:C)*XCOR-139*:C (1-:C)*YCOR-119*:C]

IF :V=3 [SETPC 12 SETXY LIST(1-:C)*XCOR+139*:C (1-:C)*YCOR-119*:C]

PD FD 1 BK 1

SIER

END

TO SIERST

CS WINDOW PU

TYPE [ENTER THE DIVISION RATIO (SAY .5)] MAKE "C READ

SETXY LIST ( 140 - RANDOM 280) (120-RANDOM 240)

PD FD 1 BK 1

SIER

END

TO SIERSQ

PU MAKE "V RANDOM 4

IF :V=1 [SETPC 1 SETXY LIST (1-:c)*XCOR-139*:C (1-:C)*YCOR+119*:C]

IF :V=2 [SETPC 4 SETXY LIST (1-:C)*XCOR+139*:C (1-:C)*YCOR+119*:C]

IF :V=3 [SETPC 9 SETXY LIST (1-:C)*XCOR-139*:C (1-:C)*YCOR-119*:C]

IF :V=4 [SETPC 12 SETXY LIST (1-:C)*XCOR+139*:C (1-:C)*YCOR-119*:C]

PD FD 1 BK 1

SIERSQ

END

TO SIERSQST

CS WINDOW PU

SETXY LIST ( 140 - RANDOM 280) (120-RANDOM 240)

PD FD 1 BK 1

TYPE [ENTER THE DIVISION RATIO (SAY .5)] MAKE "C READ

SIERSQ

END

TO JULIAPT

CS HT WINDOW SETPC 0

TYPE [ENTER THE SCALING FACTOR FOR AXES (SAY 60) ==>] MAKE "SCALE READ

TYPE [ENTER THE REAL PART OF C ==>] MAKE "CR READ

TYPE [ENTER THE IMAGINARY PART OF C ==>] MAKE "CI READ

MAKE "LIMIT 0.5 + SQRT(.25+ SQRT (:CR*:CR + :CI*:CI))

SETPC 12 DOT LIST :CR*:SCALE :CI*:SCALE

TYPE [HOW MANY ITERATIONS TO TAKE (SAY 100) ==>] MAKE "TIMES READ

GETPOINTS1

END

TO AXES

CS SETPC 0 BK 2*:SCALE TICK

REPEAT 4 [FD :SCALE TICK] BK 2*:SCALE RT 90 BK 2*:SCALE TICK

REPEAT 4 [FD :SCALE TICK] HOME

END

TO TICK

LT 90 FD 3 BK 6 FD 3 RT 90

END

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TO GETPOINTS1

TYPE [ENTER THE REAL PART OF A COMPLEX NUMBER ==>] MAKE "R0 READ

TYPE [ENTER THE IMAGINARY PART OF THE COMPLEX NUMBER ==>] MAKE "I0 READ

MAKE "RT :R0 MAKE "IT :I0

AXES

PU SETXY LIST :R0*:SCALE :I0*:SCALE PD SETPC 12

CHECKSTATUS :RT :IT

IF :J = 0 (PRINT :R0 :I0 "PRISONER)

IF :J = 1 (PRINT :R0 :I0 "ESCAPER)

GETPOINTS1END

TO CHECKSTATUS :XR :XI

MAKE "J 0

JULIA 1

END

TO JULIA :T

IF :T = :TIMES [STOP]

MAKE "R1 (:RT*:RT - :IT*:IT + :CR)

MAKE "I1 2*:RT*:IT+:CI

MAKE "DIST SQRT (:R1*:R1 + :I1*:I1)

SETXY LIST :R1*:SCALE :I1*:SCALE

(PR :R1 :I1 "DISTANCE :DIST) WAIT 10

IF :DIST > :LIMIT [MAKE "J 1 STOP]


JULIA :T+1

END

TO JULIASET

CS WINDOW HT


TYPE [ENTER THE REAL PART OF C ==>] MAKE "CR READ

TYPE [ENTER THE IMAGINARY PART OF C ==>] MAKE "CI READ

MAKE "LIMIT 0.5 + SQRT(.25+ SQRT (:CR*:CR + :CI*:CI))

SETPC 12 DOT LIST :CR*:SCALE :CI*:SCALE MARK

MAKE "TIMES 20TYPE [ENTER THE NUMBER OF DIVISIONS TO TAKE (SAY 200)==>] MAKE "D READ

MAKE "LP 3/:D

FOR "N 0 :D [MAKE "R0 (-1.5)+:N*:LP FOR "M 0 :D [MAKE "I0 (-1.5)+:M*:LP

GETPOINTSJ]]

END

TO MARK

PU SETXY LIST :CR*:SCALE :CI*:SCALE SETH 45 PD

REPEAT 4 [FD 5 BK 5 RT 90]

PU

END

TO GETPOINTSJ


CHECKSTATUSJ :RT :IT

IF :J = 0 [(PRINT :R0 :I0 "PRISONER) SETPC 0 DOT LIST :R0*:SCALE :I0*:SCALE]

;IF :J = 1 [(PRINT :R0 :I0 "ESCAPER) SETPC 13 DOT LIST :R0*:SCALE :I0*:SCALE]

END

TO CHECKSTATUSJ :XR :XI

MAKE "J 0

JULIAJ 1

END

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TO JULIAJ :T


MAKE "R1 (:RT*:RT - :IT*:IT + :CR)

MAKE "I1 2*:RT*:IT+:CI

;(PR :R1 :I1) WAIT 10 (INCLUDE THIS TO SHOW THE SEQUENCE)

IF (SQRT (:R1*:R1 + :I1*:I1)) > :LIMIT [MAKE "J 1 STOP]


JULIAJ :T+1

END

TO MANDELST

CS WINDOW HT


TYPE [ENTER THE NUMBER OF C TO CHECK (SAY 500)==>] MAKE "C READ

TYPE [ENTER THE LIMIT OF DISTANCE TO TEST (SAY 10) ==>] MAKE "LIMIT READ

TYPE [ENTER THE MAXIMUM NO. OF ITERATION (SAY 20) ==>] MAKE "TIMES READ

MAKE "LP 2/:C

FOR "N 0 :C [MAKE "CR (-1.5)+:N*:LP FOR "M 0 :C [MAKE "CI (-1)+:M*:LP GETPOINTS3]]

END

TO GETPOINTS3

MAKE "R0 0 MAKE "I0 0


CHECKSTATUS3 :RT :IT

IF :J = 0 [(PRINT :CR :CI "PRISONER) SETPC 0 DOT LIST :CR*:SCALE :CI*:SCALE]

;IF :J = 1 [(PRINT :CR :CI "ESCAPER) SETPC 13 DOT LIST :CR*:SCALE :CI*:SCALE]

END

TO CHECKSTATUS3 :XR :XI

MAKE "J 0

MANDEL 1

END

TO MANDEL :T


MAKE "R1 (:RT*:RT - :IT*:IT + 1.5*:CR)

MAKE "I1 2*:RT*:IT+1.5*:CIIF (SQRT (:R1*:R1 + :I1*:I1)) > :LIMIT [MAKE "J 1 STOP]


MANDEL :T+1

END

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Koch Curve (stage 3)

Koch Curve (stage 4)

Quadric Koch Curve (stage 3)

Koch Snowflake (stages 1, 2, 3)

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Koch Anti-Snowflake (stages 1, 2, 3)

Peano Curve (stages 1, 2, 3)

Quadric Island Curve (stages 1, 2, 3)

Sierpinski Triangle

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Julia Sets

c = 0.3 c = -0.05 + 0.745i

c = -0.6

Mandelbrot Set

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Instructions for Fractals on PC Logo

In each case, change the input values and see what happen!

Koch Curve ? cs rt 90? side1 100 0

? side1 100 1? side1 100 2

Quadric Koch Curve ? cs rt 90? side2 100 0? side2 100 1? side2 100 2

Koch Snowflake ? cs koch 100 0? koch 100 1? koch 100 2? kochst

Koch Anti-Snowflake ? cs kochanti 100 0? cs kochanti 100 1? cs kochanti 100 2? kochantist

Quadric Koch Snowflake ? cs qkoch 100 0

? qkoch 100 1? qkoch 100 2? qkochst

Peano Curve ? cs rt 90? side3 100 0? side3 100 1? side3 100 2

Quadric Island ? cs rt 90? side4 100 0? side4 100 1? side4 100 2

Sierpinski Triangle as a chaosgame

sierst (enter a value between 0 and 1)

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Fractal 5