Cynthia Lanius Fractals

8/12/2019 Cynthia Lanius Fractals

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The Math Forum

'ther Math Lessons by 5ynthia anius

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+lementary School 5ham ions2 Third "rade students in Ms/ *enner'sclass at Alief (S='s 5hambers +lementary in ouston, T> made thisthis th iteration $urassic Par! Fractal/ Wow2

And loo! at what reschoolers do2

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(ma"ine that the icture at the to of this a"e is a icture of thecoastline of Africa/Dou measure it with mile-lon" rulers and "et acertain measurement/ What if on the ne)t day you measure it withfoot-lon" rulers? Which measurement would "i%e you a lar"ermeasurement/ Since the coastline is 1a""ed, you could "et into thenoo!s and crannies better with the foot-lon" ruler, so it would yield a

"reater measurement/ .ow what if you measured it with an inch-lon"ruler? Dou could really "et into the teeniest and tiniest of cranniesthere/ So the measurement would be e%en bi""er, that is if thecoastline is 1a""ed smaller than an inch/ What if it were 1a""ed ate%ery oint on the coastline? Dou could measure it with shorter andshorter rulers, and the measurement would "et lon"er and lon"er/ Doucould e%en measure it with infinitesimally short rulers, and thecoastline would be infinitely lon"/ That's fractal/

!eo"le use them to sol+e real,%orld "roblems*

+n"ineers ha%e be"un desi"nin" and constructin" fractals in order tosol%e ractical en"ineerin" roblems/ For e)am le ta!e a loo! atAmal"amated *esearch (c/'s Fractal 5ontrol of Fluid =ynamics /

-nternet .esearch /uestions0

Dou can find the answers from the sites lin!ed abo%e/

6/ Who or"aniBed "eometry into a series of boo!s? What arethose boo!s called?

3/ What is the name of a mathematician who does researchtoday? Where does he;she wor!? What is the area ofmathematics in which he;she wor!s?

C/ Find another icture of a fractal that loo!s li!e an ob1ect innature/

(f ossible, email me your answers/

laniusEmath/rice/edu

Dou may obtain a rint %ersion of this a"e/

* Co"yri ht 1223,4556 Cynthia Lanius



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Self,Similarity

Table of Contents

Introduction

Why study fractals? What's so hot about fractals, anyway?

Makin fractals Sier ins!i Trian"le #sin" $a%a

Math &uestions Sier ins!i MeetsPascal $urassic Par! Fractal #sin" $A A (t "rows com le) *eal first iteration

+ncodin" the fractal World's ar"est

och Snowfla!e #sin" $a%a

(nfinite erimeter Finite area

Anti-Snowfla!e #sin" $a%a

Fractal !ro"erties

Self-similarity Fractional dimension

Formation by iteration

For Teachers Teachers' .otes Teacher-to-Teacher




5ats, canaries, or !an"aroos are similar if they are ali!e in some way/ (n"eometry thou"h, similar means somethin" %ery s ecific/ 9eometricfi"ures are similar if they ha%e the same sha e/ ( don't mean tworectan"les or two trian"les, but really the same sha e/ For e)am le:

The two s&uares are similar.

The two rectan"les are not similar.

0ut the two rectan"les below are similar.

oo! carefully at the last blue rectan"le and you will see that it is 3 timesas wide as the red rectan"le and 3 times as lon"/ We say that the sides arein ro ortion and the ratio or scale factorG is 3:6/ Since the corres ondin"sides are in ro ortion and the corres ondin" an"les are also of e&ualmeasureG, the fi"ures are the same sha e and are similar /

5onsider similarity in another way/ (n order for one fi"ure to be similar toanother, you must be able to ma"nify the len"th of the small fi"ure by thescale factor, and it will become e)actly the same siBe as the lar"er fi"ure/

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.ow how are fi"ures self-similar?

Many fi"ures that are not fractals are self-similar/ .otice the fi"ure tothe ri"ht/ .otice that the outline of the fi"ure is a tra eBoid/ .ow loo!inside at all the tra eBoids that ma!e u the lar"er tra eBoid/ This is ane)am le of self similarity.

Dou can also thin! of self-similarity as co ies/ +ach of the small tra eBoids is a co y ofthe lar"er/ 0elow are fi%e other e)am les of self-similarity/

Self,Similarity of Fractals

To the ri"ht is the Sier ins!i Trian"le that we ma!e in this unit/ .otice that the outlineof the fi"ure is an e&uilateral trian"le/ .ow loo! inside at all thee&uilateral trian"les/ *emember that there are infinitely many smaller andsmaller trian"les inside/ ow many different siBed trian"les can you find?All of these are similar to each other and to the ori"inal trian"le - selfsimilarity

See all the co ies of the ori"inal trian"le inside? ow many co ies do you see where theratio of the outer trian"le's sides to the inner ones is 3:6? :6? H:6? ( thin! we ha%e a

attern here/ 5an you find it?

5hec! out this %ery cool Sier ins!i animated self-similarity illustration/

/uestions on Self,Similarity

/uestion 10 (f the red ima"e is the ori"inal fi"ure,how many similar co ies of it are contained in the

blue fi"ure?

/uestion 40 Are s&uares self-similar? 5an you form bi""er s&uares out of smallerones?G Are he)a"ons? 5an you form lar"er he)a"ons out of smaller ones?G =rawe)am les to 1ustify your answer/

/uestion 70 Are circles similar? Are they self-similar? 5an you form lar"er circles outof smaller ones? =raw e)am les to 1ustify your answer/

/uestion 80 +) eriment with desi"nin" another self-similar fi"ure/

*obert =e%aney has more information on self-similarity/



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Snowfla!e #sin" $a%a

Fractal!ro"erties Self-similarity Fractionaldimension

Formation byiteration


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anius

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(t's an absolutely flat tableto e)tendin" out both ways to infinity/

S"ace: a hu"e em ty bo), has threedimensions, len"th, width, and de th,e)tendin" to infinity in all three directions/

Ib%iously this isn't a "ood re resentation of C-=/ 0esides its siBe, it's 1ust a he)a"on drawn tofool you into thin!in" it's a bo)/

Fractals can ha%e fractional (or fractal)dimension. A fractal mi"ht ha%e dimension of6/8 or 3/ / ow could that be? et's in%esti"ate

below/

$ust as the ima"es abo%e weren't %ery "ood ictures of a oint, line, lane, or s ace, thedrawin" meant to be the Sier ins!i Trian"le has limitations/ *emember as we continuethat fractals are really formed by infinitely many ste s/ So there are infinitely manysmaller and smaller trian"les inside the fi"ure, and infinitely many holes the blac!trian"lesG/

et's loo! further at what we mean bydimension / Ta!e a self-similar fi"ure li!ea line se"ment, and double its len"th/=oublin" the len"th "i%es two co ies ofthe ori"inal se"ment/



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7 ; 4 d , where d J the dimension/

0ut wait, 3 J 3 6, and J 3 3, so what number could this be? (t has to be somewhere between 6 and 3, ri"ht? et's add this to our table/

Fi ure 9imension $o* of Co"ies

ine Se"ment 6 3 J 3 6

Sier"inski(s Trian le ? 7 ; 4 ?

S&uare 3 J 3 3

5ube C H J 3 C

=oublin" Similarity d n J 3 d

So the dimension of Sier ins!i's Trian"le is between 6 and 3/ =o you thin! you couldfind a better answer? #se a calculator with an e) onent !ey the !ey usually loo!s li!ethis K G/ #se 3 as a base and e) eriment with different e) onents between 6 and 3 to see

how close you can come/ For e)am le, try 6/6/ Ty e 4


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Table of Contents

Introduction

Why study fractals? What's so hot about

fractals, anyway?

Makin fractals Sier ins!i Trian"le #sin" $a%a Math &uestions Sier ins!i Meets Pascal $urassic Par! Fractal #sin" $A A (t "rows com le) *eal first iteration

+ncodin" the fractal World's ar"est

och Snowfla!e #sin" $a%a

(nfinite erimeter Finite area Anti-Snowfla!e #sin" $a%a

Fractal !ro"erties Self-similarity Fractional dimension Formation by iteration






Fractals are often formed by what is called an iterative rocess/ ere'swhat ( mean/

To make a fractal0 Ta!e a familiar "eometric fi"ure a trian"le or linese"ment, for e)am leG and o erate on it so that the new fi"ure is more

com licated in a s ecial way/

Then in the same way, o erate on that resultin" fi"ure, and "et ane%en more com licated fi"ure/

.ow o erate on that resultin" fi"ure in the same way and "et an e%enmore com licated fi"ure/

=o it a"ain and a"ain///and a"ain/ (n fact, you ha%e to thin! of doin" itinfinitely many times/

Dou can obser%e this iterati%e rocess in all the fractals that we ma!ein this unit:

Sier ins!i's Trian"le och Snowfla!e Anti- och Snowfla!e The $urassic Par! fractal

.ot e%ery iterati%e rocess roduces a fractal/ Ta!e a line se"ment

and cho off the end/ What is the resultin" fi"ure? $ust another linese"ment - not com licated at all, and not a fractal/ Dou couldcontinue the iterati%e rocess o%er and o%er, cho in" off the end ofthe line se"ment, but it would 1ust become a shorter and shorter linese"ment, not com licated , not fractal/

0elow is a icture of a similar iterati%e o eration that is fractal/ Ta!e aline se"ment see belowG and remo%e the middle third/ What is theresultin" fi"ure? mmm/ That's a more com licated fi"ure/ (t's a linese"ment with a hole in it/

*e eat the rocess on that fi"ure/ (n other words, remo%e the middlethird of both of those sections/ This roduces an e%en morecom licated fi"ure/ .ow thin! of doin" this infinitely many times/ (nfact, this is a famous fractal called Cantor 9ust *



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