mathematical modelling lecture 14 fractals · 2012-03-02 · fractals overview of course model...
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![Page 1: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/1.jpg)
IntroductionFractals
Mathematical ModellingLecture 14 – Fractals
Phil [email protected]
Phil Hasnip Mathematical Modelling
![Page 2: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/2.jpg)
IntroductionFractals
Overview of Course
Model construction −→ dimensional analysisExperimental input −→ fittingFinding a ‘best’ answer −→ optimisationTools for constructing and manipulating models −→networks, differential equations, integrationTools for constructing and simulating models −→randomnessReal world difficulties −→ chaos and fractals
The material in these lectures is not in A First Course inMathematical Modeling.
Phil Hasnip Mathematical Modelling
![Page 3: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/3.jpg)
IntroductionFractals
Aim
To study shapes with fractional dimensions
Phil Hasnip Mathematical Modelling
![Page 4: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/4.jpg)
IntroductionFractals
Natural shapes
In our earlier discussions of scaled models we emphasised theimportance of geometrical similarity.
This is easy for man-made structures like skyscrapers andsubmarines – what about natural shapes like trees, clouds andcoastlines?
In 1982 Benoit Mandelbrot addressed these questions in ‘Howlong is the coastline of Britain?’
Phil Hasnip Mathematical Modelling
![Page 5: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/5.jpg)
IntroductionFractals
How long is a sine wave?
Before we look at our coastline, let’s tackle a simpler problem:the length of a sine wave. We’ll use the box counting method:
Draw a grid of N21 squares over the shape
Count squares needed to contain shape, S(N1)
Reduce the size of squares so now have N22 , and recount
This is like using a smaller and smaller ruler
Phil Hasnip Mathematical Modelling
![Page 6: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/6.jpg)
IntroductionFractals
Box counting method
We expect that the total length is no. boxes × size of box, i.e.
L = S(N).1N
= constant
In other words we expect:
S(N) = constant× N
Of course we must remember:
Near start, ruler is big −→ measurements inaccurateNear end, ruler is small −→ line thickness causes problems
Phil Hasnip Mathematical Modelling
![Page 7: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/7.jpg)
IntroductionFractals
How long is a sine wave?
As we shrink the size of the boxes, our estimate of the lengthconverges to the real length.
Phil Hasnip Mathematical Modelling
![Page 8: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/8.jpg)
IntroductionFractals
How long is the coastline of Britain?
This time the length does not converge, it seems to change withthe no. boxes N in each dimension.
In fact:S(N) = constant× Nd
but d is not an integer.
The coastline has a fractional dimension −→ fractal!
Phil Hasnip Mathematical Modelling
![Page 9: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/9.jpg)
IntroductionFractals
How long is the coastline of Britain?
Euclidean geometry always has integer dimensions – length isN, area N2, volume N3 and so on. Natural shapes do not.
Use box counting methodPlot on a log-log graphSlope −→ fractional dimension df
Phil Hasnip Mathematical Modelling
![Page 10: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/10.jpg)
IntroductionFractals
Generalised box counting method
We can use box counting to measure area, volume etc. too.
If we reduce the size of our box to get b times no. boxes ineach dimension, then the measured quantity m will change as:
S(bN) = bdS(N)
where d is the fractal (box counting) dimension.
E.g. halving the length of each box −→ have 2 times boxes ineach dimension −→ measured area goes up by b2 boxes.
Phil Hasnip Mathematical Modelling
![Page 11: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/11.jpg)
IntroductionFractals
Simple model – the Koch curve
Generating a Koch curve is simple. Starting with a straight line:
1 Split every straight line section into three2 Put an equilateral triangle on every middle section3 Remove the triangle’s base4 Repeat from step 1
Phil Hasnip Mathematical Modelling
![Page 12: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/12.jpg)
IntroductionFractals
Simple model – the Koch curve
Phil Hasnip Mathematical Modelling
![Page 13: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/13.jpg)
IntroductionFractals
Simple model – the Koch curve
Phil Hasnip Mathematical Modelling
![Page 14: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/14.jpg)
IntroductionFractals
Simple model – the Koch curve
Phil Hasnip Mathematical Modelling
![Page 15: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/15.jpg)
IntroductionFractals
Simple model – the Koch curve
Phil Hasnip Mathematical Modelling
![Page 16: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/16.jpg)
IntroductionFractals
Simple model – the Koch curve
Phil Hasnip Mathematical Modelling
![Page 17: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/17.jpg)
IntroductionFractals
Simple model – the Koch curve
Phil Hasnip Mathematical Modelling
![Page 18: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/18.jpg)
IntroductionFractals
Simple model – the Koch curve
Phil Hasnip Mathematical Modelling
![Page 19: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/19.jpg)
IntroductionFractals
Simple model – the Koch curve
What is its fractal dimension?
iteration L0 11 4× 1
3 = 43
2 16× 19 =
(43
)2
......
n 4n × 13n =
(43
)n
Phil Hasnip Mathematical Modelling
![Page 20: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/20.jpg)
IntroductionFractals
Simple model – the Koch curve
Now if I make my ruler 13 of its original length, I get 3 times the
boxes in each dimension, but the number of boxes I count gets4 times bigger. Remember:
S(bN) = bdS(N)
so in this case we have:
b = 3S(3N) = 4S(N)
⇒ 3d = 4
⇒ d =ln 4ln 3
Phil Hasnip Mathematical Modelling
![Page 21: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/21.jpg)
IntroductionFractals
Simple model – the Koch curve
Every time we replace a third of each line with two-thirdsi.e. each time we make length l → 4
3 lAs we keep going, l −→∞
Phil Hasnip Mathematical Modelling
![Page 22: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/22.jpg)
IntroductionFractals
Simple model – the Koch curve
Koch curve has definite start and end pointsIs infinitely long!We expected length to converge – why doesn’t it?As ruler made smaller, see more and more detailThere is always more detail to see!
Phil Hasnip Mathematical Modelling
![Page 23: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/23.jpg)
IntroductionFractals
The Koch snowflake
We can make different shapes using different starting points.
Starting from an equilateral triangle −→ the Koch snowflake.
Phil Hasnip Mathematical Modelling
![Page 24: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/24.jpg)
IntroductionFractals
The Koch snowflake
Phil Hasnip Mathematical Modelling
![Page 25: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/25.jpg)
IntroductionFractals
The Koch snowflake
The perimeter is now basically three Koch curves, so samefractal dimension as before.
−→ Infinite perimeter, but finite area!
Could also change the algorithm, e.g. replace section withsquares rather than triangles.
Phil Hasnip Mathematical Modelling
![Page 26: Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model construction ! dimensional analysis Experimental input ! fitting Finding a ‘best’](https://reader033.vdocuments.us/reader033/viewer/2022042711/5f703ad216d7ae1fef19be9d/html5/thumbnails/26.jpg)
IntroductionFractals
Summary
Fractals have unusual scaling properties −→ fractionaldimensionsPossible to have infinite perimeter, finite areaCan use the box counting method to measure fractaldimension
Phil Hasnip Mathematical Modelling