Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model...

IntroductionFractals

Mathematical ModellingLecture 14 – Fractals

Phil [email protected]

Phil Hasnip Mathematical Modelling


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.



Aim

To study shapes with fractional dimensions



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?’



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



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



How long is a sine wave?

As we shrink the size of the boxes, our estimate of the lengthconverges to the real length.



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!



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



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.



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

























What is its fractal dimension?

iteration L0 11 4× 1

3 = 43

2 16× 19 =

(43

)2

......

n 4n × 13n =

(43

)n




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




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 −→∞




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!



The Koch snowflake

We can make different shapes using different starting points.

Starting from an equilateral triangle −→ the Koch snowflake.



The Koch snowflake



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.



Summary

Fractals have unusual scaling properties −→ fractionaldimensionsPossible to have infinite perimeter, finite areaCan use the box counting method to measure fractaldimension


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Mathematical Modelling Lecture 14 Fractals · 2012-03-02 · Fractals Overview of Course Model...

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