Mathematical Modelling Lecture 14 Fractals

Transcription

1 Lecture 14

2 Overview of Course Model construction dimensional analysis Experimental input fitting Finding a best answer optimisation Tools for constructing and manipulating models networks, differential equations, integration Tools for constructing and simulating models randomness Real world difficulties chaos and fractals The material in these lectures is not in A First Course in Mathematical Modeling.

3 Aim Introduction To study shapes with fractional dimensions

4 Natural shapes Introduction In our earlier discussions of scaled models we emphasised the importance of geometrical similarity. This is easy for man-made structures like skyscrapers and submarines what about natural shapes like trees, clouds and coastlines? In 1982 Benoit Mandelbrot addressed these questions in How long is the coastline of Britain?

5 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 N1 2 squares over the shape Count squares needed to contain shape, S(N 1 ) Reduce the size of squares so now have N2 2, and recount This is like using a smaller and smaller ruler

6 Box counting method We expect that the total length is no. boxes size of box, i.e. L = S(N). 1 N = constant In other words we expect: S(N) = constant N Of course we must remember: Near start, ruler is big measurements inaccurate Near end, ruler is small line thickness causes problems

7 How long is a sine wave? As we shrink the size of the boxes, our estimate of the length converges to the real length.

8 How long is the coastline of Britain? This time the length does not converge, it seems to change with the no. boxes N in each dimension. In fact: but d is not an integer. S(N) = constant N d The coastline has a fractional dimension fractal!

9 How long is the coastline of Britain? Euclidean geometry always has integer dimensions length is N, area N 2, volume N 3 and so on. Natural shapes do not. Use box counting method Plot on a log-log graph Slope fractional dimension d f

10 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 in each dimension, then the measured quantity m will change as: S(bN) = b d S(N) where d is the fractal (box counting) dimension. E.g. halving the length of each box have 2 times boxes in each dimension measured area goes up by b 2 boxes.

11 Simple model the Koch curve Generating a Koch curve is simple. Starting with a straight line: 1 Split every straight line section into three 2 Put an equilateral triangle on every middle section 3 Remove the triangle s base 4 Repeat from step 1

12 Simple model the Koch curve

13 Simple model the Koch curve

14 Simple model the Koch curve

15 Simple model the Koch curve

16 Simple model the Koch curve

17 Simple model the Koch curve

18 Simple model the Koch curve

19 Simple model the Koch curve What is its fractal dimension? iteration L = = ( n 4 n 1 3 = ( 4 n 3 ) 2 ) n

20 Simple model the Koch curve Now if I make my ruler 1 3 of its original length, I get 3 times the boxes in each dimension, but the number of boxes I count gets 4 times bigger. Remember: so in this case we have: S(bN) = b d S(N) b = 3 S(3N) = 4S(N) 3 d = 4 d = ln 4 ln 3

21 Simple model the Koch curve Every time we replace a third of each line with two-thirds i.e. each time we make length l 4 3 l As we keep going, l

22 Simple model the Koch curve Koch curve has definite start and end points Is infinitely long! We expected length to converge why doesn t it? As ruler made smaller, see more and more detail There is always more detail to see!

23 The Koch snowflake We can make different shapes using different starting points. Starting from an equilateral triangle the Koch snowflake.

24 The Koch snowflake

25 The Koch snowflake The perimeter is now basically three Koch curves, so same fractal dimension as before. Infinite perimeter, but finite area! Could also change the algorithm, e.g. replace section with squares rather than triangles.

26 Summary Introduction have unusual scaling properties fractional dimensions Possible to have infinite perimeter, finite area Can use the box counting method to measure fractal dimension