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Fractals and Chaos
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Snowflake
We have seen that the first few stages of the snowflake are given by:

Assuming that the length of each side of the triangle is 1 unit complete the
following table:
STAGE
PERIMETER
1
3
2
3
4
5
6
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
Can you work out a formula for the perimeter at the nth stage?
Do you think the area of the snowflake curve is finite or infinite?
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Fractals and Chaos
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There be Dragons Here
We are going to make a dragon, a very complex dragon, more complex than any dragon
you may, or may not, have seen before. You will need a pencil and some (graph) paper
and perhaps an eraser. To construct the dragon you need to follow the following steps:
1. Draw a straight line, say 2cm long, but the larger this line, the more stages you
can draw. This is the start of your dragon.
2. Now draw two lines so that the original line forms the hypotenuse of an isosceles
right-angled triangle and erase the original line. This is the first stage. (See the
diagram below.)
3. For the second stage, replace each of the lines from the first stage with two new
segments at right angles so that the lines from stage one form the hypotenuse of
an isosceles right-angled triangle.
4. The new segments are placed to the left then to the right along the segments of the
first stage.
5. Continue this construction, always alternating the new segments between left and
right along the segments of the previous stage. This generates the `dragon curve'.

Check the first few stages of the dragon curve are those given below:
The solid lines represent the dragon curve at each stage, whilst the dotted lines are the
previous stage: the dotted lines are not part of the dragon curve.

(Optional) With the aid of a computer can you work out the 10th and 16th stages of
the dragon curve.
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Fractals and Chaos
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Patchwork Quilts
Consider the following the following process:
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Work out what happens to the black square after you iterate this process four
times. Make up your own template and perform the same operation. (You may
find a computer useful for this exercise.)
Now we allow rotations, so consider the following template:
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Starting with a black square what do you get when you iterate this process? Again
a computer will be useful.
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Fractals and Chaos
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Recall that if we have two coordinates: (a,b) and (c,d) then we add them as follows:
(a,b) + (c,d) = (a+c,b+d).
We can also multiply them, but it isn't quite as straight forward.
(a,b) x (c,d) = (ac-db, bc+ad).
For example, we have
(2,3) + (4,-1) = ( 2+4,3-1) = (6,2)
(4,-2) - (3,5) = ( 4-3,-2-5) = (1,-7).
(2,3) x (4,-1) = (11,10)
(2,3) x (2,-3) =(13,0).
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Calculate the following:
1. (3,5) + (7,-1),
2. (4,-1) + (3,3),
3. (2,7) + (4,-9),
Calculate the following:
1 1 1 1
1.  ,    , 
2 2 2 2
1 1 1 1 1 1
2.  ,    ,    , 
2 2 2 2 2 2
Plot the following points on graph paper:
1 1
1.  , 
2 2
 1
2.  0, 
 2
 1 1
3.   , 
 4 4
4. (3,1) x (3,0),
5. (2,-1) x (2,-1),
6. (0,1) x (0,1).
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Fractals and Chaos
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Recall that given a coordinate (a,b) we can write this number using polar coordinates.
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Write the following coordinates as polar coordinates; that is, find their distance
from the origin and the angle they make with the positive part of the x-axis.
1. (1,1),
2. (-1,1),
3. (0,-1).
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Calculate (1,1) x (1,1). Plot (1,1) and (1,1) x (1,1) on graph paper. Using a ruler
and a protractor measure, as accurately as you can, the distance of both points
from the origin and the angle they make with the positive x-axis.
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Calculate (1,2) x (1,2). Plot (1,2) and (1,2) x (1,2) on graph paper and using a
ruler and a protractor, measure as accurately as you can, the distance of both
points from the origin and the angle they make with the positive x-axis.
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Using the previous two questions as motivation, can you work out what happens
to the distance from the origin and the angle when we square a number in polar
coordinates?
We are now going to try and iterate using polar coordinates. We try something very
simple like squaring. So we select a number in polar coordinates, we square it, square the
result and so on and see what happens as we square more and more. When we tried this
with normal numbers we found that those numbers less than 1 and bigger than -1 all got
closer and closer to zero. If we selected the points 1 or -1, then they just stayed at 1.
Finally, if we selected any number larger than 1 or less than -1 then these numbers just
got bigger and bigger.
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Fractals and Chaos
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
For each of the following points in polar coordinates work out what happens if
you square them, square the results and so on.
1 
1.  ,0 
2 
1 
2.  ,10
4 

3.
4.
2,0
1,5
Can you describe what happens generally like we did above for real numbers?
You might need to do more experimentation.
Hopefully, you saw that if r<1 in the polar coordinate [r,a] then iterating this point gives a
collection of points that get close and closer to the origin. If a is non zero, then the points
spiral into the origin. If r=1 and a is non zero the points spin around a circle. Whilst if r>1
then the points get further and further away from the origin.
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Which of the following points are contained in Julia set for `squaring’ and which
are not.
1 
1.  ,45
4 
2. 1,34
3. 2,50
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
Does the point [1,45] ever come back to itself when we continue to square it?
Do all points on the Julia set come back to where they started if we square them a
sufficient number of times?
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Fractals and Chaos
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
Using a computer experiment with various different Julia sets to see the different
shapes you can generate. Can you find the values of the constant you need to
generate the following Julia sets?

Using a computer explore the Mandelbrot set pictured below
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