Answer to 9/30 (flowers) problem

Here are the answers to the 9/30/07 flowers problem

When we look at all three pictures we start seeing something in common. Look at them. Do you see that the second picture actually contains the first one and that the this picture actually contains the second? Look at them and see that this is true. This is the idea behind a "fractal": A mathematical object that contains itself within itself within itself....forever.

So how can we create this idea in SeeLogo? First let us prepare three names that will define the three pictures and we can call them A1 A2 and A3.
so we can type:
NEW A1
NEW A2
NEW A3

right now A1 A2 and A3 are empty. They have nothing in them but in few minutes we will fill in the void.

The first picture (A1) consists of a line and a circle "spun 6 times around the center"
step 1: Make a line and a circle


In the SeeLogo language there can be many ways to do it. (Click on A1 first)

For example we could type in the command line:
GOFD 40 FD 30 CIRCLE 60
JT 0 0
SPIN 6 IT

Or type directly in the editor:
SPIN 6 [FD 40 GOFD 30 CIRCLE 60 JT 0 0]
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The 2nd picture will be A2.
The idea now will be to do to steps almost like before:

Step 1: Make a line and a circle and fir A1 inside the circle using proportion.
Step 2: Bring the point back to the center and spin the picture 6 times.

We note that in order to fir the also picture A1 into the small circle we need to scale A1 to fit in it. Since the Diameter of the big picture A1 is 2*(40+60)=200 and the diameter of the small circle is 60 pixels, the scaling factor is exactly 60/200=30%

To make it in Seelogo we click on A2 we click on A2 first and then type:

FD 40 GOFD 30 CIRCLE 60 SIZE 60/200 [A1]
JT 0 0
SPIN 6 IT

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The next level will be same as level 2 except that you will use A2 instead of A1 and there is really not limit to the number of levels you can make. This process actually starts creating an infinite object: A fractal.

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Once you get the idea it you will feel some change in your brain but you cannot really force it. It has to happen naturally. This is why I do not judge you by your performance but by the effort you put into it and sometime to much pushing does not help. Still avoiding it does not help either. I suggest that you practice and I will give you a test in class (not on paper) to be able to [produce a similar picture. This will motivate you much more than a paper test and I think you will learn better. I will try to give you more examples later and/or in class to allow you to practice more but once you master it the benefit will carry much further than just this math class.

--Dani




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