Hello!

I'd like to do two things in the frame of fractal discussion - to add

one more routine to our collection and to "add some oil to the flame"

of the discussion.

The first part is a simple one. Here is a Mandelbrot Snowflake.

to mandelbrot.flake.main :length :level

cs ht

repeat 4[mandelbrot.flake.step :length :level rt 90]

end

to mandelbrot.flake.step :length :level

if :level<0 [fd :length stop]

mandelbrot.flake.step :length/4 :level-1 lt 90

mandelbrot.flake.step :length/4 :level-1 rt 90

mandelbrot.flake.step :length/4 :level-1 rt 90

mandelbrot.flake.step :length/4 :level-1

mandelbrot.flake.step :length/4 :level-1 lt 90

mandelbrot.flake.step :length/4 :level-1 lt 90

mandelbrot.flake.step :length/4 :level-1 rt 90

mandelbrot.flake.step :length/4 :level-1

end

The second part is a little more difficult, for I'm not sure in my

English. But, I think, there are a lot of clever heads there to

clarify the things I'd like to share, if they are worth discussing.

I'd like to draw your attention from the perfect beauty of the

fractals to some important mathematical questions arisen there.

Let's take for example Koch Snowflake. Can you estimate a perimeter of

the first order curve and the square inside it? Sure! And what about

second order curve? Rather simple calculations. And the third order

curve? And if we go to the infinity? And if one goes to the infinity

and accurately makes estimations there, he'd find out that the

perimeter of the Snowflake "is equal" to infinity, but the square is

just 8/5 of initial one.

The same investigations may be done with Sierpinski Triangle. The

children may even cut the triangles of the white and black pieces of

paper and... go to the infinity.

About Dragon curves.

May be, I've missed it, but I didn't see this example in the fractal

discussion. The following codes are from J. Muller wonderful book.

to dragon :size :level

ldragon :size :level

end

to ldragon :size :level

if :level=0 [fd :size stop]

ldragon :size :level-1 lt 90

rdragon :size :level-1

end

to rdragon :size :level

if :level=0 [fd :size stop]

ldragon :size :level-1 rt 90

rdragon :size :level-1

end

It might be interesting to investigate with the children this curves

with the help of the strip of paper.

Take the strip of paper and fold it several times, putting every time

the left edge on to the right one. Then unfold the strip. There will

be folds of different kind - "hills" (H) and "gaps" (G)

How many folds will be there? S(n)=2* S(n-1)+1, S(0)=0.

In the middle there always be - G

What about the folds to the left and to the right of the middle?

They are of opposite kind, that is, first to the right - H, first to

the left - G, second to the right - G, second to the left - H and so

on.

For example, after 3 foldings: H H G G H G G

after 4 - H H G H H G G G H H G G H G G

One more remarkable quality:

after every new folding the right side of the new strip repeats the

hole previous one.

All this things can be easily explained and understood.

What it's all about?

Now, change G for lt 90, H to rt 90 and you'll have the Dragon curve.

I've talked a lot today! :-> Thanks to all!

Hope, something from all this may be useful.

Olga Tuzova.

-------------------------------------------------

Olga Tuzova, Ph.D.

Computer Science teacher,

International School of General Education,

St.Petersburg, Russia,

URL http:\\www.ort.spb.ru

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