Fractal theory in the proverbial nutshell.

Blimey. You know when you get something in your head but there's no way you could explain it to anyone? I hope I've got this right! It's taken me ages to make it make some sort of sense.

So OK, do this:


Cut a piece of paper, about 2.5cm wide and about 12cm long.

Holding it in both hands, move your right hand towards your left to fold it in half.

Hold the halved paper, and repeat the right to left movement.

Do this two more times. This means your paper should be folded four times, all of those using a right to left movement. You should have a strip of paper folded a lot and looking very uninteresting.

Unfold it, making sure to crease each fold as you do, so you can clearly see where they are. Turn it on it's thin edge and move it around until it looks like this:



The easy way to find the correct layout of it is to look for the 'square'.

This is called the first iteration.

Now, if you folded it all up and continued to fold it again for the same amount of times, what you would get is this:



This is called the second iteration.

If you kept on doing it and kept on doing it, what you would eventually get is this:



This would be the fifitieth iteration.


See? That's basically how fractals work. It's easy!

If you want to go further and look at it in a tabular way, for this particular fractal sequence you should imagine that your folded paper is a road. The first turn you make is a 'right' one.



This is always the same and always in the centre of your table.

R

On the second fold, if you fold your paper out you can see that initially you've done the same thing as before; made a right turn. But then you turn right again, and then left.



So your table reads as so:

RRL

On fold three the fractal rules start to come into play.



Because each iteration always starts out with the previous iteration, all the entries to the left of the centre table (R) are the entries of the iteration (or table if it's simpler) before.

RRLR

And the entries to the right of the table are the opposite of the ones on the left.

RRLL

So table three actually looks like this:

RRLRRLL

You see? It's kind of like opposite reflections. Sort of like symmetry, but cooler.

The rule is, the last entry into the right hand side of the table should be the opposite of the first entry on the left.

So if you look at your folded paper, and you make your fourth fold, and then you fold it out, it should look like this:



To tabulate that you need to start with your centre 'right' R.

And then to the left of that all the code you have already established, the whole of the previous third iteration:

RRLRRLL

RRLRRLLR

And to the right of it, the whole of the previous iteration but reversed RRLLRLL from centre outward.

Remember, each letter is the opposite of the one on the left of the centre R.

RRRLLRLL

So the final tabulated iteration is RRLRRLLRRRLLRLL  



Isn't it beautiful?



Three rules for this fractal:

1/Start with a Right in the centre.
2/All the iterations to the left of the centre are the same as the table before.
3/All the iterations to the right of the centre are the opposite as the corresponding ones on the left.


*****


You might think this is dull, but for me it's just lovely.


So, Ha. Go forth and multiply!