Fractals in 10 minutes no. 3 – The Dragon
When first I looked through the pages of the book "Hacker's Delight", I found myself looking at the chapter about bases. There I learned a very curious fact - with the digits of 0,1 and the base of -2, you can represent any integer. Right afterwards I learned something even more interesting - with the digits of 0,1 and the base of 1-i, you can represent and number of the form a+bi where a and b are integers. Having nothing to do with this curious fact, I let the subject go.
Some time later, I was reading through Knuth's "Art of Computer Programming", and found that with the base of (1-i)^-1, and digits of 0,1 you can generate the dragon fractal!
Generating the fractal is quite simple actually:
"""calculate the dragon set, according to Knuth"""
s = set([0.0+0.0j])
for i in range(n):
new_power = (1.0-1.0j)**(-i)
s |= set(x+new_power for x in s)
return s
(By the way, can you do it better?)
The annoying part is converting the complex numbers to drawable integer points. After doing so, I used PIL to draw the jpeg.
Here's a link to the code.
Tags: base systems, dragon fractal, fractal, Knuth, Python
January 24th, 2008 at 6:28 am
actually there are Dragon curves - very interesting fractals
simplest way to get them is folding a paper string by half few times
curious fact is that 4 Dragon curves can completely fill 2d space without any interference
i was playing with .NET some time ago and have a sample program which generates Dragon curve for few steps ahead with specific folding direction - but it's horribly slow because i was using strings to represent fractal :)
January 24th, 2008 at 7:50 am
Well, because of your comment I also looked at the dragon curve page on wikipedia. It seems that:
1. Dragon curves may be produced by lsystems (lstrings). On a side note, a version of the book 'Jurassic Park' had another level of the dragon curve on the openning page of each chapter. It was drawn by using lsystems. I remember that as a kid reading this book, I really liked it (and tried to write a program to draw it).
2. As you said, about folding paper - well, that really makes me very happy to know. I never made this connection between fractals and folding. Excellent!
3. It seems what I actually drew in this post is the twin-dragon, which is also named the Davis-Knuth dragon. It is constructed from two dragon curves.
Thanks!
July 5th, 2008 at 5:05 pm
[...] pe net am găsit și un algoritm descris de Knuth în „Arta programării calculatoarelor”, iar individul al cărui blog îl indic descrie [...]