The Math Kid

dagothar:

The Jurassic Park fractal

If you read The Jurassic Park by M. Crichton (and I’m sure you did), you must have noticed how each of the chapters started with the page featuring an interesting drawing. It started with a simple set of lines, and as you progressed in the book, it got more, and more convoluted. I couldn’t figure out the rule behind it, but it fascinated me, and years later I found the explanation.

It’s a fractal, and is also called Heighway dragon, the Jurassic Park dragon, or simply a dragon curve. It’s a nice thing to doodle. There are a few ways you can do this and the choice is yours:

• the paper strip folding method:

Take a strip of paper. Start folding it: right over left, and keep repeating: right over left, right over left… You won’t be able to fold it more than 4-5 times. Now, unfold it and crease all the folds so they all are at 90 degrees angle. Lay it on a flat surface, and voila! You can make more of these and then connect them end to end to get more complicated shapes (more iterations of the fractal).

• the string (or the pen-and-paper) method:

You can pretty well just draw the dragon on the paper, starting with a straight line segment and making proper right or left right-angle turns. A nice method comes for telling the sequence of turns:

1st iteration: R (R is right turn),

2nd iteration: R_R_L (L is left turn),

3rd: RRL_R_RLL,

4th: RRLRRLL_R_RRLLRLL,

and so on… As you can see, in each of the iterations you just take the previous one, add R at the end, and then add the previous iteration again, but this time reversing it, and swapping R’s and L’s.

• the (not quite) mathematical method:

You can calculate the n-th turn direction by executing the following algorithm:

if ((n & -n) « 1) & n = 0 then: turn <- R; else: turn <- L;

& is a bitwise and operator, « is left shift and you have to remember the -n is encoded as two-complement. Also, the n-th full iteration is a sequence is a sequence of 2n-1 turns, so calculate all of these preferably! :)

For example the direction of 42nd turn is:

((00101010 & 11010110) « 1) & 00101010 =

00000100 & 00101010 = 0, so the 42nd turn is R.

Sources:

http://en.wikipedia.org/wiki/Dragon_curve

http://math.rice.edu/~lanius/frac/jurra.html

mirandamolina:

Variations of “Pythagoras tree” in space, using Structure Synth.
Check out HD versions in Youtube

Variaciones del “Árbol Pitagórico” en el espacio, usando Structure Synth.
Ver versiones HD en Youtube

(vía Geometría Dinámica » Fractales en movimiento)