Thursday, September 28, 2006

Three Faces of Pi

Fractals aren’t just shapes; fractals can be numbers as well. A fractal sequence is one that, in some sense, contains infinite copies of itself. One example of a fractal sequence is the signature of an irrational number (a number that can’t be expressed as a fraction, like π or the square root of 2). To create a fractal signature sequence for a number x, create values of y, where y = i + jx and i and j are both integers. Create y values for i and j from 1 to infinity and order them from smallest to largest. If x is irrational, then every y value will be different and the sequence of the i’s and the sequence of the j’s are both fractal sequences.

Infinity is really big, so instead, let i and j both vary from 0 to 32. Then, draw a line from (0, 0) to (32, 32), going through each point in order of the y values. Different irrational numbers will create different graphs. Even though each graph is one continuous line, the angles of the zigs and zags change, giving the appearance of rectangular blocks of different shades of gray.

The above image is composed of three separate graphs, aligned so that the entire image is one continuous line. The graphs represent three different irrational numbers which are based on π: π/2 (~ 1.5708), the natural logarithm of π (~1.145), and the square root of π (~1.772).

2 Comments:

Blogger Tim said...

Nice explanation. I'm not a math person and I've always been intrigued how something as complex and unusual as a fractal image can sprout from a few numbers.

I like the simplicity here. I looked at the large version of the image and I can actually see the start of the line and the end. It's like connect the dots, which is something I understand : )

I wonder if understanding the mathematical architecture of fractals (which I don't, really) takes away from the magic and excitement of them or adds to it?

9/29/2006 9:17 AM

 
Blogger Kerry Mitchell said...

When I taught Physics, I had a student tell me that I ruined the romance of rainbows for him by explaining how they work. :-)

I feel that understanding the math of fractals definitely adds to their excitment (although it's not magic any more)--it's like having some sense of music adds to the appreciation of opera, or knowing how to cook adds to the appreciation of good food. One thing it does do is raise one's standards--if I see a spiral with a black hole in the middle because there weren't enough iterations used, I'm generally less than pleased.

In this case, knowing something about math allows me to go places others don't go--I've been fortunate to have created types of images that I've never seen others do before.

9/30/2006 2:09 PM

 

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