Monday, March 26, 2007

Random

Walk with me for a couple of minutes.

Take the famous Fibonacci sequence which is defined as "after two starting values, each number is the sum of the two preceding numbers." As in 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...etc. As in the first two numbers 0+1 = 1, the next two numbers 1+1= 2, the next two numbers 1+2 = 3...etc. Got it?

Now, there's a whole lot going on here that has to do with the Golden Mean, how the sequence shows up in nature, rumors of its use in art, architecture and proportions of the human body among many other things, but we're not going to go there right now. Google it and/or look it up on Wikipedia later.

Instead, take that same Fibonacci sequence and introduce the element of randomness. As in flipping a coin (heads = adding the two preceding numbers together, tails = subtracting the two preceding numbers). My understanding is it has been taught among math students that in coin flipping 50% of the time it will land on heads and 50% of the time it will land on tails (this 50/50 rule has been disproved, apparently, but I'm not a math person so I don't know this for certain). Soooo we can say the probability is 1 (half the time it lands on heads/half the time it lands on tails .5+.5 = 1).

Here's an example from Science News Online of a randomized Fibonacci sequence: "[S]uccessive coin tosses H H T T T H, for example would generate the sequence 1, 1, 2, 3, -1, 4, -5, -1." If you remove the minus signs (thereby taking the absolute value of the terms), you will find that "the exponential rate at which the average value of a random Fibonacci sequence increases" is a constant which has the approximate value of 1.13198824.

1.13198824 is known as Viswanath's Constant.

Now I know almost nothing about math. I failed pre-algebra twice, I can barely add, subtract, multiply and divide and I have a severe case of mathphobia, BUT even I can be in awe of the beauty of Viswanath's constant.

What does Viswanath's Constant mean to me? It means order within chaos. It means below what appears to be completely random activity is a lovely, simple progression.

Beautiful.

P.S. Since I am not a math person, feel free to comment with corrections to the text and/or my explanation.

1 comment:

so and so said...

It's me again. I'm a non-math person and share your fascination with these equations. (I miraculously had my math requirements waived to get into a UC school.) If I could do it all over again and not quit as soon as I got frustrated, I could really get into math. It's not just another language, it's another way of thinking. I'm reading "Hyperspace" by Michio Kaku and it's blowing my mind. There's not a lot of math in it, but the author is a theoretical physicist and makes me want to get into that field (yeah right) and I don't understand a lot of it right away, but I can kind of intuit the basic concepts if I re-read a lot of it slowly. Hypercubes and 4-D geometry and the like...mindblowing.