A game of chaos

Written by Student C as part of Math 493: Mathematics Through 3D Printing

Let's play a game. A math game. A game... of chaos.

Heres how to play. Start out half-way (1/2) on our favorite interval [0,1]. Now flip a coin, heads move half way to 1 (¾), if tails move halfway to 0 (¼).

So now there were two outcomes- you are now either at ¾ or ¼. In either case, flip a coin. Lets say you are at 1/4- Heads, move half-way to 1 (5/8). Tails, move half-way to 0 (1/8). Now, if you were at 3/4 – Heads, move half-way to 1 (7/8). Tails, move half-way to 0 (3/8).

Do this a few more times to create your picture.

Now comes the time for reflection- if I say that you can do this as many times as you want, will you ever reach 0 or 1? What about ½? What about any of the points you received? That is, for every point that you've created, do you think that you will ever be able to get that number back by continuing to play this game? Remarkably, even if you play this game 100 times, 10,000 times, 100,000,000 times or even an infinite number of times, you will never be able to get back to any numbers that you've already created.

In this manner, we have discovered a set of numbers that have what is known as “measure 0”. That is, even though after doing this game an infinite number of times and generating a infinite number of numbers, the 'distance' to any two numbers is 0. That is, you cant play the game enough times to get back to any original number.

Now lets try extending this game. Let play this in 3 dimensions. Instead of looking at the two points on the number-line, lets look at the eight points on the unit cube. This game has to be slightly altered to play properly. First, number the points of the cube. Then, instead of flipping a two-sided coin, flip an 8 sided coin. When you flip the corresponding point, move half-way there. Again, do this as many times as you would like. Eventually, you will generate the Menger sponge. Several iterations of this can be viewed below:

Returning to this concept of measure 0, how does this apply to the Menger Sponge? Doing this iterative process an infinite number of times generates some interesting results. Namely, the volume of the entire object is zero. But perhaps, the more interesting notion is that the surface area is infinite. This can application can be “explored” by moving around “inside” the sponge, only to come to the conclusion that there is no way to ever find purchase of the object. In this way we can never reclaim any of the numbers we've generated. That is, measure 0 in 3 dimensions.