| [12:00] System: Chruser
has joined the chat |
| [12:00] Chruser:
I think my favorite "unexpected" result is that random walks are recurrent in one and two dimensions, but not in three or more dimensions. |
| [12:04] Chruser:
In other words, if you either move one step to the left or one step to the right with equal (positive) probability, you'll end up back at the starting point after a finite number of steps. Similarly, if you move either up, down, left, or right each step (also with equal probability), you'll end up at the starting point. You can't generalize the result if you also account for an up and down direction (or for higher dimensions). |
| [12:10] Chruser:
It has some fun counterintuitive consequences. For instance, if you flip a symmetrical coin, and add 1 to your score for each heads, and subtract 1 for each tails, you're guaranteed to end up with your starting score of 0 again after enough flips. |
| [12:11] Chruser:
If, however, you keep track of three separate scores A, B, and C, and you roll a symmetric D6, with each outcome either adding or subtracting 1 to A, B, or C (in the 6 possible ways, not accounting for exactly which number does what), you're not guaranteed to end up with a score of <A,B,C> = <0,0,0> ever again after the first throw. |
| [13:32] System: Chruser
has left the chat |
| [16:51] System: Chruser
has joined the chat |
| [16:51] Chruser:
(I haven't tried it, but you could probably find some fractional Hausdorff dimension between 2 and 3 when this symmetry breaks down. It'd probably make a neat mathematical constant.) |
| [16:53] Chruser:
(I'd probably call it the divergence number, in honor of Steins;Gate. :P) |
| [17:38] System: Chruser
has left the chat |
| [22:39] System: WetWired
has joined the chat |
| [22:39] WetWired:
Just because it can't be proven doesn't mean it's false... |
