You can modify the continued fraction slightly to make pi regular as well, but the normal continued fraction sequence doesn't give much of an insight. Other than the fact that 3 + 1/(7 + 1/16)) is a damn good approximation (7 digits, pretty good for something that can be written using only 4 digits total: [3;7,16]).
I thought that colouring the pattern by the instantaneous velocity of the ball would be an obvious improvement and might uncover further structure.
maybe there could be a database, online encyclopedia of random-walks
Rather, rational numbers awfully close (in ordinary human terms) to specific, well known irrational numbers. There are, I think, just as many irrational numbers comparably close to any rational number.
It's math. There's no such thing as "too pedantic", as long as you're being interesting and not mean about it.
> I think there are uncountably more irrational numbers close to any rational number than there are rational numbers close to an irrational number.
I think that's right.
Irrationals near a rational are almost certainly uncountable, as otherwise I think we can force all the irrationals to be countable by bucketing them. I think that concern is countered if any bucket has to be uncountable, but if it's not all that makes some rationals special in a way they probably aren't.
Rationals near an irrational is definitely countable, as all the rationals is countable.
Another choice is whether to use absolute directions, or relative to the current direction, as in Logo.
Dating gurus hate him for this one weird trick.
