I always wondered if some hidden pattern would be exposed when visualising numbers in unconventional ways in numbers with no known pattern such as Pi or prime numbers. A sort of multi-dimensional rendering that suddenly reveals a hidden pattern.
This is sort of how Fermat's last theorem was solved by Andrew Wiles (forgive me if I misrepresent this proof, my math is a few years rusty) - by creating a different kind of representation of elliptic curves, it was possible to compare them to modular forms in a way that created a contradiction that proved the theorem correct.
Well some numbers expose patterns when written as a continued fraction. In particular e becomes pretty regular.
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]).
Larger integers in continued fractions mean you get 'more information' out of the limb. That means not only is Phi "1s all the way down" it is the continued fraction that converges the slowest. If you've ever used the iterated matrix product (which is a specific edge-case of the algorithm to convert continued fractions to decimals), you'll know how slow it is!
Closest I found was https://en.wikipedia.org/wiki/Normal_number, but this seems to mean that no matter what base you choose the digits are uniformly distributed. Meaning, yes, it's random.
Would be nice to make not just a decimal direction picture, but one for other bases as well. Binary won't work as you just move back and forth along a single line, but ternary should work, and as the minimal base for 2D directions, is less arbitrary than decimal. Then I'd look at bases 4,5,6,7, and octal as well to see whether the picture depends more on the number or on the base.
Another choice is whether to use absolute directions, or relative to the current direction, as in Logo.
Also If you do this for the Square Roots of the integers you can see every integer root is special and has it's own kind of shape. And the Squares are also very interesting in that they have no shape in this viewpoint. Just a dot. So you go from infinitesimal chaotic walk patterns to a single dot depending on if the integer is a square or not.
maybe there could be a database, online encyclopedia of random-walks
> Here are some more irrational numbers expressed in this way
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.
If we want to open the floodgates on being too pedantic, I think there are uncountably more irrational numbers close to any rational number than there are rational numbers close to an irrational number. But in both cases, it's definitely a bunch.
> If we want to open the floodgates on being too pedantic
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.
The path will look entirely different depending on the base you choose, but all paths for all bases should look roughly equally "random" because it's widely believed that π is a normal number:
It is not a traditional uniform random walk because the next movement is based on the previous movement, but it is a random walk. It is a pseudo-random number generator of sorts, and has properties very close to a uniform pseudo-random number generator because PI is likely normal.
https://www.youtube.com/watch?v=pegjULYJae4