Footsteps of pi(axleos.com)
axleos.com
Footsteps of pi
https://axleos.com/footsteps-of-pi/
37 comments
Funny, I did a similar exercise in 2019, https://earth.hoyd.net/visualizing-100k-decimals-of-pi-and-t... loved how it turned out. Even planned to make it into an art piece on the wall.
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]).
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]).
Phi/golden ratio also has a cool continued fraction sequence...it's only 1's all the way down
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!
Square roots in general have periodic patterns. Which isn't too surprising, something like z = a/(b+cz) is pretty much a quadratic equation after all.
But phi is indeed especially interesting because of what its sequence implies for rational approximations of phi.
But phi is indeed especially interesting because of what its sequence implies for rational approximations of phi.
One example is Ulam spiral: https://en.wikipedia.org/wiki/Ulam_spiral
I guess the Ulam spiral is more an artefact of the cartesian plane than an "hidden pattern": https://www.youtube.com/watch?v=EK32jo7i5LQ
maybe do it in another base rather than base 10. Just because we have 10 fingers does not mean a god does.
> The colors are arbitrary, and have no deeper meaning
I thought that colouring the pattern by the instantaneous velocity of the ball would be an obvious improvement and might uncover further structure.
I thought that colouring the pattern by the instantaneous velocity of the ball would be an obvious improvement and might uncover further structure.
There's no structure here- it's a random walk.
My question is which random walk!?
pi is normal, similar to but not random.
Say more. I'm not sure I understand.
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.
Although, this SE thread: https://math.stackexchange.com/questions/51829/distribution-... seems to indicate that pi is not proven to be normal
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.
Although, this SE thread: https://math.stackexchange.com/questions/51829/distribution-... seems to indicate that pi is not proven to be normal
don't count on it! if it uncovered a structure to pi, that would be really big news
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.
Another choice is whether to use absolute directions, or relative to the current direction, as in Logo.
for binary, introduce a constant down step, and then let the line run back and forth in that space.
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
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.
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.
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.
I saw one that was similar except that it used triplets of pi digits (3.xyzxyzxyz..) and drew line segments with direction (x,y) and magnitude z.
I wonder if the path stays consistent in other (maybe very high) number bases, or if that general path is random and unique to base-10
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:
https://en.wikipedia.org/wiki/Normal_number
https://en.wikipedia.org/wiki/Normal_number
Is this not just generating a random walk from a pseudorandom number generator.
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.
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a random walk over the 10th roots of unity
> I originally created this image in early 2020 to impress and woo my now-girlfriend, who I adore.
Dating gurus hate him for this one weird trick.
Dating gurus hate him for this one weird trick.
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https://www.youtube.com/watch?v=pegjULYJae4