Footsteps of pi

by codyd51 • 3 years ago

62 33 3 years ago

AdamCraven• 3 years ago

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.

JohnMakin• 3 years ago

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.

contravariant2• 3 years ago

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]).

azeemba• 3 years ago

Phi/golden ratio also has a cool continued fraction sequence...it's only 1's all the way down

solarist• 3 years ago

One example is Ulam spiral: https://en.wikipedia.org/wiki/Ulam_spiral

curtisblaine• 3 years ago

I guess the Ulam spiral is more an artefact of the cartesian plane than an "hidden pattern": https://www.youtube.com/watch?v=EK32jo7i5LQ

boznz• 3 years ago

maybe do it in another base rather than base 10. Just because we have 10 fingers does not mean a god does.

tromp• 3 years ago

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.

knome• 3 years ago

for binary, introduce a constant down step, and then let the line run back and forth in that space.

phyllistine• 3 years ago

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

munificent• 3 years ago

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

marginalia_nu• 3 years ago

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.

TheMagicHorsey• 3 years ago

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.

empath-nirvana• 3 years ago

Is this not just generating a random walk from a pseudorandom number generator.

vouaobrasil• 3 years ago

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.

rjeli• 3 years ago

See also "π plays Pokemon Sapphire", currently at 372 24-hour segments, and still stuck in the starter town (with a level 76 Sceptile):

https://www.youtube.com/watch?v=pegjULYJae4

doubloon2• 3 years ago

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

dllthomas4• 3 years ago

I would think the squares are a line, not a dot?

doubloon2• 3 years ago

well depends on how you interpret, could take '0' to mean 'stay where you are' or you could mean it 'go west' or whatever. but yes.

dllthomas4• 3 years ago

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.

tashi• 3 years ago

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.

dllthomas4• 3 years ago

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.

deepspace• 3 years ago

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.

jovial_cavalier2• 3 years ago

There's no structure here- it's a random walk.

thechao2• 3 years ago

My question is which random walk!?

fsckboy2• 3 years ago

pi is normal, similar to but not random.

fsckboy2• 3 years ago

don't count on it! if it uncovered a structure to pi, that would be really big news

nh23423fefe• 3 years ago

a random walk over the 10th roots of unity

hoyd• 3 years ago

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.

Footsteps of pi

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