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Footsteps of pi
- AdamCraven 3y agoI 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 3y agoThis 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.
- contravariant 3y agoWell 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 3y agoPhi/golden ratio also has a cool continued fraction sequence...it's only 1's all the way down
- contravariant 3y agoSquare 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.
- thechao 3y agoLarger 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!
- solarist 3y agoOne example is Ulam spiral: https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral
- curtisblaine 3y agoI guess the Ulam spiral is more an artefact of the cartesian plane than an "hidden pattern": https://www.youtube.com/watch?v=EK32jo7i5LQ https://www.youtube.com/watch?v=EK32jo7i5LQ
- boznz 3y agomaybe do it in another base rather than base 10. Just because we have 10 fingers does not mean a god does.
- tromp 3y agoWould 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 3y agofor binary, introduce a constant down step, and then let the line run back and forth in that space.
- phyllistine 3y agoI 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 3y agoThe 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
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- marginalia_nu 3y 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.
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- TheMagicHorsey 3y agoI 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 3y agoIs this not just generating a random walk from a pseudorandom number generator.
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- vouaobrasil 3y agoIt 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 3y agoSee 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 https://www.youtube.com/watch?v=pegjULYJae4
- doubloon 3y agoAlso 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
- dllthomas 3y agoI would think the squares are a line, not a dot?
- dllthomas 3y 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 3y agoIf 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.
- dllthomas 3y 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 3y 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_cavalier 3y agoThere's no structure here- it's a random walk.
- thechao 3y agoMy question is which random walk!?
- fsckboy 3y agopi is normal, similar to but not random.
- jovial_cavalier 3y agoSay more. I'm not sure I understand. Closest I found was https://en.wikipedia.org/wiki/Normal_number 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-of-the-digits-of-pi https://math.stackexchange.com/questions/51829/distribution-... seems to indicate that pi is not proven to be normal
- fsckboy 3y agodon't count on it! if it uncovered a structure to pi, that would be really big news
- nh23423fefe 3y agoa random walk over the 10th roots of unity
- bluffingmarconi 3y ago[dead]
- hoyd 3y agoFunny, I did a similar exercise in 2019, https://earth.hoyd.net/visualizing-100k-decimals-of-pi-and-tau-2913/ 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.