[HN Gopher] Footsteps of pi
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Footsteps of pi
Author : codyd51
Score : 45 points
Date : 2023-11-14 11:57 UTC (11 hours ago)
(HTM) web link (axleos.com)
(TXT) w3m dump (axleos.com)
| AdamCraven wrote:
| 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 wrote:
| 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.
| contravariant wrote:
| 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 wrote:
| Phi/golden ratio also has a cool continued fraction
| sequence...it's only 1's all the way down
| contravariant wrote:
| 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.
| solarist wrote:
| One example is Ulam spiral:
| https://en.wikipedia.org/wiki/Ulam_spiral
| boznz wrote:
| maybe do it in another base rather than base 10. Just because
| we have 10 fingers does not mean a god does.
| tromp wrote:
| 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 wrote:
| for binary, introduce a constant down step, and then let the
| line run back and forth in that space.
| phyllistine wrote:
| 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 wrote:
| 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 p is a
| normal number:
|
| https://en.wikipedia.org/wiki/Normal_number
| marginalia_nu wrote:
| > 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 wrote:
| 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 wrote:
| Is this not just generating a random walk from a pseudorandom
| number generator.
| vouaobrasil wrote:
| 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 wrote:
| See also "p 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
| doubloon wrote:
| 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
| dllthomas wrote:
| I would think the squares are a line, not a dot?
| dllthomas wrote:
| > 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 wrote:
| 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.
| dllthomas wrote:
| > 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 wrote:
| > 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 wrote:
| There's no structure here- it's a random walk.
| nh23423fefe wrote:
| a random walk over the 10th roots of unity
| hoyd wrote:
| 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.
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