So, this infinite spirograph drawing would occur for any irrational number x in e^(x*theta), not just pi. And although it's cool and pretty, this complex graph is just showing that pi can't be represented as a ratio of two numbers. Otherwise, the lines would join up at some point.
Interestingly, asking why it "just" misses is misleading, we should be asking why it gets close in the first place. And this is because of the continued fraction of pi! 22/7 is a very good approximation of pi, thus after 22 spins it almosts repeats. The second time is after 333 spins, since 333/106 is an even better approximation of pi!
The reason it even seems to get close to its original trajectory is because of fractional approximations. After 22 spins around its axis, it gets very close because 22/7 is a good approximation of pi. The second time is after 333 spins, as 333/106 is an even better approximation of pi. It will continue to have this visual effect based on the partial continued fractions of pi. See more at https://mathworld.wolfram.com/PiApproximations.html
There are many mathematical concepts that took me a while to understand and conceptualize. I still recall how hard a time I had with logarithmic equation. But once I was able to, I could see how well the foundational concept fits with things in life.
Now you understand why this upset the Greeks so much when they figured it out. They really felt like it was an attack on their beliefs in a rational universe.
Now realize that it's going to keep doing that... forever. Even if you extended the animation eternally, it would just slightly miss every single time. Even when it looks like the whole area is filled in solid, you just zoom in further and there will be a tiny gap so it can keep on missing. Always.
For some reason, I remember someone saying that Pi does repeat, but it takes an astronomical amount of decimal places to get a repeat. It was about 15 years ago? Unless it was a thought experiment, clearly it was wrong, and I remember thinking the person was an idiot but it was someone in the mathematics world, wellish known. That or dementia is coming for me super early.
It depends on what you mean by a repeat, but it never fully repeats itself. Any particular sequence, such as 1234, will probably show up an infinite number of times, but not regularly like "exactly every one zillion digits the sequence 314159 shows up".
Oh, a separate possibility is that you might have heard about how pi has not been proven to be "normal", which would mean that each digit shows up roughly evenly. It's hypothetically possible that someone could someday prove that pi has a finite number of a particular digit. If that were the case, then after an astronomical amount of decimal places, a digit, perhaps 7, would just never show up again.
This video does a great job illustrating it but not explaining it. A rational number can be expressed by dividing two whole numbers (1/2, 3/4, 245/1036).
The video doesn't explain that it you repeated this process with a rational number, you'd get a repeating pattern, like using a Spirograph. Since pi is not rational, it never quiiiiiiite connects back to where it started.
what property would π or some other constant have to have for z(θ) to be a space-filling curve? my understanding of the infinite is shaky at best, but without any particular evidence, I would've thought π being irrational would cause z(θ) to create a disc as θ -> inf
I believe your understanding is correct. By 'the end of pi', I think they were pointing out that is only true for pi and not any practical realization of pi (which are always rational). That is, you can show the process will work, but you cannot actually do the process, even given an infinite amount of time.
Math person here, I thought about it a little and I think e^(a theta i) + e^(b theta i) would be space filling if a/b is irrational. I figure this to be the case because the period of e^(a theta i) is 2 pi / a, and if the ratio between their periods a/b is not rational then the whole function is non-periodic: causing it to always trace a new path in the area.
Note that its image will still have measure 0, so it will cover essentially none of the area, unlike what @LurkingSarcasm said
Any rational number would have a finite period, i.e. after a finite number of spins it would repeat exactly. Thus, irrationality is required (and in fact sufficient) to make a curve "dense" in the disk, in the sense that I define in my next comment. Talking about space-filling curves gets more complicated. What does it mean to take the limit of the curve as θ goes to infinity? 1/2
At least the following statement is true: If the constant is irrational, then for any point x in the disk and any epsilon, there exists θ s.t. z(θ) with distance
There's a positive correlation between my birth and continued existence, and the increase in gang activity. I strongly discourage you to think about it.
pdp1
Here is an explanation of it, it's a few posts down from the top https://www.reddit.com/r/oddlysatisfying/comments/17dif1m/visualization_of_pi_being_irrational/
SiameseDream
So, this infinite spirograph drawing would occur for any irrational number x in e^(x*theta), not just pi. And although it's cool and pretty, this complex graph is just showing that pi can't be represented as a ratio of two numbers. Otherwise, the lines would join up at some point.
OhIfIMust
seannikiforuk
'Think about it' 'We will' 'No you wont'
kaoticone
Nature's spirograph
ErniesWidow
I had one of those back in the '70's.
GreenNinja22
ccpreCookieGamer
Beautyful. Analysis rocks.
EccentricNut
lostlittletimeonthis
Would you believe 3.14?
coffee321123
ElbowDeepInAMoose
Repeatedly
WebDragonG3
awesome use of that clip.
evildadunit
repeatedly
Youwouldntstealatoaster
Interestingly, asking why it "just" misses is misleading, we should be asking why it gets close in the first place. And this is because of the continued fraction of pi!
22/7 is a very good approximation of pi, thus after 22 spins it almosts repeats. The second time is after 333 spins, since 333/106 is an even better approximation of pi!
ShadyEsperanto
as a non-math person, great explanation. in appreciation, I won't steal your toaster
OhIfIMust
Denvercoder09
"I will"
OhIfIMust
No, you won't...
GasBandit
This is worse than that bouncing DVD shit
lonelyrangerofthedreams
I think I need to throw up
okkilby
Irrational can be pretty
baecaughtmewhackin
And vice versa. Learned that a number of times in my life.
Youwouldntstealatoaster
The reason it even seems to get close to its original trajectory is because of fractional approximations. After 22 spins around its axis, it gets very close because 22/7 is a good approximation of pi. The second time is after 333 spins, as 333/106 is an even better approximation of pi. It will continue to have this visual effect based on the partial continued fractions of pi. See more at https://mathworld.wolfram.com/PiApproximations.html
BlackElkSpeaks
ButtsAreBest
If anybody else wants it, the music is Can You Hear The Music by Ludwig Göransson from the Oppenheimer soundtrack.
chewmaca2
Fritzy19
KurtopiaCreative
I-is there a Fibonacci sequence in there?
username404error
my 1st thought too. i came in comments to seek confirmation.
IrrelevantOutsideOfMyBubble
Oh, probably. The Fibonacci sequence, e, and pi naturally show up all over the place.
KurtopiaCreative
Mind. Blown.
YeastInfectedWhiskerBiscuit
It's not Pi, it's plad!
AZRAELSBLADE
They’ve gone to plaid!
chewmaca2
zoeytg
There are many mathematical concepts that took me a while to understand and conceptualize. I still recall how hard a time I had with logarithmic equation. But once I was able to, I could see how well the foundational concept fits with things in life.
iusedtodream
My entire body clenched when it just slightly missed. Now I'm irrational!
dragoonwraith
Now you understand why this upset the Greeks so much when they figured it out. They really felt like it was an attack on their beliefs in a rational universe.
arpsalott
I unlocked a new groan
EchoPMIM
Now realize that it's going to keep doing that... forever. Even if you extended the animation eternally, it would just slightly miss every single time. Even when it looks like the whole area is filled in solid, you just zoom in further and there will be a tiny gap so it can keep on missing. Always.
Tjitso
Oh my sweet OCD
iusedtodream
ahhhhhhhhhhhhhhhhhhhh!
AllisAching
Hey irrational, I'm Dad
moodystudios
Only my pi hole clenched
podkayne13
Must rectify!
somethingnotyettaken
Then relaxed again after a few when it started looking cool. Then it got to the end and I went "ah shit, here we go again..."
BananaForScaIe
I whispered "Oh noooo" like a certain cartoon character lol. Never done that in my life.
Uhhjustaskingforafriend
Just want to point out this has just as much to do with e being irrational as pi.
digzol
Just use the fill tool.
Turtlegir
For some reason, I remember someone saying that Pi does repeat, but it takes an astronomical amount of decimal places to get a repeat. It was about 15 years ago? Unless it was a thought experiment, clearly it was wrong, and I remember thinking the person was an idiot but it was someone in the mathematics world, wellish known. That or dementia is coming for me super early.
IrrelevantOutsideOfMyBubble
It depends on what you mean by a repeat, but it never fully repeats itself. Any particular sequence, such as 1234, will probably show up an infinite number of times, but not regularly like "exactly every one zillion digits the sequence 314159 shows up".
IrrelevantOutsideOfMyBubble
Oh, a separate possibility is that you might have heard about how pi has not been proven to be "normal", which would mean that each digit shows up roughly evenly. It's hypothetically possible that someone could someday prove that pi has a finite number of a particular digit. If that were the case, then after an astronomical amount of decimal places, a digit, perhaps 7, would just never show up again.
AyatollahBahloni
As an innumerate, my hat is off to people who "get" math.
allihearisnoise
Like what make a number irrational?
JeremyGabbard
This video does a great job illustrating it but not explaining it. A rational number can be expressed by dividing two whole numbers (1/2, 3/4, 245/1036).
The video doesn't explain that it you repeated this process with a rational number, you'd get a repeating pattern, like using a Spirograph. Since pi is not rational, it never quiiiiiiite connects back to where it started.
spunkydunks
QuartzPoker
I have a hypothesis that the last digit of Pi is going to be a 7.
dwolvin
Has anyone tried telling Pi to calm down?
NotTinyPancakes
Ah I say that to my mum when I want to regret living
dwolvin
Never in the history of ever have the words "calm down" had the intended effect...
ihugpeople
Yes, engineers.
Engineer: Pi is about 3.
LurkingSarcasm
But it will cover essentially the entire area which is also interesting
FiftyShadesOfBroccoli
Plus it draws a circle, and circles are what pi is all about in my very limited understanding of math
Woogyface
not with an infinite thin line unless someone find the end of pi
FeltNokia
what property would π or some other constant have to have for z(θ) to be a space-filling curve? my understanding of the infinite is shaky at best, but without any particular evidence, I would've thought π being irrational would cause z(θ) to create a disc as θ -> inf
DavidBrooker
I believe your understanding is correct. By 'the end of pi', I think they were pointing out that is only true for pi and not any practical realization of pi (which are always rational). That is, you can show the process will work, but you cannot actually do the process, even given an infinite amount of time.
spuddastardly
Math person here, I thought about it a little and I think e^(a theta i) + e^(b theta i) would be space filling if a/b is irrational. I figure this to be the case because the period of e^(a theta i) is 2 pi / a, and if the ratio between their periods a/b is not rational then the whole function is non-periodic: causing it to always trace a new path in the area.
Note that its image will still have measure 0, so it will cover essentially none of the area, unlike what @LurkingSarcasm said
Youwouldntstealatoaster
Any rational number would have a finite period, i.e. after a finite number of spins it would repeat exactly. Thus, irrationality is required (and in fact sufficient) to make a curve "dense" in the disk, in the sense that I define in my next comment. Talking about space-filling curves gets more complicated. What does it mean to take the limit of the curve as θ goes to infinity? 1/2
Youwouldntstealatoaster
At least the following statement is true: If the constant is irrational, then for any point x in the disk and any epsilon, there exists θ s.t. z(θ) with distance
aoshistark
Did you know there is a direct correlation between the decline of spirograph and the increase in gang activity. Think about it!
robertdoobies
Spirographs are directly linked back to the Venus Flytrap Theory of Atoms
cloverleafbane
No you won’t.
NopeForever
So, gangs discourage spirographs?
IAlwaysUpvoteSunsets
I came to the comments looking for precisely this quote 😁
vindik8or
There's a positive correlation between my birth and continued existence, and the increase in gang activity. I strongly discourage you to think about it.
bluebombers1411337
I will
aoshistark
No you won't...
3Davideo
Thought they both peaked in the 90s...
ProbablyWrong524
No you won't
testzero
I will.
FunkyKick
CommodusLeitdorf
No you wont.
CaptainDogWasTheBestDog
Is it making a sphere??
MrListerTheFirst
Eventually, yes. You should wait a while. No, longer.
KilroyLichking
szpet627
KilroyLichking
this whole series is great. the physics one after this is amaizing
Movietimeme
Do you have a link? I would love to know more.
szpet627
https://www.youtube.com/watch?v=B1J6Ou4q8vE
Movietimeme
PalaverQuader
mm some sauce would be nice
szpet627
Animation vs math
PalaverQuader
thx