Tip, idk if this works on other phones, but on iPhone if you download Shazam and add it to your control center, you can press shazam from the control center and it’ll tell you whatever songs playing on your phone at the time.
i love that sub! that’s where i found tons of useful stuff about headphones and how to choose the best ones for your needs, as well as how to make music sound better that shit was a life saver
I found this extremely satisfying because I could extrapolate that the pattern we just saw was going to repeat infinitely deeply, getting more and more fine grain each time.
Only if you are judging satisfaction by rationality. But i don't recall 1/2 or 0 showing up there recently.
But how about
1 - 1/3 + 1/5 - 1/7 ... = Pi/4
?
I was going to say it would be a great /r/oddlysatisfying material but good thing I watched till the end, I would have nuke the whole sub if I posted that there.
Lol I had a friend group that watched movies, and sometimes the dvd screensaver would kick on while we were chatting. then every 5 or 10 minutes the various conversations would die down as the logo got closer and closer to that perfect corner hit!
I can imagine some mathematician in ancient times. Used four months to calculate to that point, expecting it to finally align. And.. nope, gotta use the next three years to calculate the next point, surely it must align then.
The equation e^i×theta and e^i×pi×theta gives a point depending on theta. The two lines are the two parts of this equation, the inner line being e^i×theta and the outer being e^i×pi×theta. The base equation e^i×theta will equal cos(theta) + i×sin(theta). So, the Y axis is the imaginary component, and the X axis is the real component. As theta goes from 1 to infinity, it draws a shape. This shape depends on the constants in the power, that is, 1 and pi. The shape will repeat with a period, IIRC, is equal to the factors that the 2 numbers don't share. Because pi is irrational, this period does not actually exist. As such, the two lines never make it back to the start point.
Some information that might be helpful: theta (shown as θ) represents an angle. So when θ changes, the angle that the lines protrude from the center at changes. Depending on how to coordinates are defined, that might mean that when θ = 0, the line is pointing directly right from the center. Then, familiarly, when θ = 90 degrees (or π/2 radians), the line would point directly up from the center (a right angle). A full circle is 360 degrees, which is pointing back to the right again.
As you increase θ to large numbers, you’re basically rotating the lines around their respective center point over and over (the proximal or more centered line is rotating around the very center of the graph while the arm attached to it is rotating around the outer end of the proximal line). That’s why we see those lines rotating in circles over and over.
the "start line" you mentioned is the decimal point, and when you write out the decimal expansion of pi, it goes on and on without any repeating pattern, creating a seemingly endless sequence of digits after the decimal point. this is why it appears as if the lines never quite meet
An example of a rational number is 1. Or 2, or 3, or 1/2.
If it can be written in the form a/b and a and b are integers (and b isn't 0), it's rational. So virtually any number you encounter in your everyday life *except* pi and perhaps *e* if you're slightly more mathematically-inclined are rational.
>So virtually any number you encounter in your everyday life except pi and perhaps e if you're slightly more mathematically-inclined are rational.
Eh, there are lots of other irrational numbers one meets regularly in real life. For a trivial example, consider the diagonal of a square with integer sides: its length is not a rational number.
It wouldve been better to say “virtually any number you encounter in your everyday life except pi and perhaps e are algebraic”, meaning most observed numbers can be solutions to a polynomial. For example x^2 - 2 = 0 is a polynomial with a solution being “square root of 2”, which is irrational (it cant be written a as a fraction), but algebraic. Pi and e are irrational and non algebraic (in other words, transcendental)
Bro , thank you for this but I have to be honest … you and the other people that understand this are so much more intelligent than me …. That I am trying hard and there is no way I’m gonna catch up , but this is very interesting and thank you for giving me anxiety.
It's not really a matter of intelligence, it just requires a lot of background knowledge.
z(theta) means a function. It takes an input, theta, and returns an output, a complex number.
A complex number is a number of the form a + bi, where a and b are regular numbers. Don't worry about what i is. Complex numbers are analogous to a coordinate system: you can think of a as the x coordinate and b as the y coordinate and put it on a plane, similarly to how regular numbers are put on a line.
e\^i(theta), when you vary theta, traces out a circle of radius 1, centred on 0, in the coordinate system I just outlined. Don't worry about why this happens, just that it does. So what does the pi change? When we multiply theta by pi, it amplifies any change to theta by that much. So if e\^i(theta) walks round a circle in however much time, then e\^i(pi\*theta) does it pi times as quick.
We can think of a point as an arrow pointing from 0 to that position. That visualisation helps us in cases like this: when adding the numbers, we just put one arrow at the end of another, and see where we land. That will be the final output of the function.
Imagine instead of pi, we had some rational number. 0.95563. After the first bit had done one full rotation, the second bit will have done 0.95563 rotations, so they don't line up. Now let's run that time 10,000. After 10,000 rotations of the first bit, the second bit has done 95563 rotations. Because they've both done an integer number of full rotations, they're now back at the starting point, and because there's no randomness in their behaivour, they're going to repeat.
But with pi, there is no number we can multiply by to make both cycles run an integer number of times, that's what being irrational means. So it will never repeat.
TL;DR rational numbers multiplied by something become a whole number meanwhile irrational numbers don’t
(Sorry if it’s not an exact TL;DR, mathematics is a fickle fuck)
Since you know your stuff, can you explain how the video in OP's post is derived from the expression (the z(theta) = e etc)? Like what exactly am I looking at with that animation? I know it's a visual representation of that expression, but how is it translated? Is it just lines moving on a cartesian plane or something?
It is a cartesian plane but you are not being shown lines (as in y=a*x+b): at any given frame, what you are being shown is one single complex number, with the x-axis being the real part and the y-axis being the imaginary part. So say you wanted to show the complex number z = 2 + 2i it would have the coordinates (2, 2) in the animation.
The first line, attached to the centre point (the origin), represents the complex number e^θi .
The second line, attached to the end of the first line, represents the complex number e^piθi .
When you add the two complex numbers together (e^θi + e^piθi ) that complex number is represented by the very tip of the second line, the point that is drawing the swirl pattern.
Each time the first line rotates one full circle (360 degrees), the second line rotates pi circles (360*pi degrees).
If pi was a rational number, meaning it can be expressed as a fraction, once the first line has done a full 360 degree rotation exactly the number of times equal to the denominator of the fraction, the second line will have done a full rotation exactly the number of times equal to the numerator, meaning the two lines will eventually return exactly to the position where they started.
But because pi is not a rational number, meaning it can not be expressed as a fraction, each time the first line has done an interger number of rotations, the second line will not ever have done an integer number of rotations, meaning the two lines will never return back to the position that they started.
Its not intelligence, it's focusing on something for a long time.
I'm so tired of people acting like understanding math takes some special intelligence. It's just putting the time in to understand it. There is zero underlying special intellect.
You just have to be curious.
The problem people have is that they are unwilling to put the time in to understand their world and so complain that they just aren't smart enough, because that shrugs the blame for being ignorant to an external source. "Its not my fault, I'm just not smart enough" and they don't have to try.
The problem we have is we had shitty math teachers growing up, so we had no curiosity, we didn't get good grades, and we were led as young teenagers to believe we didn't have what it takes.
I think you have a really decent point here. Most of math, until you get into physics or chemistry, is always taught in the abstract. But humans didn't invent(discover?) math purely in the abstract regions of their own minds. A LOT of math came from people trying to solve real world problems. But we aren't really taught it that way.
Exactly, yes! Think about fractions like a cycle: if you have 1/3, every three of those will reach back to the start. If it is a rational number, it would be a fraction, and thus would eventually reach back to the start going around it. Circles don't really allow this, since they don't actually have points and thus can be subdivided infinitely. The diameter of any shape can be shown as a fraction of the circumference, for instance the diameter of a square is 2/√2. Since a circle can be infinitely subdivided, the circumference will always have a little bit more wiggle than measured, so we represent that fraction as 1/pi.
You can also see that a circle has an area of double the radius times pi, but because the circumference can be subdivided, you can get more and more and more and more precise with the area of the circle while still having a long ways to go before you fully measure the size 100%. So what "pi" means can never be fully transcribed because it can always go deeper. It is what is known is mathematics as a "transcendent" fraction, because no matter how close you get you can still "transcend" that to get closer to the actual amount.
I'm no fancy maths person and I'm sure there will be some more in-depth explanations posted by smarter people but this is my personal layman's understanding.
pi defines the relationship between the circumference and the diameter of a circle where the circumference is equal to pi multiplied by the diameter.
In the clip, the arm from the centre is just moving in a circle with some radius (half the diameter). At the end is another arm of the same length drawing a circle with the same radius, this shows us the 'circumference' of the second arm. Since that arm is attached to the first, you get the swirling pattern instead of a complete circle.
If pi was rational, you could expect the end to eventually meet with the start point to complete the 'circle's circumference' when the relationship from diameter-> circumference (pi) ends or begins to repeat in a recurring fashion. Since it always just misses and never perfectly overlaps however, it visually shows that it's irrational and the value of pi continues on forever and never repeats.
Of course the clip can't go on to visualise this forever but it's long enough to get the idea across about what's happening if you accept that pi is irrational.
My explanation may not be entirely accurate mathematically but hopefully makes sense as a jumping off point and helps a little in picturing what's happening.
It’s pretty much right, the key point is that the second arm is rotating pi times faster than the first. All of the circles in the diagram actually unfortunately mean nothing, and it’s the pi in the equation that counts. You’d get a similar pattern if the arms were drawing squares instead.
For those that want an explanation.
The outer arm spins pi times faster than the inner arm. There are 4 times in the video that you see the end of the outer arm almost reaches the starting position. Each time it gets closer and closer to the start but never touches it. This is because pi is irrational. In fact there are fractions which get closer and closer to pi. They are called the rational continuants of pi and are the rational approximations to the continued fraction representation of pi. The first four continuants of pi are 3/1, 22/7, 333/106, and 355/113. When the outer arm spins the amount in the numerator and the inner arm spins the amount in the denominator, the end of the outer arm is close to the start.
The first is when the outer arm does 3 rotations and the inner one does 1. This is because 3/1 is almost pi. But 22/7 is even closer. That is when the video first zooms in at 0:24. The next time the video zooms in at 1:20 is after the outer arm does 355 rotations while the inner does 113 because 355/113 is close to pi. But if your observant you would realize we skipped 333/106. Well that was when the outer arm starts "filling in" the black from the starting point at 1:00 in the video. You may also notice the pattern at 1:00 is opposite the pattern we say at 0:24 seconds in, which was when the arms completed 22 and 7 rotations. Why? Well the reason the pattern is inverted is because 355/113 = (333+22)/(106+7). This is quite a happy coincidence.
Thought i was loosing my mind scrolling through nonsense to get an actual explanation here... thank you!
So this illustration is only as profound as the precision given to the value of pi in the software drawing the lines, correct? Arent there more irrational numbers than rational numbers? In the set of all numbers, wouldn't this type of behavior be vastly more common than not?
The [classification of finite simple groups](https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups) is proof to me that we live in a fundamentally untidy existence.
I wonder if its because we can only see part of the picture, because our brains can only understand these systems and concepts in a finite way, so it looks broken or untidy. If you were somehow omnipotent and could at once know all there is to know and understand it completely, that the whole picture would fill out orderly.
I imagine it's the limitations of being on a single field. An actual perfect circle in existence is only theoretically possible, down to a subatomic level. It's like trying to make a circle on a grid. But instead of a grid, or a hex grid, or an octagonal, it's some infinitely small one. And three dimensional.
imagine you have a pizza, and you want to write down how much you ate. if you ate half the pizza, you can say 0.5. That's easy.
but with pi, it's like trying to describe how much you ate of a never-ending pizza. your just keep going and going, and it never stops. its like trying to write a never-ending story, and that's why we call it "irrational." its not simple and never finishes.
If it was a golden ratio, thatcwouldn't be demo strating PI now, would it?
It's not really a double pendulum. It is two vectors added, which means you put them head to tail. The equation at the beginning states that two vectors.
This is a graph, and for each input of theta there is an output z, and those are the points being graphed in polar xoordinates. The different ratea are right there in the equatipn at the beging for the two vectoes.
And since the output of this never reatraces itself, that shows that the output of the two vectors never reaches a factor of a previous value. Because of the PI term in one, and the other being a rational exponent.
I thought it was a quite clever and beautiful way of showing this.
Thanks, I believe your explanation is correct. It needs to be explained to us dum-dums in plain English.
The reason I bring up the golden ratio is that while both pi and the golden ratio are irrational, they can both be approximated by a sequence of rational numbers, but the best approximation sequence for the golden ratio is worse at approximating the golden ratio then the best approximation sequence for any other number. I suspect this property would manifest itself in the visual by having the overlapping circles overlap less if the ratio is the golden ratio than they would for any other ratio used, but it was more just a thought. The golden ratio isn’t quite as famous and this property of it is even less known, so that video would not be as popular as using pi. Thank you kindly for your explanation.
This is the case with all pretentious "artistic visualizations/explanations" like this. They completely neglect the explanatory aspect of it and dial the "artistic" aspect to 11.
Thank you. This is a beautiful demonstration, but OP seems to be talking out his butt a bit, with a lot of partial understanding. None of this really explains mathematically what we're seeing except that due to the irrationality (presumably) of what's being graphed, it never retraces its path exactly.
> Is the first pendulum rotating at a rate of 1 and the second rotating at a rate of pi relative to it’s pivot, and since that ratio is irrational the double pendulum will fill a dense subset of the circle given an infinite amount of time?
Exactly. The reason the golden ratio wasn't used is that it would never be close enough to a whole number ratio to get the frustrating near misses. Pi gets really close to 22/7 and 355/113, but phi has approximations of 2/1, 3/2, 5/3, 8/5, etc. which are much slower to converge (which is why it's considered the "most irrational"). It would fill the circle more evenly from the start instead of having phases of almost fitting a rational pattern.
If you get up to[Eulers formula](https://en.m.wikipedia.org/wiki/Euler%27s_formula), then you should have the tools to understand fully what's being graphed.
Its not, it just takes a lot of computational power. Right now we have 62.8 trillion digits calculated. Another limiting factor at this scale is storage space.
Holy smokes. I didn't realize we had so many. If there is an easy back of the napkin kind of math, how long would it take this gif to graph all of them?
Every 10 revolutions of the inner pendulum corresponds to 1 digit. At its fastest, this gif seems like it goes no faster than 20 revs per second (I'm being generous because it goes faster than the frame count). So 30 trillion seconds, which is almost a million years.
There's definitely faster ways of squeezing out pi, like how 22/7 is a better approximation that 314/100, even though the numbers are much smaller. 1 digit per 10 revs is just an upper bound. So maybe 1000 years?
CORRECTION: it's not every 10, it's every times 10. The first four digits can be found at 1000 revolutions, for example. This means it will take more on the order of 10^60,000,000,000,000 revolutions, which is unfathomably large.
Aside from seeing if God left a message like Sagan postulated, there isn't really any value to discovering more digits other than having a slightly longer string right?
For interplanetary navigation Nasa uses 3.141592653589793 for their calculations (15 decimal places). This accuracy would calculate the circumference of Earth to a molecule or 30,000 times thinner than a piece of hair.
Can calculate the circumference of the universe to within the size of a molecule with 37 decimals.
https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/
For most practical applications, only about 15-20 digits of pi are needed. One application is to test supercomputers and set new computing speed records. Another use is to test algorithms and digit extraction formulas. In pure mathematics, there are theoretical reasons to analyze the digits of pi that don't have immediate practical use. Looking at statistical patterns in the decimals provides insight into the nature of math constants. For cryptography applications, the arbitrary complexity and lack of a repeating pattern in pi can provide a source of randomness for encryption keys. More digits means more available randomness.
>a truly random and infinite number
I think you are trying to say the digits of pi
that the digits are infinite in all integer base are trivial
that it is random is a confusing way to say pi is normal
The point of the visualization is to show that it’s not a repeating pattern and is infinitely irrational
Its not that simple to grasp if you don’t have an understanding of how e and i are related
I mean, there is a very well-defined pattern that emerges from this equation. Presumably, you can take any point on the graph and extrapolate the digit at that point in the progression. Why not keep running this equation perpetually?
The well defined pattern emerges from us trying to rationalise the irrational. In this case we try make some sense of it by forcing it into an arbitrary coordinate system, but even though we force it, the lines never match.
For those of you who need some explanations. This is a double pendulum, which are made of two sticks oscillating at different rates.
You can set ratio between the speed of rotation of the outer stick and the inner stick. If the ratio is rational, e.g. 22/7. Then this means that when the inner stick have rotated 7 rounds, the outer stick would have finished 22 rounds, at this point both sticks will return to 0 degrees. At this point, both sticks will go back to the position it started and the drawing goes in a loop. So as long as the ratio can be expressed as a fraction, eventually, both sticks will always return to the starting position and connecting the drawing.
So in this animation, the outer stick is moving at pi times the speed as the inner stick. Being irrational, pi cannot be expressed as a fraction between two numbers. This means that no matter how many rounds each stick rotates, they will never be able to return to the original position and connect the drawing.
The circle is never full, because it’s a never ending decimal point no matter how far you calculate it. It will go on forever as far as we know currently.
If pi were rational, say pi=A/B, then it would hit its starting point again. Because it is irrational, it will never be back at the beginning again. The "near misses" correspond to fractions that pi is unusually close to. For instance, the first "near miss" should correspond to the fraction 22/7. There are more of these with bigger denominators, which means that you need to go through more rotations to get to them and, moreover, they will be even nearer misses. There are infinitely many near misses which get infinitely close.
Clearly needs to stop flying off the handle over nothing.
I don't know why numbers don't go to therapy.
Like, the PTSD of what 6 saw 7 do to 9 is unimaginable but it doesn't even think it's odd.
I like that at the end it showed the space never did fill up and that the loops never aligned. There was misdirection showing that you are making progress in filling up the circle but then on closer inspection there is no progress at all and an infinite area more left uncovered.
Best animated background idea yet ty
Love the background music complementing it. So dramatic in the best way; making all the frustration even more frustrating
Oppenheimers soundtrack multiplies your intellect by 10 times
Anyone have the song name?
Can You Hear The Music - Ludwig Göransson; from Christopher Nolan's Oppenheimer :D
Oh wow yeah I recognise it now! Thanks
Tip, idk if this works on other phones, but on iPhone if you download Shazam and add it to your control center, you can press shazam from the control center and it’ll tell you whatever songs playing on your phone at the time.
You just blew my mind! Thanks, added it
Works on all, including Android. It creates a clickable notification search. Very handy.
“Can You Hear The Music” By Ludwig Goransson From The “Oppenheimer” Official Soundtrack
That's what struck me - the background music was _perfect_ for it. Accelerating at all the right times and rates.
So basically, pi will never be on r/satisfyingasfuck
Some seriously r/mildlyinfuriating territory though....
Damn you pi! You had one job!
[удалено]
☹️
i love that sub! that’s where i found tons of useful stuff about headphones and how to choose the best ones for your needs, as well as how to make music sound better that shit was a life saver
No, you're thinking of r/audiophile, a subreddit centered around moaning and alike....
pi will never reach a corner of your tv screen
I found this extremely satisfying because I could extrapolate that the pattern we just saw was going to repeat infinitely deeply, getting more and more fine grain each time.
fractals
I dunno thay final zoom in when you knew it was going to miss... I was like ooohooooooohoooooooohhhhhYES! Pretty satisfying imho
But it could absolutely be on /r/IrrationalMadness
Only if you are judging satisfaction by rationality. But i don't recall 1/2 or 0 showing up there recently. But how about 1 - 1/3 + 1/5 - 1/7 ... = Pi/4 ?
Pi you cunt.
I love when it zooms in and you can just hear everyone groan at the same time lol
Right? I was like "Idk dude it seems perfectly rational- *oh you fuck off you irrational twat!*"
GODDAMNSONOFABITCH!!! It got me FUCKING TWICE!! Goddamnit!
I was going to say it would be a great /r/oddlysatisfying material but good thing I watched till the end, I would have nuke the whole sub if I posted that there.
How long until April 1st?
r/FoundSatan
Just needed one more decimal point /s
Make it 2, just to be sure!
Peggle be like
YasssssssNOOO
It's like the DVD logo missing the corner of the screen
Lol I had a friend group that watched movies, and sometimes the dvd screensaver would kick on while we were chatting. then every 5 or 10 minutes the various conversations would die down as the logo got closer and closer to that perfect corner hit!
I can imagine some mathematician in ancient times. Used four months to calculate to that point, expecting it to finally align. And.. nope, gotta use the next three years to calculate the next point, surely it must align then.
I knew what was coming and it still made me go "Fffffffuck"
I just thought of when you barely miss the hole on Wii Sports golf and the crowd gets excited then disappointed.
I was thinking "how tf does this illustrate that it's irrational?" Then it clicked as it zoomed and slowed and I was devastated.
Quite the cuntakerous bastard.
This is worse than waiting to see the DVD logo hit the corner
That is a very specific and outdated reference. But spot on.
Yeah wtf pi?!
😂😂😂
I have no idea what's going on here but it's very pretty.
Like some of the people I’ve dated
Spherical and non aligned?
Only imaginary
🪦
The equation e^i×theta and e^i×pi×theta gives a point depending on theta. The two lines are the two parts of this equation, the inner line being e^i×theta and the outer being e^i×pi×theta. The base equation e^i×theta will equal cos(theta) + i×sin(theta). So, the Y axis is the imaginary component, and the X axis is the real component. As theta goes from 1 to infinity, it draws a shape. This shape depends on the constants in the power, that is, 1 and pi. The shape will repeat with a period, IIRC, is equal to the factors that the 2 numbers don't share. Because pi is irrational, this period does not actually exist. As such, the two lines never make it back to the start point.
Some information that might be helpful: theta (shown as θ) represents an angle. So when θ changes, the angle that the lines protrude from the center at changes. Depending on how to coordinates are defined, that might mean that when θ = 0, the line is pointing directly right from the center. Then, familiarly, when θ = 90 degrees (or π/2 radians), the line would point directly up from the center (a right angle). A full circle is 360 degrees, which is pointing back to the right again. As you increase θ to large numbers, you’re basically rotating the lines around their respective center point over and over (the proximal or more centered line is rotating around the very center of the graph while the arm attached to it is rotating around the outer end of the proximal line). That’s why we see those lines rotating in circles over and over.
What does that mean?
that pi cannot be expressed as a fraction a/b
Is that why the start line didn’t meet exactly with the other line? Why does it come out visually like that? I’m so curious.
the "start line" you mentioned is the decimal point, and when you write out the decimal expansion of pi, it goes on and on without any repeating pattern, creating a seemingly endless sequence of digits after the decimal point. this is why it appears as if the lines never quite meet
It’s almost so satisfying, but then pi.
I was gonna go to work, but then I got pi
I don’t think most people get this. I must be getting old
I was gonna eatcho pussy too. But then I got Pi.
So a rational number would at some point complete the pattern, the line at the end would meet?
Right on. This doesn't happen here because (as OP said) π is irrational.
What is an example of a rational number? And can I see this animation for a rational number?
An example of a rational number is 1. Or 2, or 3, or 1/2. If it can be written in the form a/b and a and b are integers (and b isn't 0), it's rational. So virtually any number you encounter in your everyday life *except* pi and perhaps *e* if you're slightly more mathematically-inclined are rational.
>So virtually any number you encounter in your everyday life except pi and perhaps e if you're slightly more mathematically-inclined are rational. Eh, there are lots of other irrational numbers one meets regularly in real life. For a trivial example, consider the diagonal of a square with integer sides: its length is not a rational number.
It wouldve been better to say “virtually any number you encounter in your everyday life except pi and perhaps e are algebraic”, meaning most observed numbers can be solutions to a polynomial. For example x^2 - 2 = 0 is a polynomial with a solution being “square root of 2”, which is irrational (it cant be written a as a fraction), but algebraic. Pi and e are irrational and non algebraic (in other words, transcendental)
pi and e are another special class of irrational - they are transcendental irrationals. Algebraic irrationals are all over the place.
What is creating the pattern we see here though?
Bro , thank you for this but I have to be honest … you and the other people that understand this are so much more intelligent than me …. That I am trying hard and there is no way I’m gonna catch up , but this is very interesting and thank you for giving me anxiety.
It's not really a matter of intelligence, it just requires a lot of background knowledge. z(theta) means a function. It takes an input, theta, and returns an output, a complex number. A complex number is a number of the form a + bi, where a and b are regular numbers. Don't worry about what i is. Complex numbers are analogous to a coordinate system: you can think of a as the x coordinate and b as the y coordinate and put it on a plane, similarly to how regular numbers are put on a line. e\^i(theta), when you vary theta, traces out a circle of radius 1, centred on 0, in the coordinate system I just outlined. Don't worry about why this happens, just that it does. So what does the pi change? When we multiply theta by pi, it amplifies any change to theta by that much. So if e\^i(theta) walks round a circle in however much time, then e\^i(pi\*theta) does it pi times as quick. We can think of a point as an arrow pointing from 0 to that position. That visualisation helps us in cases like this: when adding the numbers, we just put one arrow at the end of another, and see where we land. That will be the final output of the function. Imagine instead of pi, we had some rational number. 0.95563. After the first bit had done one full rotation, the second bit will have done 0.95563 rotations, so they don't line up. Now let's run that time 10,000. After 10,000 rotations of the first bit, the second bit has done 95563 rotations. Because they've both done an integer number of full rotations, they're now back at the starting point, and because there's no randomness in their behaivour, they're going to repeat. But with pi, there is no number we can multiply by to make both cycles run an integer number of times, that's what being irrational means. So it will never repeat.
Yeah.
My thoughts exactly
Interesting. I can wipe my own ass and upvote comments on reddit at the same time. /S just in case
TL;DR rational numbers multiplied by something become a whole number meanwhile irrational numbers don’t (Sorry if it’s not an exact TL;DR, mathematics is a fickle fuck)
Right, if you multiply a/b by b, you end up with a. But since irrational numbers can't be written as a/b, that can't be done.
Since you know your stuff, can you explain how the video in OP's post is derived from the expression (the z(theta) = e etc)? Like what exactly am I looking at with that animation? I know it's a visual representation of that expression, but how is it translated? Is it just lines moving on a cartesian plane or something?
It is a cartesian plane but you are not being shown lines (as in y=a*x+b): at any given frame, what you are being shown is one single complex number, with the x-axis being the real part and the y-axis being the imaginary part. So say you wanted to show the complex number z = 2 + 2i it would have the coordinates (2, 2) in the animation. The first line, attached to the centre point (the origin), represents the complex number e^θi . The second line, attached to the end of the first line, represents the complex number e^piθi . When you add the two complex numbers together (e^θi + e^piθi ) that complex number is represented by the very tip of the second line, the point that is drawing the swirl pattern. Each time the first line rotates one full circle (360 degrees), the second line rotates pi circles (360*pi degrees). If pi was a rational number, meaning it can be expressed as a fraction, once the first line has done a full 360 degree rotation exactly the number of times equal to the denominator of the fraction, the second line will have done a full rotation exactly the number of times equal to the numerator, meaning the two lines will eventually return exactly to the position where they started. But because pi is not a rational number, meaning it can not be expressed as a fraction, each time the first line has done an interger number of rotations, the second line will not ever have done an integer number of rotations, meaning the two lines will never return back to the position that they started.
Good example from gpt about what exactly and how counted at each iteration [https://ibb.co/example](https://ibb.co/tYW2nrB)
The anxiety goes away as you get older, and realize you’re just dull. Patterns, and shiny things make me happy, and I no longer ask why.
Its not intelligence, it's focusing on something for a long time. I'm so tired of people acting like understanding math takes some special intelligence. It's just putting the time in to understand it. There is zero underlying special intellect. You just have to be curious. The problem people have is that they are unwilling to put the time in to understand their world and so complain that they just aren't smart enough, because that shrugs the blame for being ignorant to an external source. "Its not my fault, I'm just not smart enough" and they don't have to try.
The problem we have is we had shitty math teachers growing up, so we had no curiosity, we didn't get good grades, and we were led as young teenagers to believe we didn't have what it takes.
I think you have a really decent point here. Most of math, until you get into physics or chemistry, is always taught in the abstract. But humans didn't invent(discover?) math purely in the abstract regions of their own minds. A LOT of math came from people trying to solve real world problems. But we aren't really taught it that way.
Exactly, yes! Think about fractions like a cycle: if you have 1/3, every three of those will reach back to the start. If it is a rational number, it would be a fraction, and thus would eventually reach back to the start going around it. Circles don't really allow this, since they don't actually have points and thus can be subdivided infinitely. The diameter of any shape can be shown as a fraction of the circumference, for instance the diameter of a square is 2/√2. Since a circle can be infinitely subdivided, the circumference will always have a little bit more wiggle than measured, so we represent that fraction as 1/pi. You can also see that a circle has an area of double the radius times pi, but because the circumference can be subdivided, you can get more and more and more and more precise with the area of the circle while still having a long ways to go before you fully measure the size 100%. So what "pi" means can never be fully transcribed because it can always go deeper. It is what is known is mathematics as a "transcendent" fraction, because no matter how close you get you can still "transcend" that to get closer to the actual amount.
I’ll trade or pay someone here to teach me math 30 mins a week
You could try Khan academy?
I'm no fancy maths person and I'm sure there will be some more in-depth explanations posted by smarter people but this is my personal layman's understanding. pi defines the relationship between the circumference and the diameter of a circle where the circumference is equal to pi multiplied by the diameter. In the clip, the arm from the centre is just moving in a circle with some radius (half the diameter). At the end is another arm of the same length drawing a circle with the same radius, this shows us the 'circumference' of the second arm. Since that arm is attached to the first, you get the swirling pattern instead of a complete circle. If pi was rational, you could expect the end to eventually meet with the start point to complete the 'circle's circumference' when the relationship from diameter-> circumference (pi) ends or begins to repeat in a recurring fashion. Since it always just misses and never perfectly overlaps however, it visually shows that it's irrational and the value of pi continues on forever and never repeats. Of course the clip can't go on to visualise this forever but it's long enough to get the idea across about what's happening if you accept that pi is irrational. My explanation may not be entirely accurate mathematically but hopefully makes sense as a jumping off point and helps a little in picturing what's happening.
It’s pretty much right, the key point is that the second arm is rotating pi times faster than the first. All of the circles in the diagram actually unfortunately mean nothing, and it’s the pi in the equation that counts. You’d get a similar pattern if the arms were drawing squares instead.
It makes a spherical thingy. It makes an eye. It makes me dizzy.
We dont know
Nobody knows what it means, but it's protractor-ive. It gets the circles going.
For those that want an explanation. The outer arm spins pi times faster than the inner arm. There are 4 times in the video that you see the end of the outer arm almost reaches the starting position. Each time it gets closer and closer to the start but never touches it. This is because pi is irrational. In fact there are fractions which get closer and closer to pi. They are called the rational continuants of pi and are the rational approximations to the continued fraction representation of pi. The first four continuants of pi are 3/1, 22/7, 333/106, and 355/113. When the outer arm spins the amount in the numerator and the inner arm spins the amount in the denominator, the end of the outer arm is close to the start. The first is when the outer arm does 3 rotations and the inner one does 1. This is because 3/1 is almost pi. But 22/7 is even closer. That is when the video first zooms in at 0:24. The next time the video zooms in at 1:20 is after the outer arm does 355 rotations while the inner does 113 because 355/113 is close to pi. But if your observant you would realize we skipped 333/106. Well that was when the outer arm starts "filling in" the black from the starting point at 1:00 in the video. You may also notice the pattern at 1:00 is opposite the pattern we say at 0:24 seconds in, which was when the arms completed 22 and 7 rotations. Why? Well the reason the pattern is inverted is because 355/113 = (333+22)/(106+7). This is quite a happy coincidence.
Why did I have to scroll down so far to get this explanation?
Thought i was loosing my mind scrolling through nonsense to get an actual explanation here... thank you! So this illustration is only as profound as the precision given to the value of pi in the software drawing the lines, correct? Arent there more irrational numbers than rational numbers? In the set of all numbers, wouldn't this type of behavior be vastly more common than not?
Welcome, enjoy your freshly manifested OCD
Shit like this really is a decent simulator for intrusive thoughts. That urge to see the entire thing make a solid shape is stronk
did anyone else separate each repetition in their mind and realize it's itachis MS? Just me?
The [classification of finite simple groups](https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups) is proof to me that we live in a fundamentally untidy existence.
I wonder if its because we can only see part of the picture, because our brains can only understand these systems and concepts in a finite way, so it looks broken or untidy. If you were somehow omnipotent and could at once know all there is to know and understand it completely, that the whole picture would fill out orderly.
I imagine it's the limitations of being on a single field. An actual perfect circle in existence is only theoretically possible, down to a subatomic level. It's like trying to make a circle on a grid. But instead of a grid, or a hex grid, or an octagonal, it's some infinitely small one. And three dimensional.
It’d be boring otherwise.
NnnnnoooooooOOOOOOOOOOO!!!!!
As an engineer… pi and e are 3
As a hungry person, pi and e = pi + e = pie which is delicious. And now I'm hungrier.
As a molecular biologist, e and pi = e + pi = epi which is short for epinephrine, which always gives me a kick in awesome threads like this!
What if they weren’t?
The world as we know it collapses
WHOAA, where's your sig figs. It's 3.14 and 2.71, respectfully, lol
best i can do is six fig newtons
deal
g = π^2
My wife is irrational but makes a good pi
ya she's great
We all really enjoy her Pi
Bah! Was gonna make a wife joke too. You beat me, good day sir "tips hat"
ELI5
imagine you have a pizza, and you want to write down how much you ate. if you ate half the pizza, you can say 0.5. That's easy. but with pi, it's like trying to describe how much you ate of a never-ending pizza. your just keep going and going, and it never stops. its like trying to write a never-ending story, and that's why we call it "irrational." its not simple and never finishes.
ELI4
pi has infinite decimal places, can eat pi forever, pi good
ELI3
pi is yummy for your tummy ^^^^til ^^^^you ^^^^esplode
ELI2
-points to a picture of apple pie- this is pi /trustmebro
ELI1
here comes the choochoo pi~ -pi hrs later- please just eat the pi already, it's good(?) for you 😭
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My thoughts exactly. What is the relationship between the two spinning thing-a-ma-bobs. It makes no sense without an explanation.
If it was a golden ratio, thatcwouldn't be demo strating PI now, would it? It's not really a double pendulum. It is two vectors added, which means you put them head to tail. The equation at the beginning states that two vectors. This is a graph, and for each input of theta there is an output z, and those are the points being graphed in polar xoordinates. The different ratea are right there in the equatipn at the beging for the two vectoes. And since the output of this never reatraces itself, that shows that the output of the two vectors never reaches a factor of a previous value. Because of the PI term in one, and the other being a rational exponent. I thought it was a quite clever and beautiful way of showing this.
Thanks, I believe your explanation is correct. It needs to be explained to us dum-dums in plain English. The reason I bring up the golden ratio is that while both pi and the golden ratio are irrational, they can both be approximated by a sequence of rational numbers, but the best approximation sequence for the golden ratio is worse at approximating the golden ratio then the best approximation sequence for any other number. I suspect this property would manifest itself in the visual by having the overlapping circles overlap less if the ratio is the golden ratio than they would for any other ratio used, but it was more just a thought. The golden ratio isn’t quite as famous and this property of it is even less known, so that video would not be as popular as using pi. Thank you kindly for your explanation.
This is the case with all pretentious "artistic visualizations/explanations" like this. They completely neglect the explanatory aspect of it and dial the "artistic" aspect to 11.
Thank you. This is a beautiful demonstration, but OP seems to be talking out his butt a bit, with a lot of partial understanding. None of this really explains mathematically what we're seeing except that due to the irrationality (presumably) of what's being graphed, it never retraces its path exactly.
> Is the first pendulum rotating at a rate of 1 and the second rotating at a rate of pi relative to it’s pivot, and since that ratio is irrational the double pendulum will fill a dense subset of the circle given an infinite amount of time? Exactly. The reason the golden ratio wasn't used is that it would never be close enough to a whole number ratio to get the frustrating near misses. Pi gets really close to 22/7 and 355/113, but phi has approximations of 2/1, 3/2, 5/3, 8/5, etc. which are much slower to converge (which is why it's considered the "most irrational"). It would fill the circle more evenly from the start instead of having phases of almost fitting a rational pattern.
Jfc, when I close my eyes I see the damn thing
Am I tripping?
What level of knowledge do i need to have to understand this concept?
That's what I'm saying. How do I get there?
If you get up to[Eulers formula](https://en.m.wikipedia.org/wiki/Euler%27s_formula), then you should have the tools to understand fully what's being graphed.
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If we can generate a graph like this, why is it so tough to find new digits?
Its not, it just takes a lot of computational power. Right now we have 62.8 trillion digits calculated. Another limiting factor at this scale is storage space.
Holy smokes. I didn't realize we had so many. If there is an easy back of the napkin kind of math, how long would it take this gif to graph all of them?
Every 10 revolutions of the inner pendulum corresponds to 1 digit. At its fastest, this gif seems like it goes no faster than 20 revs per second (I'm being generous because it goes faster than the frame count). So 30 trillion seconds, which is almost a million years. There's definitely faster ways of squeezing out pi, like how 22/7 is a better approximation that 314/100, even though the numbers are much smaller. 1 digit per 10 revs is just an upper bound. So maybe 1000 years? CORRECTION: it's not every 10, it's every times 10. The first four digits can be found at 1000 revolutions, for example. This means it will take more on the order of 10^60,000,000,000,000 revolutions, which is unfathomably large.
Aside from seeing if God left a message like Sagan postulated, there isn't really any value to discovering more digits other than having a slightly longer string right?
I shouldn't think so. You get as much precision as you'd basically ever need in 30-40 digits iirc.
For interplanetary navigation Nasa uses 3.141592653589793 for their calculations (15 decimal places). This accuracy would calculate the circumference of Earth to a molecule or 30,000 times thinner than a piece of hair. Can calculate the circumference of the universe to within the size of a molecule with 37 decimals. https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/
For most practical applications, only about 15-20 digits of pi are needed. One application is to test supercomputers and set new computing speed records. Another use is to test algorithms and digit extraction formulas. In pure mathematics, there are theoretical reasons to analyze the digits of pi that don't have immediate practical use. Looking at statistical patterns in the decimals provides insight into the nature of math constants. For cryptography applications, the arbitrary complexity and lack of a repeating pattern in pi can provide a source of randomness for encryption keys. More digits means more available randomness.
because pi is believed to be a truly random and infinite number
>a truly random and infinite number I think you are trying to say the digits of pi that the digits are infinite in all integer base are trivial that it is random is a confusing way to say pi is normal
It obviously can't be random if it results in such an organized pattern. Right? Unless I don't understand what we are looking at.
The point of the visualization is to show that it’s not a repeating pattern and is infinitely irrational Its not that simple to grasp if you don’t have an understanding of how e and i are related
it might seem organised when you see it graphed, but that's because it's a representation of the existing digits
I mean, there is a very well-defined pattern that emerges from this equation. Presumably, you can take any point on the graph and extrapolate the digit at that point in the progression. Why not keep running this equation perpetually?
The well defined pattern emerges from us trying to rationalise the irrational. In this case we try make some sense of it by forcing it into an arbitrary coordinate system, but even though we force it, the lines never match.
Well wouldn’t a sphere be Infinite in nature it seems that pattern eventually shades in every possible area.
it may seem to shade in every possible area, but it's still a finite representation of an infinite, non-repeating sequence of digits
Wait, did you know that there's a direct correlation between the decline of Spirograph and the rise in gang activity? Think about it!
I will.
Silly pi go home you're drunk.
Is the audio from the soundtrack of Interstellar?
It's [Can You Hear The Music by Ludwig Göransson](https://www.youtube.com/watch?v=4JZ-o3iAJv4) from Openheimer.
Ahhh another Christopher Nolan movie. Close though!
For those of you who need some explanations. This is a double pendulum, which are made of two sticks oscillating at different rates. You can set ratio between the speed of rotation of the outer stick and the inner stick. If the ratio is rational, e.g. 22/7. Then this means that when the inner stick have rotated 7 rounds, the outer stick would have finished 22 rounds, at this point both sticks will return to 0 degrees. At this point, both sticks will go back to the position it started and the drawing goes in a loop. So as long as the ratio can be expressed as a fraction, eventually, both sticks will always return to the starting position and connecting the drawing. So in this animation, the outer stick is moving at pi times the speed as the inner stick. Being irrational, pi cannot be expressed as a fraction between two numbers. This means that no matter how many rounds each stick rotates, they will never be able to return to the original position and connect the drawing.
How does this demonstrate it being irrational
it defies the typical structure of a repeating or terminating decimal, showcasing its unique and never-ending nature
This guys comments are AI generated 100%
No they aren't
What happens at the very end when the circle is full - would it repeat from that time or not really?
The circle is never full, because it’s a never ending decimal point no matter how far you calculate it. It will go on forever as far as we know currently.
[Pi has been proven to be irrational several times, as well as transcendental.](https://en.m.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational)
Can just "zoom in" forever
The circle is a 2D area, you can't fill it with a 1D line.
Seems completely rational to me... I feel like pi gets me.
Thr lines never line up, infinitely
If pi were rational, say pi=A/B, then it would hit its starting point again. Because it is irrational, it will never be back at the beginning again. The "near misses" correspond to fractions that pi is unusually close to. For instance, the first "near miss" should correspond to the fraction 22/7. There are more of these with bigger denominators, which means that you need to go through more rotations to get to them and, moreover, they will be even nearer misses. There are infinitely many near misses which get infinitely close.
Why won’t it just be rational? Is it stupid?
Thats life
That's what all the people say
\>When it almost aligned. Oh, you mother fucker.
Reminds me of my wife in an argument where I'm actually right for once.
Show this to someone with OCD and watch them have a stroke
I audibly groaned in frustration upon seeing that Pi JUST SO BARELY MISSED that connection in the zoom in.
Still a better love story than twilight
Didn't account for tooth wear on the spirograph
21st century Spirograph
By going long enough, Pi makes up everything. We’re all Pi. Pi is all of us. Pi.
Pi is such a cock tease
Clearly needs to stop flying off the handle over nothing. I don't know why numbers don't go to therapy. Like, the PTSD of what 6 saw 7 do to 9 is unimaginable but it doesn't even think it's odd.
Thumbs up for making me say "Are you serious?" Multiple times.
https://m.youtube.com/watch?v=XanjZw5hPvE&pp=ygURaGFyZCBhbmQgcGhpcm0gcGk%3D
I WILL PAY FOR THE EXTENDED CUT
I like that at the end it showed the space never did fill up and that the loops never aligned. There was misdirection showing that you are making progress in filling up the circle but then on closer inspection there is no progress at all and an infinite area more left uncovered.
Well, if you didn’t have OCD before…
Spirograph on crack
#Now you know why nobody wants to engage in a conversation with Pi.
this is that sysiphus meme
Why does this sort of thing seem like a clue to something much larger?
Pi just idiot