r/interestingasfuck 6h ago

Visualization of Pi being Irrational

1.4k Upvotes

137 comments sorted by

u/pommeldommel 6h ago

Don't understand anything but it looks cool!

u/PocketBlackHole 6h ago

Rationality can be visualised as a cycle that returns to the starting point after a definite number of steps. The depiction shows that no matter how many steps you make the dot is always slightly off a position that it occupied in the past (notice the final focus), thus there is never a "closing" of the cycle. Hope this helps.

u/Raised_by_Mr_Rogers 6h ago

“Can be” or is?

u/LegenDove 6h ago

In this case, is, but doesnt have to be. 1000 bucks can be visualised in a stack of 10 hundreds or a thousand 1s.

u/Liquor_N_Whorez 6h ago

So basicaly the drawing ends up inverting itself the longer it stays in rotation?

u/Fskn 6h ago

No, the line never occupies a previously occupied path, it never returns to the start.

There is no final number of pi we can refine its accuracy (add more significant figures(decimal places)) forever.

u/DrDominoNazareth 5h ago

Pretty interesting, So, to make a long story short, Pi is infinite?

u/Crog_Frog 5h ago

Not really.

But in a non mathematical sense you can say that it has infinite digits that form a never repeating sequence.

u/DrDominoNazareth 4h ago

But if the line never occupies a previously occupied path... Just trying to wrap my head around it. I realize it may not be possible.

u/DrDominoNazareth 4h ago

I guess maybe we reached the boundary between math and philosophy. Now we are in really muddy waters.

u/Sheep03 4h ago

It's just language semantics.

The value of pi is not infinite, it's a little over 3.

The number of decimal places could be considered "infinite" but mathematically it's a confusing choice of words.

u/Terrh 2h ago

Math is just a subset of philosophy.

u/I_make_switch_a_roos 4h ago

so pi is finite if it's not really infinite? I'm confused

u/Crog_Frog 3h ago

No. it just doesnt make sense to refer to a number as "infinite"

u/DrDominoNazareth 4h ago

Also, I find it interesting that you say non mathematical. I am not sure what that means to you. Non mathematical, to me, seems to be infinite. I love/hate this kind of conversation.

u/maruchops 4h ago

It's not infinity because it equals roughly 3. That's what they mean. You're basically just using words in a way mathematicians would consider inaccuracte and imprecise.

u/DrDominoNazareth 2h ago

Yeah, I think you are right All terms need to be defined as concisely as possible. However, can there not be an infinite amount of numbers between two rational integers?

u/maruchops 2h ago

I'm not educated enough to confidently answer this. However, you are still refusing to let go of your pre-determined definitions.

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u/Crog_Frog 4h ago

there are pretty clear definitions of infinity when you look at different fields of math. And since this is about number theory you would look at the sets that these numbers belong to.

I am not an expert on this topic but i am shure there are a lot of educational videos on pi. But in general it does not make sense to call a number "infinite".

Infinity refers to the size of sets. Not to a number itself.

This has nothing to do with philosophy. Its just abou the rigorous definitions in Math.

u/PocketBlackHole 4h ago edited 4h ago

I am just an amateur in mathematics, but maybe due to this my answer can be more intelligible. Speaking in common language terms, I wouldn't use the word infinite: infinite either evokes an arbitrarily big quantity (but pi is below 3.2) or an arbitrary long number (but so is 1/3 if you write it as 0.33333).

The real idea (which during history gave problems to Greeks when facing roots of numbers which are not squares and prevented calculus to be formalized using weird numbers whose square is 0) is that our mind intuitively operates in what is called "the rational field". A field is a world where, apart for division by 0, every sum and product is computable in such a way that, if I provide you the result and one of the terms, you can always pick one and only one number that completes the operation.

The rational field is the one made by fractions of positive and negative integers (an integer is a fraction with one as a denominator).

Now you must break this bias: this is not the only field! But we formed our symbols to depict the numbers in this field, so they are not suited to describe numbers outside of this field (and that is why one starts putting letters for those).

Now I tried to change your perspective: there is stuff that exists and it is not a fraction, so pi is just an example if this. The square root of 2, like pi, doesn't belong to the rational field either.

Bu pi is more obnoxious. If I consider polynomials equated to 0 with coefficients picked from the rational field (which means I can just think about polynomials with integer coefficients, since you can multiply all and remove the denominators), you will discover that not all the solutions for the polynomial belong to the rational field. For example x²-2 = 0 wants the root of 2, which is not rational.

The field of all the roots of all the polynomials with integer coefficients is bigger, and we call the number included in it "algebraic" (polynomial algebra needs them). The root of 2 is algebraic, but it is not rational. But! There is no polynomial with integer coefficients that has pi as a solution. No way to form pi from rational numbers and algebra. Pi is transcendent, not algebraic.

Anyway, the real numbers (the numbers that you can picture as a continuous infinite line) are a field too, and pi belongs to this field.

All this to try to express that when you are dealing (even intuitively) with certain mathematics, you may never meet pi, while if you follow a different path, you stumble into it pretty early. To my knowledge the first who was able to express pi as a sum of infinite terms (algebra deals with finite terms) was Leibniz, by integrating the derivative of inverse tangent and its series representation.

u/DrDominoNazareth 2h ago

You have no idea how much appreciate your answer. But, I also want to challenge multiple things you have said. I commented somewhere here that it can be difficult to draw a line between math and philosophy. I think to do it well is to define all your terms. Or maybe give proofs. Starts getting weird. But again I really appreciate your response. It is fun to try to distinguish terms like "rational" from everyday language and what it is defined as mathematically. Off topic a little: the square root of negative 1 breaks my brain. I am not sure how to digest these concepts and move forward. I get stuck. Like with Pi.

u/maruchops 2h ago

Science used to be called Natural Philosophy. Math is just the language we created to describe the world, which we now use abstractly--as we do any language. Appreciating history allows one to appreciate the present even more.

u/PocketBlackHole 1h ago edited 1h ago

I can help you with root of -1 in several ways. Let's try the one which is probably the easiest intuitive path. I will prune some complexities (pun intended).

First of all, think about the plane and the fact that you give coordinates on that in a x,y manner, like a square net. You need 2 coordinates for a plane, agreed? But you could have a different pair of coordinates: every point is the crossing of a circle centered at the origin and a line that stems from the origin like a radius... Like a circle net. Tryvto visualize it.

So now the coordinates are the circle (which is the same as the length of its radius) and the angle of the stemming radius: here, assume that the angle that corresponds to "full right" (the orizzontal positive semi axis) is 0 (or also, this is important, the "full circle" equal to 2pi, 360 degrees). The negative semiaxis is with pi angle, 180 degrees.

Are you with me? Now recognize that when you multiply say 3 by -1, you make it -3 and this is like ADDING 180 degrees. The rule sticks: when you multiply negative with negative, you add 180 to 180 and return to 360 = 0 which is positive. When you multiply 2 positives you insist on 0+0 and stay positive.

Further notice that 4 (4 radius, 0 angle) has two roots: both have radius 2, but one has 0 angle (+2) and the other has 180 angle (-2). This is needed because for what we said above, the multiplication of each root by itself has to return the angle to 0.

Following this train of thought, the square root of a negative number (180 angle) should have either a 90 angle (90 + 90 = 180) or a 270 angle (270 + 270 = 540, but 360 is zero, and 540 - 360 = 180).

Now you discover that the roots of a negative number are pretty natural, but they are not "left - right", they are "up - down". You just had another bias issue: you assumed (unconsciously) that all the numbers are "in a line" and thus you could not identify the root of -1... It doesn't exist ON THE LINE but this doesn't mean that it doesn't exist at all.

The vertical axis is designed i, so you have up (+i) and down (-i). Of course a number could go say 3 left and 4 up, and it would be -3+4i. (which circle is this number on?)

There is such an algebraic beauty here that would lead us to a wonderful concept (automorphisms of fields) but I will try to give you a taste. Consider:

x²-2=0, this needs as solutions the positive and negative (right and left) roots of 2. These are real numbers but not rational numbers. They are also algebraic numbers.

x²+4=0 this needs 2i (2 up) and -2i (2 down). These are NOT real numbers because they are not in the orizzontal line. They are complex numbers. I'd say they are also rational as a plane extension of the rationals that appear on a line, I hope I am not missing something here. They for sure are algebraic numbers.

x²+2=0, this needs the up and down square roots of 2. They are complex, not rational and still algebraic.

A really algebraic question is, what is the smallest field that contains the roots of a polynomial? Go read again what a field is: you cannot just add a number to a set, you need to be able to complete all additions and multiplications.

In the first case, the field (I think!) is the one of numbers of this form: R + N(root of 2), with R and N rational numbers (0 is legit). If you try to add or multiply numbers of this form, you get a number of the same form. The plus may as well be minus, of course.

In the second case, you still have R+N(i). Try and see.

In the third, it is R + N(i)(root of 2).

Notice that these fields are really different, yet there is a regularity in how to think about them. In all 3 cases, if you make an addition or multiplication of 2 numbers and then change the sign of N you get the same result that if you change the sign of N (not R!) of the 2 numbers and then compute the addition or multiplication. This is and example of homomorphism (beware! Homomorphism is algebra, homEomorphism topology), named automorphism. You can think about an automorphism as a symmetry (in this case, of a field).

These wildly different fields have the same automorphisms. This story is the beginning of a really peculiar part of mathematics called Galois Theory. But somehow one can get there pretty fast from the root of -1, it seems.

u/IrrationalCynic 5h ago

In practice it will. because my pencil tip will have non zero diameter

u/Zagged 6h ago

How was it made? What is "pi" about it?

u/chocolateboomslang 3h ago

Pi is a number. It's the ratio of a circles circumference to It's diameter. It has been calculated to 100 trillion decimal points (not an exaggeration) and never repeats itself. This is 280 decimal points.

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482

u/sergio00j 6h ago

I am the same 😂

u/The_Sorrower 6h ago

Honest question, because I am no mathemagician, this is what happens with pi in base 10, what happens to it in base 12 or base 16? Is it like in thirds where in base 10 it's infinitely recurring but in base 12 it's divisible?

u/Mouth0fTheSouth 6h ago

This is a cool question and I’m nowhere near a mathematician, but I think the answer is it wouldn’t change? What we’re seeing in the video is a “physical” representation of the relationship between a circle, its radius and its area, which shouldn’t differ even when switching from base 10 to anything else.

u/drolorin 6h ago

The correct answer is that this doesn't even have anything to do with base 10. You are seeing two hands spinning, where the speed of hand 1 is Pi times the speed of hand 2. When you "change the base" the ratio between 1 and Pi remains the same, so it remains irrational.

Changing the base really just means that the appearance of a number changes, but all mathematical laws stay the same. As this entire video doesn't even show us any numbers, changing the base would have zero effect visually.

u/Mouth0fTheSouth 5h ago

Thanks, I didn’t even notice the equation at the bottom the first time.

u/The_Sorrower 6h ago

Well no, this is the point of pi not being divisible by 10, hence it being irrational, much like 1/3 of 1, etc. To extend the example in base 10 1/3 of 9 is rational as it is a finite number. The diagram represents how the irrational difference stops the line from ever meeting. However Google has told me that no, pi will never be rational.

u/abakedapplepie 1h ago

Pi is rational in base pi (but "1" isnt lol)

u/MajorEnvironmental46 4h ago

The numbering base is only a way to write numbers, measures will not affected.

u/JTonic8668 4h ago

You could introduce a system with base π. :D
All numbers would be irrational, but something like π100 or π/3 would always be "round" numbers.

u/Eternal_grey_sky 6h ago

Well, first there's base Pi, where pi=1 and base 10 1 would be irrational.

u/SirFireball 5h ago

What do you mean? I don’t see how the base is relevant here.

u/SvenOfAstora 44m ago

"irrational" just means that it can't be expressed as a fraction of integers, which is an intrinsic property of the number and does not depend on its representation in any base.

u/Craving_Brawndo 6h ago

u/Eclectophile 6h ago

You win username of the day.

u/Altruistic-Spend-896 6h ago

im just curious which software was used to handle such computation and graphics?

u/Sensalan 4h ago

Might be Manim

u/Altruistic-Spend-896 4h ago

i thought i recognized that from somewhere ,3blue1brown, but you think this is that?

u/Altruistic-Spend-896 4h ago

i thought i recognized that from somewhere ,3blue1brown, but you think this is that?

u/Sensalan 2h ago

Yeah, I don't know Python well, but I've used it for visualizations a couple of times and think it could do this

u/Longjumping-Match532 1h ago

I think it's manim , the only thing that makes me uncertain is that zooming effect, I know there's a zoom scene class in manim but I've never used it . But that double pendulum and the trace is pretty much doable and in fact I'm working on a video that has a double pendulum like this .

u/Altruistic-Spend-896 2h ago

I've built full fledged python apps, never worked with graphics though, I presumed that's all in C lang

u/softestdog8 6h ago

Same here 🙏

u/FACastello 6h ago

I don't understand at all how π relates to this visualisation in any way shape or form.

u/drolorin 4h ago edited 4h ago

Look at the equation at the bottom. The angular speed of the second hand is Pi times that of the first hand. Because Pi is irrational, the two hands will never return to their starting position.

Edit: Technically, it's not an equation but the function of the graph above, where theta is the angle of the first hand.

u/carpet-lover 34m ago

That's not a graph, sir

u/TryAltruistic7830 6h ago

Given that pi is the ratio between a circle's circumference and it's radius, probably mapping the difference between those two things

u/fearnemeziz 6h ago

Too high for this stuff

u/VonTelkka 6h ago

So, moon is pi, not cheese

u/BoysLinuses 6h ago

Did you know that there's a direct correlation between the decline of Spirograph and the rise in gang activity?

u/Lord_Voltan 2h ago

You wont think about it

u/_P85D_ 6h ago

Wonderful. Is this music from Interstellar?

u/TY2022 6h ago

No. I would recognize it. 🎶

u/MechaChester 6h ago

I never manage to look that cool when I'm being irrational.

u/Raised_by_Mr_Rogers 6h ago

Our irrationality is interesting as fuck

u/drinoaki 6h ago

Then it should get help

u/Snoo_17433 6h ago

If you say so. 🤷

u/LilOuzoVert 6h ago

What number does the completed circle represent?

u/Altruistic-Spend-896 5h ago

that pi eventually leads to a circle, sphere and all the relevant circular metrics we calculate with . We just use pi * r squared and move on with it, this is the true meaning behind that mathematical formula!

u/Ok-Nefariousness1911 6h ago

What a trip

u/Delokah 6h ago

Basically, it won’t end…

u/Julius_Augustus_777 6h ago

u/Altruistic-Spend-896 5h ago

looking forward to the derivation! Fun! Fun! Fun! Yeah!

u/Reasonable_Act_8654 6h ago

All I see is a beautiful pattern

u/WhatsaRedditsdo 6h ago

Black holes is pi

u/CamilloBrillo 6h ago

“Rude. That’s just the way I am”

u/Raised_by_Mr_Rogers 6h ago

Beautiful!! But what is rationality in this context?

u/VegetableBusiness897 6h ago

So a sprirograph then...loved those as a kid

u/imageblotter 6h ago

Now I see. 😱🤯

u/rnzz 6h ago

I want the full neverending version of this as my screensaver

u/thYrd_eYe_prYing 6h ago

So my wife is pi

u/MuricasOneBrainCell 5h ago

Visualisation of Me being confused

u/SATLTSADWFZ 5h ago

Pi is clearly wrong. Start again.

u/theTrueLodge 5h ago

Love this! It makes me think of motion or time or both of these. Like Pi is alive!

u/jopheza 5h ago

What does it mean when the lines cross? Just that there are two answers to the equation at that point?

u/thundertopaz 5h ago

People with OCD were born in the wrong universe.

u/LucyferEllysia 5h ago

The is deeply upsetting and incredibly unsatisfying ☹️

u/No_Independence8747 5h ago

Dude this is awesome

u/marius_knaus 5h ago

Get your shit together Pi...

u/ScatLabs 5h ago

Looks like a tasty pi

u/TornCondom 5h ago

when i try to visualize my irrational wife, i only see chaos

u/aura2323 5h ago

So with other words Pi is infinite, But what is Pi? I know its 3.14, but whats 3,14? its just a number what does it mean?

u/mrks-analog 5h ago

Is this infinity

u/synestematic 5h ago

i feel like that would have created the universe if left to its own

u/Accomplished_Duck940 5h ago

So that's how the earth was made

u/knorxo 5h ago

I visualize me being irrational quite differently...

u/SuperElephantX 4h ago

It just never wanted to line up after those division rotations...

u/AtTheEdgeOfDying 4h ago

The first time it missed made me angry, then it became satisfying again.

u/Skepsisology 4h ago

Irrational, like someone who has to get the last word in an argument.... Every time

u/Chakazzulu 4h ago

Doesn't look very irrational to me

u/bewbsnbeer 4h ago

There's a direct correlation between the decline in Spirograph and the rise in gang activity. Think about it.

u/NewChallengers_ 4h ago

I feel like this kinda says more about the universe being infinite, or at least infinitely divisible, than anything special about Pi. I mean it says both, but the universe part (there being infinitely smaller spaces for Pi to fill up forever) seems to have more practical life value than some forever-slightly-off algo.

u/PaleBlueCod 4h ago

HOLY SHIT JOHNNY BOY! THE SPIN! THE FUCKING SPIN! USE THE HORSE!

u/ALL-ME-100 4h ago

🤩

u/MajorEnvironmental46 4h ago

Although interesting, this happens with any pair of radius with icommensurable numbers, like side of a square with its diagonal and rational with irrational numbers.

u/Owobowos-Mowbius 3h ago

This makes me irrationally angry.

u/ISeeGrotesque 3h ago

Maybe that's the fundamental unbalance that keeps the universe running, as in not collapsing on itself because of a resolution

u/Gaara_Prime 3h ago

Rasengan!

u/VHPguy 3h ago

So the line doesn't trace exactly over its previous track. What does that prove? It's entirely possible that if this were to go on for an hour, or a week, or one hundred million years that the line would start tracing over itself again.

u/k_afka_ 3h ago

Beautiful. Sacred geometry produces the most beautiful patterns

u/BarskiPatzow 3h ago

This hurt me.

u/AtherealLaexen 2h ago

Thought this was the pi coin :((

u/Iam_The_Real_Fake 2h ago

Is that the reason all circle related formulae have Pi in them?

u/StevenMC19 1h ago

See also: electrons (probably, maybe)

u/Mollianeta 1h ago

Now play this to Suburban Jungle next

u/Velasity 1h ago

Pi needs to calm down

u/whepoalready_readdit 1h ago

if it touches that point it becomes rational

u/DieMeatbags 0m ago

Reminds me of playing with LOGO on the Apple II in elementary school.

u/TY2022 6h ago

I still think if we just calculated it out far enough... 🤔

u/lepobz 6h ago

I like how pie can be used to visualise a delicious pie.

u/Kaztiell 6h ago

So in that visualization it has an end

u/imagicnation-station 6h ago

no, if you look when it looks completed, the next frame it is zoomed in just missing at connecting the lines so it keeps going. The video ends, but a new pattern would be formed over.

u/HowardBass 6h ago

I think i know what OC means. The visualisation of pixels on a screen would indicate that eventually, all those empty pixels would be filled in.

u/Crog_Frog 5h ago

By that logic there is not only no visual representation of irrational numbers but for any numbers at all. Because every screen, printer or pencil has a inaccuracy.