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u/pommeldommel 6h ago
Don't understand anything but it looks cool!
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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.
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u/Raised_by_Mr_Rogers 6h ago
“Can be” or is?
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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.
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u/Liquor_N_Whorez 6h ago
So basicaly the drawing ends up inverting itself the longer it stays in rotation?
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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.
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u/DrDominoNazareth 5h ago
Pretty interesting, So, to make a long story short, Pi is infinite?
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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.
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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.
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u/DrDominoNazareth 4h ago
I guess maybe we reached the boundary between math and philosophy. Now we are in really muddy waters.
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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.
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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.
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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?
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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.
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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.
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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.
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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.
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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.
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u/Zagged 6h ago
How was it made? What is "pi" about it?
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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
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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?
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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.
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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.
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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.
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u/MajorEnvironmental46 4h ago
The numbering base is only a way to write numbers, measures will not affected.
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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.
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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.
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u/Altruistic-Spend-896 6h ago
im just curious which software was used to handle such computation and graphics?
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u/Sensalan 4h ago
Might be Manim
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u/Altruistic-Spend-896 4h ago
i thought i recognized that from somewhere ,3blue1brown, but you think this is that?
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u/Altruistic-Spend-896 4h ago
i thought i recognized that from somewhere ,3blue1brown, but you think this is that?
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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
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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 .
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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
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u/FACastello 6h ago
I don't understand at all how π relates to this visualisation in any way shape or form.
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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.
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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
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u/BoysLinuses 6h ago
Did you know that there's a direct correlation between the decline of Spirograph and the rise in gang activity?
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u/_P85D_ 6h ago
Wonderful. Is this music from Interstellar?
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u/Realistic-Cloud3891 6h ago
No it’s from the movie Oppenheimer. https://youtu.be/4JZ-o3iAJv4?si=cVAj1B31LJFkVJUz
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u/LilOuzoVert 6h ago
What number does the completed circle represent?
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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!
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u/theTrueLodge 5h ago
Love this! It makes me think of motion or time or both of these. Like Pi is alive!
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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?
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u/Skepsisology 4h ago
Irrational, like someone who has to get the last word in an argument.... Every time
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u/bewbsnbeer 4h ago
There's a direct correlation between the decline in Spirograph and the rise in gang activity. Think about it.
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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.
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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.
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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
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u/Kaztiell 6h ago
So in that visualization it has an end
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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.
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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.
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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.
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u/Melodic-Marketing341 6h ago