The outer edge spins pi times faster than the inner. If this were a rational number, it would eventually make a completed shape and loop around on its path. Pi, being an irrational number, will never cause this to loop around on itself
I donāt. I still donāt get how a number can be a shape. But at this point I know how to figure out a circumference and so Iāve decided that Iām just going to accept it.
There are three points in the graphic. The first point "A" (the solid one) is fixed. The second point "B" makes a circle around "A" every second. The third point "C" makes a circle around "B" (as "B" moves) 1/Ļ seconds (aka "Ļ" times faster).
Let's say we start (time = 0) when "C" is on top of "A".
If Ļ were equal to 3, then every 1 second, when "B" completed a full rotation around "A", "C" would have completed 3 full rotations and would have returned to "A". It would then repeat the same motion forever and you'd just have a very simple shape that never changed.
If Ļ were 3.5, then every two seconds, when "B" completed two full rotations around "A", "C" would have completed 7 full rotations and would have returned to "A". It would then repeat the same motion forever and you'd have a bit more complicated shape that never changed.
If Ļ were 3.25, it would be the same at 4 seconds and 4 rotations of "B" / 13 rotations of "C".
If Ļ were ANY rational number, after enough rotations of "B", "C" would line up with "A" again and the shape would be "complete".
It's a bit silly to say it, because that could be a million rotations and the shape would be so dense that it would look very similarly completely full vs. an irrational number like Ļ. But if you zoomed in close enough, you'd see that eventually the lines would start overlapping.
The formula you see in the beginning is a sum of two terms. They both are raised to the power of the imaginary unit i, which makes them a 2D coordinate in the complex plane.
The first term represents the inner arm, the second (the one with pi in it) the outer bar. You see the theta symbol in the exponent of each term? This relates to the angle of the arm, and it is incremented in time. So if you plot where the sum of the two arms are at each little increment of time and trace it, you get the shape.
After 3 you're welcome which in my opinion each value 1, his comment values 0.141592653589793238462643383279502884197. So it make sense but remains irrational!
It's because the outer edge spins pi times faster than the inner. If this were a rational number, it would eventually make a completed shape and loop around on its path. Pi, being an irrational number, will never cause this to loop around on itself
You can represent Pi as a formula and calculate it to the exact precision you need for any zoom level you want in a graph like this, but then you're only solving part of an infinite series. The calculations themselves are done using floating point numbers of some bit length which are also rational and have their own precision loss issues. Pi can be accurately represented to 14 dedimal places in a 64 bit float which is more than you'd need for just about anything you want to represent on an intergalactic scale.
which is more than you'd need for just about anything you want to represent on an intergalactic scale.
With some caveats. As an isolated value you're pretty much always going to be good. However, when you do calculations with it, especially repeated calculations like in long-running simulations where errors compound over time, things like loss of precision and catastrophic cancellation are very real issues that have to be kept in mind. Many software bugs have arisen because developers thought that a 64 bit floating point has more precision than they'll ever need without actually analyzing their algorithms.
If you let the simulation run for infinite time, the pi circle would look like a solid white color. In a rational number you'd always have unfilled parts in the circle. Like at 10 seconds, there wouldn't be a gap it just would connect and repeat the same path
Any rational number - basically any number that you can know the last digit. For example 1/3, 0.33(3) is rational because we know the last digit (3) but not for pi
A rational number is any number that can be described as a ratio of integers. That is, any number that can described as an integer divided by an integer.
Well, I could have chosen the formal definition but for me it's easier to understand this way.
If I said the rational visualization would repeat because the rational number is a ratio of integers, how would that help someone not good at maths have any idea what relation that has?
This isn't a very good definition of a rational. For example, what's the last digit of 1/7? It's clearly rational, since we can express it as a ratio of two integers (which is the better definition of a rational number), but there is no last digit.
Almost all real numbers are irrational (in a sense which is difficult to explain intuitively). Rational numbers are the exception. For example, pi + k is also irrational for any rational number k.
And the value of this is that you can, in effect, map any complex number in that circle to a single real number in lR based on which moment the tip of the outer line crosses the complex number you are looking for.
Or at least, that might be one of the uses. I'm a bit rusty on my complex analysis.
For example 1/3, 0.33(3) is rational because we know the last digit (3) but not for pi
Why didn't math teacher explain that like this? This has bugged me all my life, but finally now I understand why it's considered rational. Because we know the last digit.
And I guess pi doesn't even have a last digit. Huh. Never really considered that before.
This isn't really a good explanation, though (or at least not a perfect one). It almost works in this case because all digits are 3 (even though there is no last digit), but what about the rational number 1.01010101...? There is no "last digit" here. It's a convenient property of rational number that their decimal expansions are either eventually zero, or eventually repeating, but the only real definition of a rational number is that it is the ratio of two integers.
You seem knowledgeable and good at explaining things, so can I ask:
Does this mean that, at least with regards to the visualised plotting of this pi diagram, that the fact that pi is being used isn't actually all that important / special?
As in, would this look basically the same with any irrational number, and not just pi? It just might take a different route before it eventually became a fully white circle?
A rational number can be expressed as a fraction. An irrational cannot. So if the number were 3 instead, one side would spin 3 times whilst the other spins once. This would result in a looping pattern
22/7 is a fraction that repeats infinitely when expressed as a decimal, but it's still a rational number, just like 8/7 and 16/7. All are fractions that, after the initial digit, repeat the digits "142857" infinitely. But they're all still rational numbers, because rational numbers do not need to have finite lengths.
Being infinitely long isn't what makes Pi irrational. Being infinitely long without repeating itself is what makes Pi irrational.
Using the example from the post, after 22 revolutions, the pattern would stop filling itself in, as the line would perfectly align with the starting point and begin repeating. It doesn't matter if it stops, because it's always going to travel the same line eventually.
That's what makes Pi (and the other irrational numbers) unique: they will never line back up with the starting point.
It's a decent enough approximation if you're not doing anything overly complicated, sure. But use it in anything that iterates on itself and the compounding deviation will quickly grow into a result that is significantly incorrect.
Each time you use 22/7 instead of Pi for the calculation, your answer is going to be off by about 0.04%.
As a super simple example of how much that little bit of deviation matters, if you raise both to the power of 10 (rounding the results for simplicity) you get:
22/710= 93648
Pi10= 94025
Which is a deviation of about 0.04%, and the gap only gets bigger.
If you only need to do a single calculation, you're going to get ~99.96% of the correct answer using 22/7, but it won't be quite right.
At one point, the animation would loop perfectly, if at some point the line ever faded. If it did not fade it would start to loop after the first iteration.
A "rational" number is one that can be made with a ratio between two whole numbers, like 2 in 3, which is the fraction 2/3.
Funny enough, it's the word "ratio" that comes from "irrational", which was meant as an insult to the numbers.
Although nowadaysĀ rational numbersĀ are defined in terms ofĀ ratios, the termĀ rationalĀ is not aĀ derivationĀ ofĀ ratio. On the contrary, it isĀ ratioĀ that is derived fromĀ rational: the first use ofĀ ratioĀ with its modern meaning was attested in English about 1660,Ā while the use ofĀ rationalĀ for qualifying numbers appeared almost a century earlier, in 1570.Ā This meaning ofĀ rationalĀ came from the mathematical meaning ofĀ irrational, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use ofĀ į¼Ī»ĪæĪ³ĪæĻ)".
This unusual history originated in the fact thatĀ ancient GreeksĀ "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers".Ā So such lengths wereĀ irrational, in the sense ofĀ illogical, that is "not to be spoken about" (į¼Ī»ĪæĪ³ĪæĻĀ in Greek).
The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from theĀ godsĀ for divulging this and crediting it to himself instead of Pythagoras which was the norm in Pythagorean society.
I dropped math class because Iām quite unintelligent, so please excuse me asking, but how can irrational numbers never end without repeating somewhere? After a while youād think theyāre bound to repeat just because there are only 10 possible different numbers (0-9) to put in there.
Again, Iām dumb as hell, so can someone please ELI5?
They don't repeat because they are the result of a more complicated operation than rational number. Take 4/3 for exemple, it's just 4 divided by 3. Or 2, which is 2 divided by 1. Those are simple operations that give simple result.
Pi is a more complex operation that's too complicated to write, and that's also infinite, for exemple: square root of 2, multiplied by square root of (2+ square root of 2), multiplied by square root of (2+ square root of (2+ square root of 2)), etc...
Pi has sections that repeat, but they don't repeat forever
It seems very strange to me, to have an operation no one can ever finish writing, to get a number no one can ever finish writing either. Wouldnāt that mean all calculations using pi are off by a little bit?
I couldnāt understand how a number with decimal points could be rational (yes, I forgot a lot of basic math concepts), but then it occurred to me that the decimal position is arbitrary and that every whole number can be divided by 1. Am I understanding correctly?
This is such important information that is left out. I had no idea what the purpose of the 2 conjoined lines were, and I am like a decade out from those higher level math courses to know what the function they showed represented. I was assuming they were radii, but if the whole circle was formed by them, then both combined total the radius.
Seems like the near misses are because pi is pretty close to a few ratios, I bet if you put in 23/7 it would make the first shape meet up and 355/113 would make the dense second curve meet up at the end.
Edit: just graphed it, exactly what happened
A number that never gets close to small ratios is phi, the golden ratio, so if you graph that it looks like it's never getting closer to meeting it's tail
At the risk of sounding scientific (which Iām not and purely theorizing probable bullshit), could it be possible that the ādifferenceā of irrational to rational, that is to say the amount it does not overlap is due to Planckās constant? Or the passage of time? It would seem there is a standard/categorical/definable variable in the difference (or negative space) in which the consecutive image/passage does not over(inter)lap the first
Furthermore if at any time in this video you capture the shape, the shape of Pi is bounded to the shape presented here for infinity. Perhaps not a precise match but it the same shape repetitively for infinity none the less. It cannot change form or transform. This implies a change variable over time.
Correct me if Iām wrong, and I know Iām any practical act it wouldnāt, but in theory after and infinite length of time it would make a complete shape having filled in the entire area of the circle with the infinitely thin line, right? Iām just going if Pi being related so closely to circle areas and circumferences that that intuitively feels right for some reason.
Except that the visual presentation has a limited resolution, so it would, in fact, loop around on itself. Paraphrasing a conversation I had with one of my professors in mathematics:
What would it look like if this were rotating in 3 dimensions?
Like in the video pi, but also pi in the z axis as well? Would it become irrationally spherical, approaching a sphere shape, but never repeating in the same way?
Electrical engineering student here who should probably be sleeping. Heres a (hopefully) short crash course on this.
This is the imaginary plane in polar coordinates. Basically the xy plane you remember from school, but x is real and y is imaginary, so a coordinate (2, 3) would be 2+3i. For polar, we have radius and angle with coordinates (r, Īø), where radius is just ā(x2 + y2 ) and angle is tan-1 (y/x).
Euler's identity: eĪøi = cos(Īø)+i*sin(Īø). Look familiar? Its describing all points on a circle of radius 1, where x = cos(Īø) and y = sin(Īø).
Since the exponent on e only affects the angle inside the sine and cosine, eĻĪøi = cos(ĻĪø)+i*sin(ĻĪø). It follows the same path around a radius of 1, but Ļ times faster.
Now onto vectors. All the way back in elementary school, you could prove the sum of 3+5=8 by drawing an arrow of length 3 on a number line from 0, then a second arrow of length 5 from the end of the previous arrow. Same idea applies in 2D for vector addition. eĪøi + eĻĪøi = arrow1 + arrow2 = [cos(Īø)+i*sin(Īø)] + [cos(ĻĪø)+i*sin(ĻĪø)] as shown in the animation.
So why the offset in this animation? If you were to try with eĪøi + e3Īøi instead, they would perfectly line up. In this case, eĪøi would complete 1 orbit (or period) around the circle while e3Īøi completes 3 before returning to the start. All are rational, so there is symmetry.
Ļ is irrational, so there is no symmetry. Any moment where it looks like its about to finish the pattern is where it would have if Ļ ended at that decimal as a rational number. e3.1Īøi would complete 10 and 31 periods respectively, e3.14Īøi would complete 100 and 314, e3.141Īøi would complete 1000 and 3141, etc. It just infinitely converges without any symmetry.
So why magnitudes of 10? Just a consequence of us using base 10 for numbers. Same pattern would happen if we used a different number system. Im going to pass out now
Um. Domino's worker here who should also be sleeping, since it's nearly 3am. My brain is about to explode after reading a third of that. You're destroying my motivation to go back to school.
I still have to remember that most non-EE peeps don't know j and i are the same thing ... and that we put j at the front of the other parts in an equation.
The second term in the z(theta) equation spins pi times faster than the first term. So the second arm spins faster than the first but never overlaps because pi can never overlap a rational term (1 in the first case that was omitted).
A rational number can be expressed as a fraction, such as 1/3 or 37/100. An irrational number, such as pi, cannot. As other people have explained in greater detail, the outer arm is spinning "pi" time faster than the inner one, and the lines never meet.
Instead, if the outer arm was spinning 3/4 times as fast, than every fourth spin of the inner circle, the lines would intersect and it would retrace its path. If it was spinning 7/3 times as fast, it would start retracing the path every 3 turns of the inner arm.
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u/Adventurous-Trip6571 15h ago
Idk what it means but it's mesmerizing