Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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It’s not “only related to a circle”. It figures heavily in Trigonometry, and triangles/angles are absolutely everywhere in science/nature.
It’s not “only related to a circle”. It figures heavily in Trigonometry, and triangles/angles are absolutely everywhere in science/nature.
Any angle that exists can be interpreted as part of a circle so you always have a reference to PI
Any angle that exists can be interpreted as part of a circle so you always have a reference to PI
Okay, other people have already given you nice, simple answers about how circles are fundamental objects, so I hope you’ll permit me to stretch the ELI5 requirement until it writhes in pain.
The true, deep, fundamental reason that pi shows up everywhere is that the *exponential function* shows up everywhere.
The exponential function is a function that is equal to it’s own derivative. If you fix exp(0) = 1, it’s *the only* function equal to it’s own derivative.
This is *huge*. I can’t emphasize enough how fundamental of a fact this is. The exponential function is a *fixed point*, an eigenvector of sorts, of the all-important differentiation operator. That means whenever you’re dealing with *anything* differentiable, any relationship between derivatives of functions, this function exp magically appears. It is often called “the most important function in mathematics”.
The most amazing thing? This exponential function is perfectly well defined on the entire complex plane. Look on the real axis, you get exponential growth/decay. Look on the imaginary axis, you get *cyclic motion*. This function *repeats*, on the imaginary axis, with a period of 2(pi)(i). This period is a fundamental constant of a deeply fundamental function, so therefore pi is a fundamental constant in math.
(The familiar sin and cos functions pop right out from the complex exponential. This is where the famous e^pi ^i +1 = 0 comes from.)
I would say *that’s* the real reason why pi is so universal. That’s why it shows up *everywhere*. It’s because such a vast amount of math boils down to manipulations of the complex exponential function. Sure, pi is also about circles, but that’s because the exponential function is the basic building block of the more fundamental idea of *cyclic motion*, for which circles are a convenient visualization.
Okay, other people have already given you nice, simple answers about how circles are fundamental objects, so I hope you’ll permit me to stretch the ELI5 requirement until it writhes in pain.
The true, deep, fundamental reason that pi shows up everywhere is that the *exponential function* shows up everywhere.
The exponential function is a function that is equal to it’s own derivative. If you fix exp(0) = 1, it’s *the only* function equal to it’s own derivative.
This is *huge*. I can’t emphasize enough how fundamental of a fact this is. The exponential function is a *fixed point*, an eigenvector of sorts, of the all-important differentiation operator. That means whenever you’re dealing with *anything* differentiable, any relationship between derivatives of functions, this function exp magically appears. It is often called “the most important function in mathematics”.
The most amazing thing? This exponential function is perfectly well defined on the entire complex plane. Look on the real axis, you get exponential growth/decay. Look on the imaginary axis, you get *cyclic motion*. This function *repeats*, on the imaginary axis, with a period of 2(pi)(i). This period is a fundamental constant of a deeply fundamental function, so therefore pi is a fundamental constant in math.
(The familiar sin and cos functions pop right out from the complex exponential. This is where the famous e^pi ^i +1 = 0 comes from.)
I would say *that’s* the real reason why pi is so universal. That’s why it shows up *everywhere*. It’s because such a vast amount of math boils down to manipulations of the complex exponential function. Sure, pi is also about circles, but that’s because the exponential function is the basic building block of the more fundamental idea of *cyclic motion*, for which circles are a convenient visualization.
The truly short answer is that π is not just related to circles. It’s a magic number that shows up everywhere in the laws of mathematics, along with e. You just happen to know it best from circles, unless you study more advanced math.
The truly short answer is that π is not just related to circles. It’s a magic number that shows up everywhere in the laws of mathematics, along with e. You just happen to know it best from circles, unless you study more advanced math.
This is extremely tricky to answer as ELI5. The answers you get with “circles are kinda everywhere,” are good, but I think they leave out a significant amount of the mystery.
The ones I saw trying to explain using Euler were not wrong, but quickly left ELI5 territory as well.
I think it’s worth acknowledging that the fact that Pi shows up quite as often as it does *is* surprising. Those of us in mathematics or related fields have gotten so used to the idea that we just take it as a given. That should not take anything away from the strange ability for Pi to show up in the absolutely weirdest places.
For instance, if we were to take the sum of the following numbers: 1/1 + 1/4 + 1/9 + 1/16…, we get the answer: π^(2)/6. I believe that any sane person looking at this should be wondering: where the hell did pi come from? If you are interested in this, you can look up the Basel Problem and find any number of wonderful rabbit holes to go down.
But that is beside the point for the moment. It is just plain *weird* that pi shows up there. Sure, once you start tearing the problem apart, you can see where it eventually creeps in, but I still think it is magical.
So what is the answer? I’m not sure there is one; at least, not one better than the answers saying that circles are everywhere.
It is legitimately strange that i ends up creating a relationship between π and e. It’s there. We can work it out. But it is not like we created i to do this. It just sort of happened. And I sometimes like to sit back and simply marvel at the fact that it did this.
My apologies for not really offering an answer as such. But I really want to emphasize just how wonderful and mysterious that π does end up everywhere, even in spots that would not seem to have anything to do with circles at all. It sometimes makes me wonder if we have mistakenly linked π to circles, simply because this was the first place we ran across it.
This is extremely tricky to answer as ELI5. The answers you get with “circles are kinda everywhere,” are good, but I think they leave out a significant amount of the mystery.
The ones I saw trying to explain using Euler were not wrong, but quickly left ELI5 territory as well.
I think it’s worth acknowledging that the fact that Pi shows up quite as often as it does *is* surprising. Those of us in mathematics or related fields have gotten so used to the idea that we just take it as a given. That should not take anything away from the strange ability for Pi to show up in the absolutely weirdest places.
For instance, if we were to take the sum of the following numbers: 1/1 + 1/4 + 1/9 + 1/16…, we get the answer: π^(2)/6. I believe that any sane person looking at this should be wondering: where the hell did pi come from? If you are interested in this, you can look up the Basel Problem and find any number of wonderful rabbit holes to go down.
But that is beside the point for the moment. It is just plain *weird* that pi shows up there. Sure, once you start tearing the problem apart, you can see where it eventually creeps in, but I still think it is magical.
So what is the answer? I’m not sure there is one; at least, not one better than the answers saying that circles are everywhere.
It is legitimately strange that i ends up creating a relationship between π and e. It’s there. We can work it out. But it is not like we created i to do this. It just sort of happened. And I sometimes like to sit back and simply marvel at the fact that it did this.
My apologies for not really offering an answer as such. But I really want to emphasize just how wonderful and mysterious that π does end up everywhere, even in spots that would not seem to have anything to do with circles at all. It sometimes makes me wonder if we have mistakenly linked π to circles, simply because this was the first place we ran across it.
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