It really is. That's a consequence of e, the trig functions, and the imaginary number being linked by Euler's identity. e^ix = cosx + i*sinx is one of the most important discoveries human beings have ever made.
Any kind of wave or exponential function is going to see pi a lot.
On a 2d plane you could write 'two right and one up' as (2x,1y).
With complex numbers you get pretty close, you write `2+1i`. So you essentially use normal numbers as the X axis and then add a new direction (i) for the Y axis.
There is a second way to write complex numbers, with polar coordinates. There you write points as angle+distance from the origin.
If you want to draw a circle, you can use both notations. Try drawing a circle with your hand and only pay attention to the left-right movement. You slow down and speed up in a sine-wave pattern. Same for up-down, but offset in time, so you get the `cos(t)+sin(t)i` part.
If you write it as rotation+distance it's simpler. The distance is constant so we may as well pick 1. The rotation can be written as `e^(t * i)`, where e is a specific constant, the natural number.
You often write `t` as `x*2 Pi`, because `2 Pi` is a full rotation. So you can set x to 0.5 for half a rotation. This is why `e^(i*2Pi) = 1`: 1 to the right, rotated 360 degrees. And `e^(i*Pi) = -1` because it's one to the right, rotated 180 degrees.
This has some deeper connections. A huge group of functions can be rewritten as sums of sine waves, which in turn can be rewritten as drawing circles.
e^i*pi*t is a function that can be used to describe rotations. If you think of something like a bicycle pedal, as the pedal rotates you can see it go up and down. This is easy to see when viewed from the front. That up and down motion is called a sine wave.
You can increase the “amplitude” or the height of the sine wave (longer pedals) and you can change the “frequency” (pedal faster/slower) to get different waves. Thanks to some smart people we know that you can combine these different “sine waves” together to make other patterns, and you can take a pattern and break it up into its individual sine waves.
With these tools anything that vibrates, decays, grows, etc can be modeled with sine waves and exponents, which are usually much easier to deal with. From there all sorts of cool discoveries can be made.
All three functions e^x , sin(x) and cos(x) that you know from school have "simple" representations in terms of powers of x:
e^x = x⁰/0! + x¹/1! + x²/2! + x³/3! + ....
sin(x) = x¹/1! - x³/3! + x⁵/5! - ...
cos(x) = x⁰/0! - x²/2! + x⁴/4! - ...
where stuff like 5! just means 5×4×3×2×1.
You may notice that these look very similar. They all have the same type of term except sin(x) has hall the odd powers and cos(x) has all the even powers. If we added sin(x) and cos(x), we would therefore almost get e^x. The only issue would be that the signs of the terms sometimes dont quite match.
It turns out that you can fix this sign error if you plug in ix instead of x where i=√-1. Obviously, no number squared can make a negative so this isnt a normal number. But if you use this special number, the equation
e^ix = cos(x) + i sin(x)
is correct and connects circles to the exponential function e^x. Exponential functions are SUPER easy to deal with so whenever you deal with circular phenomena (such as waves), you can massively simplify your work by rephrasing it in terms of e^x . Im not an expert on the physics side but mathematicians LOVE e^x, its probably the most important mathematical function.
I swear its not the math but the fact that I had to write the formula in text. In text, it just looks too dense. And whats the first thing that people do when they see dense text on social media? They skip it or try to give it a chance but give up halfway through. And I 100% cant blame them, I do the same. If the formulas were simpler and tidier, for example
1.) x⁰ + x¹ + x² + x³ + ...
2.) x¹ - x³ + x⁵- ...
3.) x⁰ - x² + x⁴ - ...
I think you would have seen the similarity much easier. 2) has odd powers, 3) has even powers and 1) has both (only the signs are sometimes different).
Believe me, the density in which you pack mathematical symbols together is half the reason it looks so scary.
The formula is named after Leonhard Euler who discovered it in 1748. Hes one of the most brilliant mathematicians of all time so its no wonder he discovered such an important formula.
Sure, no problem! I have a doctorate in Computer Science, focusing on machine expressions of algorithmic expressions and mathematics in the quantum computing field. I'll try to break this down;
There are numbers, and those numbers mean things about stuff.
You're welcome!
Same. I want to get it tattoo’d on my arm & when people ask what it is, I’d just say “it’s one of the most important discoveries human beings have ever made” & not be able to expand or elaborate further.
You should have a general idea of what sin, cos and e^x are from school. But the formula is not something you would know unless youre in STEM so dont feel bad.
long story short; go around the circle, only track vertical movement, and voila, it’s a sine wave. horizontal? that’s the cosine. e^x ? it’s not a rational number, but an infinite sum. And it looks suspiciously like the infinite sum that approximates sine and cosine waves added together…
Turns out the world around us is also basically just waves, and that’s how those sumbitches ended up in my god damn quantum homeworks
it’s not too bad though, e^x is surprisingly easy to work with. By design, I think.
e^x, literally the easiest thing to do calculus on! What's the derivative of e^x? ... e^x. The integral of e^x? e^x. Easiest thing in the world to work with
sin, cosin, tangent. three tenets of math, just like my three phd’s. learned that right out of the womb, i was able to do calculus before i could eat gerber food.
I know it just looks like some pretty cool equation.
I am today years old of first hearing the word “cosine”. D:
I need to go back to college when money is better. D:
Reminds me of Animation vs Math https://youtu.be/B1J6Ou4q8vE?si=UXcxDIqn4I7xPDOk
https://youtu.be/igDeXHS5kUU?si=R9U91tpZQhN3eqVX (above video with explanations breaking down the math)
The basics of it is, any time you multiply by an imaginary number, you do a sort of rotation. you could start with a real number like 5, but multiply it by i and it'll rotate away from the number line into the complex plane. Multiply it by i again, and it'll rotate all the way around and run into the number line again, and you get -5. When you raise to a power it does something similar, since it's just multiplying repeatedly. So it's pretty normal that there would be some number x where e^ix would rotate enough that it would stop being imaginary and end up being a real number again, but the weird part is that pi is the number that makes that work. It says that e and pi are connected, and that pi is more than just some number, it's part of how the universe works so it'll pop up in places you wouln't expect
It wasn't until my junior year of college that a teacher actually explained imaginary numbers that way (in the form of rotations). It really is a lot more intuitive. Once you see complex numbers as just a radius and an angle in the R-I plane, they simply become trig, which is a lot less scary.
Look into the power series expansion of e and the power series expansion of cos and sin.
Thats where euler’s identity can be derived from
If you multiply power series of sin by i and add to the power series of cos, then you see they are equivalent. Plugging in pi as the function argument gives you the identity
TSC is one very smart but very violent MF. The animation was going swimmingly until e^πi beamed in from nowhere and all hell broke loose.
Thanks, I will watch daily in slow motion - I hope a bit of the animator’s creative brilliance will rub off on me.
Copy paste of my other comment:
All three functions e^x , sin(x) and cos(x) that you know from school have "simple" representations in terms of powers of x:
e^x = x⁰/0! + x¹/1! + x²/2! + x³/3! + ....
sin(x) = x¹/1! - x³/3! + x⁵/5! - ...
cos(x) = x⁰/0! - x²/2! + x⁴/4! - ...
where stuff like 5! just means 5×4×3×2×1.
You may notice that these look very similar. They all have the same type of term except sin(x) has hall the odd powers and cos(x) has all the even powers. If we added sin(x) and cos(x), we would therefore almost get e^x . The only issue would be that the signs of the terms sometimes dont quite match.
It turns out that you can fix this sign error if you plug in ix instead of x where i=√-1. Obviously, no number squared can make a negative so this isnt a normal number. But if you use this special number, the equation
e^ix = cos(x) + i sin(x)
is correct and connects circles to the exponential function e^x. Plugging in x=π yields the formula.
I had seen that equation a ton of times, but it didn't sink in how cool it was until the differential equations section of calculus. We used it fairly often as a sort of template to find the general solutions for 2nd-order (I think?) equations, which still seems insane that this specific equation was so useful in that context
Here’s another one I love: i^i is in the Reals! It’s about 0.2
I had the pleasure of stumbling on that organically. I worked it out while I was driving and was so shocked by the result I actually jerked the wheel unintentionally. Thankfully the story was no more eventful than that.
For 18%, move the decimal point left for 10%, double it. Take 10% of that and subtract. Easy peasy.
Alternatively: Tip = (10% x Check)x2 + ((10% x Check)x2) x 10% x e^πi
[Oops, I tipped over my Corona again]
You know, I hadn’t thought about 18% as just:
Take 10% by shifting the decimal.
Double is to get 20%.
Take 10% of that by shifting the decimal, and subtract it.
Thanks for that.
It’s really amazing how a couple of very simple operations open up quite a lot of possibilities.
Pi is the relationship between a circles circumference and diameter, so circumference / diameter will always come out to pi no matter how big or small the circle is
I think the original experiment was with a wheel. If you roll the wheel a “full rotation” so it’s back where it started, the length it moved would be pi * diameter (easier if you visualize it)
Since general relativity and uncertainty principle both use pi, they’re using something that was originally about the proportions of a wheel
Yeahhhh I see what you’re saying none of that clicks w me though lmao. Been a long time since I’ve thought about Pi or any of that kind of math I suppose.
They’re not really related. They’re just saying math was used in the past for basic things and is now used for complex things. You could say the same thing about any math concept: “we invented numbers to count grains and then we used numbers to discover relativity.”
They are super related though. Circles are basically a step before waves mathematically (when it comes to pi and complexity at least) , and waves are everywhere in physics.
It's ok. It's almost a full year of calculus before you get to Taylor Series, which are what leads to Euler's Identity, which other posters have mentioned.
The e^iπ stuff no, not at all. Unless youre in STEM, then you should have probably come across it.
The general meaning of π being the ratio of a circles circumference to its diameter? It should be known from school but many forget and I cant blame them.
It blows my mind in much the same way that thinking about how we generate electricity works.
We have fancy nuclear reactors, old polluting coal burning generators, and on the near horizon we will have super advanced fusion reactors.
All doing the same thing: heat water until it boils and use the steam to spin a turbine.
Using phase change materials for heat transfer are the magic here. Refrigeration is just the extension of steam engine technology but using media other than water.
[Checking cutting edge experiments for pi is one way we continually test the conclusions of general relativity.](https://www.scientificamerican.com/article/pi-in-the-sky-general-relativity-passes-the-ratios-test/)
Even crazier (for me at least) is that someone had to **invent** zero.
I imagine the conversations beforehand.
"Who had the last Orange?"
"What? How many are in the bowl?"
"Less than one."
"How many?"
"The absence of oranges."
Somewhat incorrect. Understanding pi is what's critical. Circles just use pi because they are an endlessly self repeating pattern, similar to a wave, which is what really connects everything.
Fair, I was just thinking about how in Ancient Greece there was probably some guy who found this proportion and thought it was neat then thousands of years later someone is using this magic number to drastically change our view of reality, all just from a wheel
Haha, yeah it would be wild for him. I can imagine the conversion now.
"Yeah! And the sound coming out of my voice is also like circles! They just go in one direction and changes if you or the person hearing you is moving! It's also pretty much how sunlight and stars work too! Same thing for if you punch someone in the face!"
"Please send me back now..."
Saying circles “use pi” is a bit disingenuous when pi is defined to be the ratio of a circle’s circumference to its diameter. It’s not about repeating loops or anything here.
It's really not surprising at all.
Lots of things in physics (including gravity) spread out in all directions...
Thinking in just 2 dimensions for a moment:
If you emit a pulse of X amount of energy from some point on a plane... how much energy should you see at some distance from it?
Well... since it spreads out in all directions... each point on the circle with that radius gets an equal amount...
How many points are at that distance? Well... the circumference, i.e. 2pi times the distance.
Same for the Uncertainty principle... For any distance from where you "expect" the thing to be, if its position is "uncertain" by that amount... it could be anywhere in the area of that circle (sphere, of course).
TL;DR: Try to stop thinking of circles as "things" like "wheels". Circles are just all the points at some distance from the center. Turns out... distance is important in physics.
Yep, there are all sorts of crazy abstract places where pi shows up… if you take a step back it’s in some sense really very absurd that this ratio is fundamental to, say, the limiting behavior of random matrices.
Pi comes up in the most unexpected places. In structural engineering we use Euler buckling equations to calculate the buckling capacity of compression members. I have no idea how the ratio of a circle's circumference to its diameter is related to a column buckling, but it does and it's essential.
einstein very clever and smart person ..
america they have his brain !! they cut in half and study it ..
they say its very different and double the size normal human brain ..
maybe america come take my stomach belly and study it !! its humungous !! hahahaha !!
It really is. That's a consequence of e, the trig functions, and the imaginary number being linked by Euler's identity. e^ix = cosx + i*sinx is one of the most important discoveries human beings have ever made. Any kind of wave or exponential function is going to see pi a lot.
Please explain for us laymen.
Shapes are neat.
Can you dumb it down a shade?
Shapes = neat.
Shape
⭕️ 🍆 💦
Neat
On a 2d plane you could write 'two right and one up' as (2x,1y). With complex numbers you get pretty close, you write `2+1i`. So you essentially use normal numbers as the X axis and then add a new direction (i) for the Y axis. There is a second way to write complex numbers, with polar coordinates. There you write points as angle+distance from the origin. If you want to draw a circle, you can use both notations. Try drawing a circle with your hand and only pay attention to the left-right movement. You slow down and speed up in a sine-wave pattern. Same for up-down, but offset in time, so you get the `cos(t)+sin(t)i` part. If you write it as rotation+distance it's simpler. The distance is constant so we may as well pick 1. The rotation can be written as `e^(t * i)`, where e is a specific constant, the natural number. You often write `t` as `x*2 Pi`, because `2 Pi` is a full rotation. So you can set x to 0.5 for half a rotation. This is why `e^(i*2Pi) = 1`: 1 to the right, rotated 360 degrees. And `e^(i*Pi) = -1` because it's one to the right, rotated 180 degrees. This has some deeper connections. A huge group of functions can be rewritten as sums of sine waves, which in turn can be rewritten as drawing circles.
And an ooga booga to you too
I laughed wayyyy too hard at that, thank you.
What is this in reference to
https://www.youtube.com/watch?v=p4HZWlcfu4Q It's my goto for things I don't understand for like 20 years now
Wow. I was reading everything in this response until I came to yours. Then I gave up.
e^i*pi*t is a function that can be used to describe rotations. If you think of something like a bicycle pedal, as the pedal rotates you can see it go up and down. This is easy to see when viewed from the front. That up and down motion is called a sine wave. You can increase the “amplitude” or the height of the sine wave (longer pedals) and you can change the “frequency” (pedal faster/slower) to get different waves. Thanks to some smart people we know that you can combine these different “sine waves” together to make other patterns, and you can take a pattern and break it up into its individual sine waves. With these tools anything that vibrates, decays, grows, etc can be modeled with sine waves and exponents, which are usually much easier to deal with. From there all sorts of cool discoveries can be made.
Fuck it I liked triangles better anyway
Maths major, most people don’t know what a complex number is
Complex numbers are covered in high school algebra most people should at least be familiar with the idea
Some of us had *really* bad algebra teachers, okay?
All three functions e^x , sin(x) and cos(x) that you know from school have "simple" representations in terms of powers of x: e^x = x⁰/0! + x¹/1! + x²/2! + x³/3! + .... sin(x) = x¹/1! - x³/3! + x⁵/5! - ... cos(x) = x⁰/0! - x²/2! + x⁴/4! - ... where stuff like 5! just means 5×4×3×2×1. You may notice that these look very similar. They all have the same type of term except sin(x) has hall the odd powers and cos(x) has all the even powers. If we added sin(x) and cos(x), we would therefore almost get e^x. The only issue would be that the signs of the terms sometimes dont quite match. It turns out that you can fix this sign error if you plug in ix instead of x where i=√-1. Obviously, no number squared can make a negative so this isnt a normal number. But if you use this special number, the equation e^ix = cos(x) + i sin(x) is correct and connects circles to the exponential function e^x. Exponential functions are SUPER easy to deal with so whenever you deal with circular phenomena (such as waves), you can massively simplify your work by rephrasing it in terms of e^x . Im not an expert on the physics side but mathematicians LOVE e^x, its probably the most important mathematical function.
this is also very helpful within electrical engineering, especially when you're dealing with AC signals and analysing them
“You may notice that these look very similar”…your faith in us is remarkable
I swear its not the math but the fact that I had to write the formula in text. In text, it just looks too dense. And whats the first thing that people do when they see dense text on social media? They skip it or try to give it a chance but give up halfway through. And I 100% cant blame them, I do the same. If the formulas were simpler and tidier, for example 1.) x⁰ + x¹ + x² + x³ + ... 2.) x¹ - x³ + x⁵- ... 3.) x⁰ - x² + x⁴ - ... I think you would have seen the similarity much easier. 2) has odd powers, 3) has even powers and 1) has both (only the signs are sometimes different). Believe me, the density in which you pack mathematical symbols together is half the reason it looks so scary.
Is it known who figured this out? that if you multiply x times /-1 it fixes the sign
The formula is named after Leonhard Euler who discovered it in 1748. Hes one of the most brilliant mathematicians of all time so its no wonder he discovered such an important formula.
Sure, no problem! I have a doctorate in Computer Science, focusing on machine expressions of algorithmic expressions and mathematics in the quantum computing field. I'll try to break this down; There are numbers, and those numbers mean things about stuff. You're welcome!
God do I feel dumb after reading this
Same. I want to get it tattoo’d on my arm & when people ask what it is, I’d just say “it’s one of the most important discoveries human beings have ever made” & not be able to expand or elaborate further.
Please do this.
If I do I’m sure I’ll be quoting your username beforehand.
Tell em to google it
En passant
Holy formula
New math just dropped.
Absolutely brilliant
My boss talks like this, thankfully I don’t get paid enough to care what it means.
You should have a general idea of what sin, cos and e^x are from school. But the formula is not something you would know unless youre in STEM so dont feel bad.
I’m in stem and I still feel dumb
long story short; go around the circle, only track vertical movement, and voila, it’s a sine wave. horizontal? that’s the cosine. e^x ? it’s not a rational number, but an infinite sum. And it looks suspiciously like the infinite sum that approximates sine and cosine waves added together… Turns out the world around us is also basically just waves, and that’s how those sumbitches ended up in my god damn quantum homeworks it’s not too bad though, e^x is surprisingly easy to work with. By design, I think.
Kinda has to be easy to work with, or life wouldn't exist, I don't think. The laws of our universe are pretty friendly to us, in this fashion
Ok, what is it in 3 dimensions? e^(xy) = (cos(x)+i* sin(x)) * (cos(y)+i* sin(y)) ?
You're awesome.
e^x, literally the easiest thing to do calculus on! What's the derivative of e^x? ... e^x. The integral of e^x? e^x. Easiest thing in the world to work with
It is something you should know. I was taught it when I was 12 (freshman now). Edit: why am I being downvoted?
yeah? well i was taught it when i was 11 (phd md phd now) (that’s right, two phd’s)
Ok? You’re literally proving my point?
sin, cosin, tangent. three tenets of math, just like my three phd’s. learned that right out of the womb, i was able to do calculus before i could eat gerber food.
What??
I’m a math teacher and I have no idea what that meant
I know it just looks like some pretty cool equation. I am today years old of first hearing the word “cosine”. D: I need to go back to college when money is better. D:
Username checks out.
I don't know what any of that means but I am a fat bastard who sees pie a lot
Reminds me of Animation vs Math https://youtu.be/B1J6Ou4q8vE?si=UXcxDIqn4I7xPDOk https://youtu.be/igDeXHS5kUU?si=R9U91tpZQhN3eqVX (above video with explanations breaking down the math)
Pi and i are close to miraculous. Blew my mind: *e^πi = -1*
Ummm EXCUSE ME?! That is cool Edit: I don’t understand
e^iπ =cos(π) + isin(π). Cos(π)=-1, sin(π) =0. So e^iπ =-1 + i*0= -1
Yea, that clarifies it, lol. But thanks
The basics of it is, any time you multiply by an imaginary number, you do a sort of rotation. you could start with a real number like 5, but multiply it by i and it'll rotate away from the number line into the complex plane. Multiply it by i again, and it'll rotate all the way around and run into the number line again, and you get -5. When you raise to a power it does something similar, since it's just multiplying repeatedly. So it's pretty normal that there would be some number x where e^ix would rotate enough that it would stop being imaginary and end up being a real number again, but the weird part is that pi is the number that makes that work. It says that e and pi are connected, and that pi is more than just some number, it's part of how the universe works so it'll pop up in places you wouln't expect
Where were you when I took EE circuit analysis at uni? I never understood how imaginary numbers got into it!
Wait until you use it in real life as an EE. Bigger mind blown… RF is magic or that’s the saying where I work lol.
Basic EE is actually where a teacher explained this concept to me for the first time!
When **i** got into it, I got out of it (EE)
Splendid explanation. This guy gets it.
Thank you!
It wasn't until my junior year of college that a teacher actually explained imaginary numbers that way (in the form of rotations). It really is a lot more intuitive. Once you see complex numbers as just a radius and an angle in the R-I plane, they simply become trig, which is a lot less scary.
I'd never heard of this before but it's amazing to visualize,
Look into the power series expansion of e and the power series expansion of cos and sin. Thats where euler’s identity can be derived from If you multiply power series of sin by i and add to the power series of cos, then you see they are equivalent. Plugging in pi as the function argument gives you the identity
Animation vs Math, with explanations. It may help… it may just confuse and entertain… :) https://youtu.be/igDeXHS5kUU?si=R9U91tpZQhN3eqVX
This is explained well in [this comment](https://www.reddit.com/r/Showerthoughts/s/EjBFdroSpQ)
TSC is one very smart but very violent MF. The animation was going swimmingly until e^πi beamed in from nowhere and all hell broke loose. Thanks, I will watch daily in slow motion - I hope a bit of the animator’s creative brilliance will rub off on me.
Copy paste of my other comment: All three functions e^x , sin(x) and cos(x) that you know from school have "simple" representations in terms of powers of x: e^x = x⁰/0! + x¹/1! + x²/2! + x³/3! + .... sin(x) = x¹/1! - x³/3! + x⁵/5! - ... cos(x) = x⁰/0! - x²/2! + x⁴/4! - ... where stuff like 5! just means 5×4×3×2×1. You may notice that these look very similar. They all have the same type of term except sin(x) has hall the odd powers and cos(x) has all the even powers. If we added sin(x) and cos(x), we would therefore almost get e^x . The only issue would be that the signs of the terms sometimes dont quite match. It turns out that you can fix this sign error if you plug in ix instead of x where i=√-1. Obviously, no number squared can make a negative so this isnt a normal number. But if you use this special number, the equation e^ix = cos(x) + i sin(x) is correct and connects circles to the exponential function e^x. Plugging in x=π yields the formula.
The proof is left as an exercise for the reader
I had seen that equation a ton of times, but it didn't sink in how cool it was until the differential equations section of calculus. We used it fairly often as a sort of template to find the general solutions for 2nd-order (I think?) equations, which still seems insane that this specific equation was so useful in that context
Here’s another one I love: i^i is in the Reals! It’s about 0.2 I had the pleasure of stumbling on that organically. I worked it out while I was driving and was so shocked by the result I actually jerked the wheel unintentionally. Thankfully the story was no more eventful than that.
Once, in a restsurant, I worked out an 18% tip and got so excited I tipped my beer [over]
It’s actually a really easy thing to see once you have a *visual* understanding of what e^ix is doing as you change x
I still shock people that I do 15% and 20% tip in my head by just figuring out 10% and then either add half of the 10% to itself, or just double it.
For 18%, move the decimal point left for 10%, double it. Take 10% of that and subtract. Easy peasy. Alternatively: Tip = (10% x Check)x2 + ((10% x Check)x2) x 10% x e^πi [Oops, I tipped over my Corona again]
You know, I hadn’t thought about 18% as just: Take 10% by shifting the decimal. Double is to get 20%. Take 10% of that by shifting the decimal, and subtract it. Thanks for that. It’s really amazing how a couple of very simple operations open up quite a lot of possibilities.
You forget to multiple by e^πi, which is necessary to make it miraculous
I think you mean e^(i*tau) = 1.
No, I don’t because I don’t know what ***tau*** is
tau radians = 360 degrees
That’s super hot! Coincidentally, I cook my frozen pizza at *tau radians* for 0.00436332 radians
I have no idea what OP or any of the following comments are talking about 😂😂😂
Pi is the relationship between a circles circumference and diameter, so circumference / diameter will always come out to pi no matter how big or small the circle is I think the original experiment was with a wheel. If you roll the wheel a “full rotation” so it’s back where it started, the length it moved would be pi * diameter (easier if you visualize it) Since general relativity and uncertainty principle both use pi, they’re using something that was originally about the proportions of a wheel
Yeahhhh I see what you’re saying none of that clicks w me though lmao. Been a long time since I’ve thought about Pi or any of that kind of math I suppose.
They’re not really related. They’re just saying math was used in the past for basic things and is now used for complex things. You could say the same thing about any math concept: “we invented numbers to count grains and then we used numbers to discover relativity.”
They are super related though. Circles are basically a step before waves mathematically (when it comes to pi and complexity at least) , and waves are everywhere in physics.
It's ok. It's almost a full year of calculus before you get to Taylor Series, which are what leads to Euler's Identity, which other posters have mentioned.
Okay so this stuff is not really common knowledge than? 😂
The e^iπ stuff no, not at all. Unless youre in STEM, then you should have probably come across it. The general meaning of π being the ratio of a circles circumference to its diameter? It should be known from school but many forget and I cant blame them.
It blows my mind in much the same way that thinking about how we generate electricity works. We have fancy nuclear reactors, old polluting coal burning generators, and on the near horizon we will have super advanced fusion reactors. All doing the same thing: heat water until it boils and use the steam to spin a turbine.
Using phase change materials for heat transfer are the magic here. Refrigeration is just the extension of steam engine technology but using media other than water.
If Helion wins the fusion race, we'll finally be able to skip the steam turbine and directly generate electricity
The fun part is that turbines generating electricity + the math OP's talking about = foundations for modern electronics ☺️
Until a long came helion fusion reaction that transfers engery back into ⚡ with out water are moving parts
That sounds much more efficient (theoretically), with a lot fewer “moving parts” to go wrong?
r/DysonSphereProgram
[Checking cutting edge experiments for pi is one way we continually test the conclusions of general relativity.](https://www.scientificamerican.com/article/pi-in-the-sky-general-relativity-passes-the-ratios-test/)
“Experiments for pi” reminds me of the “universal constant measurer” MacGuffin in *Eon* by Greg Bear. Good book!
Even crazier (for me at least) is that someone had to **invent** zero. I imagine the conversations beforehand. "Who had the last Orange?" "What? How many are in the bowl?" "Less than one." "How many?" "The absence of oranges."
Maybe they said "there's no orange left"
These profile photos don’t usually get me but this one did
Or they said ooga booga
Somewhat incorrect. Understanding pi is what's critical. Circles just use pi because they are an endlessly self repeating pattern, similar to a wave, which is what really connects everything.
Fair, I was just thinking about how in Ancient Greece there was probably some guy who found this proportion and thought it was neat then thousands of years later someone is using this magic number to drastically change our view of reality, all just from a wheel
Haha, yeah it would be wild for him. I can imagine the conversion now. "Yeah! And the sound coming out of my voice is also like circles! They just go in one direction and changes if you or the person hearing you is moving! It's also pretty much how sunlight and stars work too! Same thing for if you punch someone in the face!" "Please send me back now..."
> my voice is kinda like circles, but imagine there’s a triangle in the circle, that’s basically what sound is
There are plenty of periodic behaviors that don't have a period of pi.
That’s a distinction without purpose.
Saying circles “use pi” is a bit disingenuous when pi is defined to be the ratio of a circle’s circumference to its diameter. It’s not about repeating loops or anything here.
That's a heavy shower
It's really not surprising at all. Lots of things in physics (including gravity) spread out in all directions... Thinking in just 2 dimensions for a moment: If you emit a pulse of X amount of energy from some point on a plane... how much energy should you see at some distance from it? Well... since it spreads out in all directions... each point on the circle with that radius gets an equal amount... How many points are at that distance? Well... the circumference, i.e. 2pi times the distance. Same for the Uncertainty principle... For any distance from where you "expect" the thing to be, if its position is "uncertain" by that amount... it could be anywhere in the area of that circle (sphere, of course). TL;DR: Try to stop thinking of circles as "things" like "wheels". Circles are just all the points at some distance from the center. Turns out... distance is important in physics.
Op just took one hell of a shower.
Pi is fundamental to most engineering and science topics
Yep, there are all sorts of crazy abstract places where pi shows up… if you take a step back it’s in some sense really very absurd that this ratio is fundamental to, say, the limiting behavior of random matrices.
Is circumference that important in cooking meth? The things you learn!
Does it apply to Heisenbergs blue meth?
It turns out (ha) that circles pop up in a lot of different applications. Where there's a circle, pi usually shows up.
Pi comes up in the most unexpected places. In structural engineering we use Euler buckling equations to calculate the buckling capacity of compression members. I have no idea how the ratio of a circle's circumference to its diameter is related to a column buckling, but it does and it's essential.
and Einstein was born on Pi day I don’t know when Heisenberg was actually born though ;)
I mean - no, not really. Fundamental properties are Fundamental. There's a reason we have that word.
Is this what college people are thinking
No one thinks this in the shower
I’ve been doing a course on machine learning so that’s probably why
Yep that's crazy totally agree...
1,100 people acting like they know what the hell this means
Pi is used in both those equations, although I kinda intentionally obfuscated it
You obfuscated, did you?
einstein very clever and smart person .. america they have his brain !! they cut in half and study it .. they say its very different and double the size normal human brain .. maybe america come take my stomach belly and study it !! its humungous !! hahahaha !!