Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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“go straight, turn left, go straight, turn left, and repeat” the ratio between how much to go straight and turn is half of pi. And because the ratio of going straight and turning appears in a lot of equations pi appears in a lot of equations.
If you were to “go straight, turn left, go straight, turn right, and repeat” that’s parabolic, which also shows up a lot.
“go straight, turn left, go straight, turn left, and repeat” the ratio between how much to go straight and turn is half of pi. And because the ratio of going straight and turning appears in a lot of equations pi appears in a lot of equations.
If you were to “go straight, turn left, go straight, turn right, and repeat” that’s parabolic, which also shows up a lot.
There was a very influential mathematician, named [Euler](https://en.wikipedia.org/wiki/Leonhard_Euler), that helped write down much of the fundamentals that we use in modern calculus. One of his contributions was using *i* for the imaginary numbers, and he was also the guy that started actually using pi (the letter) to represent pi (the number).
In his work, he used circles to simplify many of the complex parts of imaginary numbers. This allowed him to use functions that work with circles (sine, cosine, tangent) to relate the two different types of numbers, even if it required him to use pi all over the place. It’s how we know that 3 numbers that are either irrational or imaginary combine into -1: [e^(pi*i)+1=0](https://en.wikipedia.org/wiki/Euler%27s_identity). Because it was so effective, the formulas he came up with are still in use even when people aren’t using circles or imaginary numbers. You see this as pi being everywhere without a circle, but back in the day he needed the circle to come first.
There was a very influential mathematician, named [Euler](https://en.wikipedia.org/wiki/Leonhard_Euler), that helped write down much of the fundamentals that we use in modern calculus. One of his contributions was using *i* for the imaginary numbers, and he was also the guy that started actually using pi (the letter) to represent pi (the number).
In his work, he used circles to simplify many of the complex parts of imaginary numbers. This allowed him to use functions that work with circles (sine, cosine, tangent) to relate the two different types of numbers, even if it required him to use pi all over the place. It’s how we know that 3 numbers that are either irrational or imaginary combine into -1: [e^(pi*i)+1=0](https://en.wikipedia.org/wiki/Euler%27s_identity). Because it was so effective, the formulas he came up with are still in use even when people aren’t using circles or imaginary numbers. You see this as pi being everywhere without a circle, but back in the day he needed the circle to come first.
If pi shows up in the solution to a problem, this means that you can somehow transform the problem into a problem about circles.
In the famous Euler’s formula (e^(i*pi) = -1), the circle is quite obvious: Travelling halfway around the unit circle on the complex plane lands you at -1. (though the tricky part is showing why the e^i means travelling along the unit circle in the first place). For [other problems that involve pi](https://youtu.be/d-o3eB9sfls) the circle is a bit harder to find – but it’s there.
If pi shows up in the solution to a problem, this means that you can somehow transform the problem into a problem about circles.
In the famous Euler’s formula (e^(i*pi) = -1), the circle is quite obvious: Travelling halfway around the unit circle on the complex plane lands you at -1. (though the tricky part is showing why the e^i means travelling along the unit circle in the first place). For [other problems that involve pi](https://youtu.be/d-o3eB9sfls) the circle is a bit harder to find – but it’s there.
I have used 3.14 for calculating the volume of a cylindrical tank in cubic feet, and then Google converts it over to gallons.
Also, when something cylindrical was hard to measure its diameter, I wrapped a string around it, then measured the distance between the marks, divide by 3.14, and that was the diameter.
I have used 3.14 for calculating the volume of a cylindrical tank in cubic feet, and then Google converts it over to gallons.
Also, when something cylindrical was hard to measure its diameter, I wrapped a string around it, then measured the distance between the marks, divide by 3.14, and that was the diameter.
One big reason is that the universe is essentially radially symmetric. In the grand scheme of things, forces will act in all directions without preference. This behavior favors spheres as the ideal shape in the universe to balance against forces.
It’s not “only related to a circle”. It figures heavily in Trigonometry, and triangles/angles are absolutely everywhere in science/nature.
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