Why did the first person even ask what it was?
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I don’t think it’s something specifically known, but typically with these things the first use of mathematics would come from construction.
For example, if you’re building a new theatre that has a 100m diameter, how much material do you need to build the walls? You would need to know the approximate perimeter to predict the usage.
Pi is useful to know because lots of things are circular. Like, if you have a granary that is cylindrical in shape, and you fill it to a certain depth, about how much grain is that? How many baskets of a known size will it fill? This could be a pretty important question if, for example, you assess taxes in grain. If part of the King’s palace is to be built in a semi-circle, how many evenly-spaced columns are needed to complete the colonnade?
But for these uses you just need an approximation of pi. We know from ancient egyptian and babylonian sources that they basically just said “It’s a bit more than 3” and called it close enough. I honestly think that Archimedes – who we know as the first person to use a method to fairly accurately calculate pi – just wanted to know what Pi was. It’s an interesting riddle, and solving it yields a mathematical constant that really would have seemed like a secret of the universe at that time.
Try to find the area of a circle without using π.
You can draw a square inside of the circle and measure it, but there is space left over on the sides, then draw small ones in the open space and measure those, with increasingly small squares you run into Zeno’s Dichotomy Paradox:
[https://www.youtube.com/watch?v=EfqVnj-sgcc](https://www.youtube.com/watch?v=EfqVnj-sgcc)
and still never find the area of a circle. there will always be some remainder at the edges.
What boggles my mind is how often Pi show up in formulas outside of geometry, electrical, physics, chemistry, etc, etc. it is everywhere at the base of nearly every process in our universe.
A circle is one of the simplest geometry shapes defined by only a radius. This means that any property of a circle will only depend on its radius. True for a sphere as well both volume and surface area. These properties of these shapes will be proportional to a power of the radius. Now I don’t think I need to adress why circles and spheres are special, these symmetries are quite common.
So we know that the circumference is some a×r and the radius is r. If we look at ar/r = a. So measuring the circumference anf dividing by radius gives us a the proportionality constant. This constant does not depend on r its a constant.
Ever heard of a surveyor’s wheel? This is a tool that’s been used for thousands of years. The simplest version consists of just a wheel with a mark on one side. The circumference of the wheel is a certain distance, such as a meter or a foot or whatever you like, so long as everyone agrees what that distance is. As the wheel rolls, the mark on the side rotates with it and the crew count how many times it did so. A much more elegant method than using a ruler when dealing with long distances.
In order to properly manufacture them, it’s important to know the ratio of the diameter to the circumference. Early engineers noticed this ratio and recorded it. What’s weird is that, over the centuries, this same number started showing up in other places and mathematicians were and sometimes still are stumped as to why it’s there. It seems to be fundamental to the universe itself and it’s a bit of a mystery as to why.
Imagine you are building a circular fence. Knowing what diameter you want it to be will tell you how much material you will need for the circumference.
Do you want to know how fast your car is going? The circumference of the wheel is part of the setting up of the speedo.
A circle is the shortest way of enclosing a given space so circles can be very useful.
The ratio is fascinating because it doesn’t matter how big it small the circle is, the ratio is constant. You can’t make a circle such that the ratio isn’t dependent on pi.
Think of a simple mechanical hand crank, like the one used to reel in a bucket from a water well. For each revolution of the crank, the rope will wrap around the axle by one revolution also. The circumference of that axle is going to be the same as the distance that the bucket moved up (not counting the thickness of the rope). A larger diameter axle is going to get you more distance per revolution. Basically it gives us a way to convert radial distance into linear distance, and vice versa. But to measure that exact distance, that’s why it’s important.
Because compared to other n-sided shapes. Circles are strange. They seem to break the rules. Yes you can ignore circles for a while. But eventually you will need to figure out how how to measure circles too.
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