Find π For Yourself

Insights from the number line

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I did ok in ➖ maths ➕ when I took it in highschool. From what I can tell though, new stuff keeps getting figured out. And I can’t find where to download this years update. This article is a part of the series, Papers & Pictures, where I track my learning progress on things like particles, black holes, abstract algebra and a bunch of other stuff.
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📝🖼
Papers & Pictures Season : 1
Episode : 1 : The Difference Between Squares
Episode : 2 : The Summing of Squares
Episode : 3 : Find π For Yourself : Pi Part-1
Episode : 4 : …Another Slice of π : Pi Part-2

*******************************************************************😴 TL;DR 🥱

The number π (pronounced pie 🍰) is the ratio between the perimeter of a circle, also called the circumference, and the width of the circle, also called the diameter.

Approximations for pi have been defined geometrically with polygons and algebraically with infinite series (products and sums).

We’ll add a definition for π using our formulas from The Summing of Squares to land on π=N/2*√(2 * (1-cosθ)) ; where N is the number of subdivision for a circle and θ is whatever unit you use to define an angle in a circle divided by N.

😪 TL;DR 🙄
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Humans like to look for patterns. Give us a pile of numbers and we’ll find how they’re related. So pi, the ratio between the periphery of a circle and its width, has been know about for a while. What’s really interesting though is how popular this ratio is showing up in astronomical equations, electromagnetism and even relativity.

All of that is great and all but really what does any of it even mean? And how is pi so many places at once? 🤔

You might’ve heard about Archimedes, a super smarty 🤓, who used polygons on the outside and inside of a circle to find that pi landed somewhere in the neighborhood of 223/71 & 22/7 ( 3.1408 & 3.1429). The motivation for this approach is to find a greater degree of accuracy for the value of pi that isn’t dependent on the accuracy of the measuring tool. What we want to do is become acquainted with the value pi first before we get rid of our measuring tools.

Archimedes used polygons with multiples of 6. Here we’re showing polygons as we increase the edges.

Gaining a new perspective usually requires a little movement. We won’t break a sweat but if you’re up for it 😎 let’s get to know pi for ourselves!

We’re going to start with a regular ‘ol stick.

A single stick of length 1

The length of the stick , we’re going to decide, is equal to 1. One what? A centimeter, a foot 🦶, an angstrum ⚛️ or a light year💡. Whatever length you like to think about, that’s how long the measuring stick 📏 is going to be! I like measuring in units so I’m going to say my stick is 1 unit long. (Most measurements have an earthy flavor, that means that someone on earth made it up for convenience and one day we decided, sure good enough and stuck to it. )

Let’s add another stick to our first stick, now we have two sticks! we can keep adding sticks forever, but who’s got time for that! We’ ll keep it to ten sticks for now that should be enough.

Adding sticks from 1 to 10

To keep track of everything we’re doing we’ll store the lengths in a table. Each number below corresponds to lengths 1 through 10.

Now if we spin a stick about the middle and we track the two ends we get our first circle. Great job 👍🏾

Rotating our stick to create a circle

We have 10 sticks so lets go ahead and make circles for each 1 to 10.

Next we’re going to need some string 🧶 so grab the biggest ball of yarn you have lying around. We’re going to pretend that a string is just a curvy stick. This means we can use it to measure things because we can lay it down straight and flat.

** Side note, it’s important to remind ourselves that Archimedes didn’t use a measuring stick for a reason. The precision we’ll arrive at shortly is going to be good enough to get to know π and we’ll build on that understanding. **

We’re going to wrap every circle in string and cut the string so the it sits snug on each circle. Once we do that we’ll measure how long our string is and add it to our table. (I’m using a program called Rhinoceros to do the measurement but you can use real sticks and strings and this should work as well)

Wrapping our circle in some string

Our table of stick lengths and string lengths

A big part about learning things is keeping track of what things change when we change things. In our table the lengths of our sticks is the thing we are changing. We changed the length by adding one stick each time from 1–10. The thing that changed is the length of the string. The thing we are learning is how these two lengths are related to each other.

So with the information in our table what we do next might seem a little random but playing around with numbers is how we got here in the first place so let’s give it a go. We’re going to divide the lengths of the strings by the lengths of the sticks.

🎉 When we compare the numbers on the bottom row we can see that for every string length divided by the stick length we get close to the same number, our approximation for pi 🥧 ! Even if we continue to increase the length of each stick to infinity and beyond we will always get pi.

Which is kind of neat, that this number is hiding in there the whole time 👀 . In every circle.

Next we’ll look at what it could mean for us to solve this without the need for measuring sticks and strings now that we have a better understanding of what we’re looking for.

Till next time!

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Previous Article
👈🏾 The Summing of Squares

Next Article
… Another Slice of Pi👉🏾

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🔗 Links to Stuff 🔗
Archimedes
Pi