…Another Slice of π
Insights from the number line.
*******************************************************************
I’m tracking my progress as I learn about maths and physics. My background is in neither discipline. The more I learn about anything really the more it leads me back to maths and physics. I decided to dive in head first! Follow me on my journey and let’s see what happens! 😉
*******************************************************************
📝🖼
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 🥱
We build on on our understanding to arrive at the formula :
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 🙄
*******************************************************************
Hi! If you’re here for the first time welcome! We’ll be building on some formulas we develop in previous articles, it might be worth checking them out. If you’re back for some more mathin’ it’s nice to see you again! 👋🏾
We’re going to start off with a kind of obvious claim that is going to be important later. You can subdivide a whole into as many different equal parts as you want. Duh!
Literally any whole value can be subdivided into smaller equal parts. Again, Duh!🙄
The circle has been subdivided in different ways throughout history for different reasons. Angles, gons, radians and turns all add up to a complete circle. Each method of subdividing the circle was defined for specific reasons but ultimately add up to the same thing 🟣. It’s like measuring a banana in inches or centimeters. When you’re done measuring you still get a banana length 🍌. Banana for scale 😂. Angles, gons, radians and turns all add up to a full circle. That being said I think it would be helpful to go through three ways circles have been subdivided. Two common and one not so much, degrees, radians and gons.
Degrees : This is the oldest way of subdividing a circle, it’s so olde that I wrote old with an e for emphasis . 📜 There is no written record for why the number 360 was chosen to subdivide the circle but the theory is that 360 is based on the number of days in a year. 360 is also a really easy number to work with. You can divide 360 by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360 and get a whole number back . This makes doing calculations with 360 really convenient. I like to think about degrees living on the inside of the circle.
Radians : Originally called the circular measure the radian is defined as the number of radii it takes to walk around the full circle. If we have a circle with radius of 1. Then the number of radians it takes to walk around the circle is 2π radii. Which is also really convenient in its own way since it is based on nothing else but a defining characteristic of the circle, its radius. I think about radians as living on the edges of circles.
Gons : Dropping into the scene Jean-Charles de Borda introduced the the idea of a gradian by subdividing an entire circle into 400 equal units. Making 90 degrees and π/2 radians into 100 gradians. (All those numbers equal the exact same thing 1/4 of a pizza) An entire circle in terms of gons is 400 gradians. This way of measuring a whole circle didn’t really catch on everywhere but found uses in surveying and geology.
I want to emphasize here that each method is slicing 🍕 up an entire circle, if you add up all of the slices you get the back the full circle. That’s it. We’re working with something that is whole. Which brings us back to our earlier point. You can subdivide a whole into as many equal parts as you like. (or keep the whole cake 🎂 for yourself!) Think about it as how different languages have a different word for the same thing. Soda, pop, coke, fizzy drink drink🥤
As a refresher we’re going to remind ourselves of the formula we came up with as an alternative for the LaW oF CosInE :
2ab(1 - cosθ) +m²
What we want to do next is set a couple of constraints.
1.) We want our equation has to equal c² : 2ab(1-cosθ) +m² = c²
2.) We also want to set a=b : Doing this sets m=0 since m = b - a . Simplifying our equation we arrive at : 2a²(1-cosθ) = c²
Now what we want to do is solve for c by squaring both sides. This gets us to √(2a²(1-cosθ) = √c²
It’s a funny thing about notation. A lot of advancements in maths and physics and communication 🗣 has been because of better notation. Original algebraic equations were problems stated as paragraphs. We have the benefit of symbols that quickly get to the point of what we’re trying to do. The power of symbols and notation is expressed here as you read combinations of shapes on a screen. What I find amazing is that any understanding can happen in the first place! Wow us! I find it helpful to think about numbers represented below. Where the root symbol is like a little house 🛖 that the number lives in.
And when (m) matches (n) the number steps out of its house.
When we multiply numbers together we can give each number its own house 🏘. On the left side we are multiplying (2) and (a²) and (1 - cosθ) so we can put each of them under their own square root symbol. √2√a²√(1 - cosθ) = √c² . This is great because now we can see that on the left side (a) is raised to 2 and on the right side (c) is also rased to 2. So both of these numbers can come out.
Now our equation looks like this : a√2√(1 - cosθ) = c
It’s good practice manipulating equations like this. So we’re going to do one more change before we’re happy. We’re going to put 2 and (1- cos θ ) back into the same house, er, square root symbol.
a√2(1 - cosθ) = c
So what does any of this mean? lol 😂. As a reminder, what we’ve done is set two conditions and simplify. We said that a = b and the equation had to equal c² . It’s helpful to remind ourselves that equations have shapes. I like to say that formulas have forms and forms have formulas.
Seeing the labeled triangle we can see that the reason a = b is because they are both radii of the same cirlce! We’re going to define a new condition. a=b=1 and anything multiplied by 1 is just itself. That’s what makes the number one so special, it can tags along but always wants you to be yourself!
So now we’ll come full ⭕️ circle 🙄😂 and remember that we can subdivide a whole into any number of equal parts. Let’s use θ to help us understand what that means. If we subdivide a circle into 4 parts θ = 90 degrees or 100 gradians or π/2 radians. These all mean the exact same thing. You’re getting a pretty big slice of cake. 🍰.
For practice let’s go ahead an subdivide our circle into 5 parts. This is going to makes θ = 72 degrees or 2π/5 radians. We’re building an intuitive understanding here so really these numbers don’t matter. All we have to remember is that when we subdivide a whole circle into any number of even slices, there is always going to be a corresponding inside angle and a corresponding portion of the circumference.
Now the fun can start!
Since θ defines the angle we make by subdividing the circle N times N*θ gives us the entire circle back. √2(1 - cosθ) = c gives us the length of the side opposite to θ. To get an approximation of the circumference of a full circle again all we have to do is multiply (c) by the number of slices we made, N.
N√2(1 - cosθ) = Nc
We’ll bring the chart with lengths of sticks and string lengths back from the previous article Find Pi For Yourself .
As the value of N increases towards infinity θ gets closer and closer to 0. Since we set the radius of our circle equal to one we’re going to have to look at the width of 2. Which is equal to 2pi. so if we divide our equation by 2 we arrive at our definition for pi! As N gets bigger and bigger we get a closer approximation for the value of π
🎉 π = N/2√2(1 - cosθ) ; as N gets towards infinity.
If you’ve made it this far Amazing! We’re going places so stick around!
Till next time.
*******************************************************************
Previous Article
👈🏾 Find Pi For Yourself
*******************************************************************
🔗 Links to Stuff 🔗
