Mathematical Donuts: What if 𝝅 = 4?
A Fake Mathematical Proof
What first comes to your mind when you look at the image above? What if I can show you proof that 𝝅 is actually 4? Can you spot the mistake in my reasoning?
Without further ado, let’s dive in!
Step 1: A Unit Circle within A Square
Since the unit circle has a radius of 1, the square that encompasses it has a perimeter of 8.
Step 2: A Zig-Zag Path by Removing Corners
We then remove corners from the original black square and get a zig-zag shape made of blue segments. We repeat this process indefinitely.
If we zoom in on one of the corners, we should be able to see that at each iteration, the perimeter of the new shape is the same as the previous one.
Purple = Orange = Blue = Black
In fact, if we repeat this up to infinity so that the zig-zag paths are infinitely smooth, the zig-zag paths become the circle.
Step 3: A Strange Result
Recall the formula for the circumference of a circle. We equate this to the perimeter of the square, which is 8.
We have proved that 𝝅 = 4!
The above is obviously faulty proof. But do you know why? In fact, the above is correct in showing that pi is less than or equal to 4.
This is because the zig-zag shape thing only gives only an upper bound for the circumference of the circle. Except, the zig-zag paths never actually become a circle because the sharp corners are not differentiable. Yet, we know that the circle is differentiable.
Our zig-zag paths just suddenly turn from something non-differentiable to something differentiable!
I am sure there are more technical reasons and rigour that can explain this illusion. But that’s beyond me for now.
If you want to know more about this, I recommend checking out the following video by 3B1B on YouTube.
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Happy reading,
Barry 🍩