Pick's theorem: The wrong, amazing proof

A video on what proofs in mathematics are for, using Pick's theorem as an example.

[LookingGlassUniverse]

Wow, I love this proof, and I love what you said about being the importance of “wrong” proofs. What a wonderful video! I hope you continue making more

[PearlofArms]

Dude, I independently discovered Pick's Theorem while in elementary school. We had to find the area of shapes on a grid and I distinctly remember discovering this trick and being very exited about it. I didn't even know algebra yet, so I didn't write up a formula or think it would be special to anyone else though. What's even more amazing to me though is that I discovered this by randomly stumbling on the "wrong" proof. I didn't think about it with water, but rather realised that, after I had counted up the squares entirely contained within a shape, the squares which had been cut in two typically had another square right next to them which was inside the total shape enough for both squares to count for 1 whole square inside the entire shape. It took a little fiddling to get from there to caring about points. Your video has turned this from a cool memory into one of my fondest. Thank you.

[johnchessant]

Great video! You articulated an idea I've had which I've been calling "textbook" vs. "classroom" proofs. The dry, often unintuitive proofs aren't really suited for the classroom lecture format; yes, they follow logically from one step to the next but they take time to decipher and they're meant to answer "how is this true" more than "why is this true". The 'wrong' proofs are not only way more helpful, they capture better what it means to do math.

[pmnt_]

My favorite proof is even more wrong. It's so wrong that it stuck and helped me to remember the surface area formula of a sphere.

"The surface of a sphere is quite easy to derive. (draws a sphere). We are enclosing the sphere with a cube (draws a cube) and project the sphere on the cube's surfaces (draws a circle on one surface). This circles' area is r²π. Because the sum of all projected circles equals the surface of a sphere and because a cube has 4 surfaces (draws circles on the 3 visible surfaces and one on a hidden surface), we get the surface of a sphere equal to 4r²π."

"But a cube has 6 sides"

"Yes but then the area formula would be wrong."

[PedroPaulo]

Congralulation on being one of the winners of SoME, I havent watched this one yet, glad to see it now

[francesco]

Don't get me wrong, this is amazing!
But I think that his proof is not necessarily unrigorous to the core. It can be formalized with a bit of effort(maybe instead of considering units of water you can consider circles!).
"Unrigorous ideas" are beautiful and they are the moving force of mathematics, but they are not "wizardry": if they are indeed correct, they can be formalized. And when you don't manage to formalize your idea, that may be an indicator that it's wrong: intuitive reasoning is risky!

That's why rigour is important in mathematics: it stops the mathematician from flying too close to the sun.

I wrote this comment because often there is the misuderstanding:

-rigour=boring
-ideas=amazing

Formalizing an intuitive idea is often a beautiful challenge that may help you understand even more about that idea.

:)

[michaellin]

I thought a similar idea where B/2 represents how one side of the boundary contributes to the inside (and the other, outside) . The "-1" would be necessary to account for how the boundary line loops in on itself, which is like one full rotation when tracing the perimeter.

[Manabender]

As I'm watching this, the video reportedly has 102 views, yet 225 likes.

Congratulations on being one of the winners of SoME! It's quite well deserved.

[pjohnson]

This technology is apparent in origami. I have been folding paper for about 9 years now. We use a method called " circle packing" to help make crease patterns for complex models. You first figure out how many flaps the base of the origami will have, and then you translate that information into a bunch of circles on square piece of paper. Fascinating art. Thank you for the video!

[cwalker]

Congrats lad, I'm astounded at the simplicity.

[kda_]

I used to dislike this formula because it was taught to us as some sort of trick for exam problems without explanation or proof. Now, having an intuition of how it works, I will certainly use it and share my knowlegde of such an amazing proof. Thanks to you :)

[Lotschi]

Hi, I come from the 3Blue1Broen video.
I really liked the video!
The „wrong proofs“ are what makes math interesting.

[nidalapisme]

I was going to vote for the previous video since the technical aspects were better than your video. But after watching your video, I see a very important message that resonates with me and i think this video deserves to reach more people.

[ericbright]

Rigorous proofs can explain that something *is* true. Proofs like this can explain *why* something is true.

[SlyRocko]

In maths, there are two kinds of valid proofs:
One is rigorous, self contained, and covers every argument that could support or shut down the idea.
The other is something I'm able to explain to my neighbor's kid across the street.

Both are fine for their different situations. In all honesty, I'd say trying to find both kind of proofs for the many ideas in maths would probably make the subject more fun while also being serious when the time is right.

[PedroTricking]

If an argument is well known despite it being wrong, then it's surely a really interesting one. I'd love to see a compilation of wrong proofs.

Also only 3k views yet? I assume that'll change soon.

[robertzarfas]

I miss teaching this theorem. It’s not in most geometry classes here in the us, but it was covered in one curriculum that I used to teach from.

[stevenlundy]

This is the best of SoME1 that I’ve watched. I hope you go on to make those videos hinted at the end!

[bingodeagle]

really enjoyed the format of this video. Succinct enough to be never boring yet never confusing, and with enough detail "left as an exercise" to give me something to think about. :D

[nikanj]

Great video and a fascinating proof I've never seen before.

Also I really miss PBS Infinite Series. There are some really spectacular maths channels on YouTube but Infinite Series was one of the best for the short time it lasted. At least we still have Space Time.