Cannons and Sparrows - Numberphile

"And 6, is 2 x 3" - My only  moment of comfort during this video :P

Just the comparison "Cannons and Sparrows" makes me think of trying to help my kid to their math homework a few years ago. "Dad, how do I do this?" "Well, you take the integral of....... You know, you're only 8 years old, I doubt they want you to solve that with calculus... Let me think about that for a bit..."

"Firing cannons at sparrows" (In German: Mit Kanonen auf Spatzen schießen) could have been translated to "Using a sledgehammer to crack a nut." A related German proverb (also about sparrows) is to "prefer the sparrow in hand to the pigeon on the roof," which is kind of contradictory and dangerous if you know people fire cannons at sparrows. Welcome to the realm of German proverb wisdom.

I am glad you're uploading more complex stuff from time to time. It's fascinating even if you can't understand everything! Thank you.

Well it depends. Is it an African or European sparrow?

it's been 2 years and no one mentioned how firing cannons at sparrows seem just a bit similar to the concept of the angry birds game

As much as I've worked with Pascal's Triangle. This really just blew my mind as it points out a property of mathematics I never even realized existed. Not just as the example stated but as a very basic truth about how numbers work at a very low level.

The first thing that comes in my head is Arsenal and Spurs.

Holy shit. So this means that (imagine the largest prime you know of, even the ones with millions of digits) you can split a polygon into that many slices with equal area and perimeter. That's crazy

That Pascal triangle is taken from The Number Devil! I love that book!

i love this canon slowly moving into a frame.

I still get surprised by how you can in any given moment of your proof find a solution which is on a completely different place of this Math world, and how complex it can get to enstablish a link between two "simple" things like splitting a polygon in equal area and perimeter pieces and Pascal's triangle. My mind gets blown away every time

This is one of my favourite numberphile videos. Günter Ziegler should feature in numberphile videos more often.

Thanks for reassuring us of that there won't be too much math in a video about math.

Wait I thought we were going to talk about cannons and birds... Aaah! You tricked me again!

I really enjoy these longer form videos.

I'm reminded of a previous episode with a similar solution for leveling a table at the biergarten by rotating it.

It's amazing how something so smoothly geometrical related to areas and perimeters (and the machinery context we use to partially solve it, with topological argument and voronoi partition) reveals itself to have a link with something so numerical than prime numbers.

Remember, the perimeters are parameters.

Neat. Some other commenters thought that because the conjecture is proven for all primes, it must generalize directly to all composites. Well, it doesn't, but I have an incomplete workaround, and any counterexample to it would be fascinating in itself. Say we're constructing an answer for 2N, where N is some number already proven. Divide your starting convex shape into 2 equal-area halves as before with line L, but don't keep track of their perimeters this time. Instead, find all solutions for dividing each half into N pieces of the same area and perimeter. The smaller areas are guaranteed to match, which leaves only the smaller perimeters; these will match on either side of L, though maybe not across L. As you rotate L through 180 degrees, the perimeters may go from being A and B to being B and A. So if certain conditions hold, some orientation of L will have a solution where all shapes have not only the same area, but also the same perimeter. I can't just say I proved it, though. A given half might well have more than one solution with many values for the perimeter. I picture the perimeter values as multiple continuous segments and some points, not contiguous with each other--no clear reason those values can't completely dance around each other, but it would require some strange behavior. More details to come tomorrow.