I'm currently a math faculty at Proof School, who really enjoys aesthetically pleasing math visuals. I made a video for the first 3b1b SOME contest, and might at some point find time to make another one.
What about summing cubes of Fibonacci numbers to get another? -1³ + 2³ +3³ = 34, -2³ + 3³ + 5³= 144, -F(n-2)³ + F(n-1)³ + F(n)³ = F(3n-1). I "discovered" that, but can't find it stated anywhere. Anyway, where is the 4th equation hiding?
In highschool I had a text file where I was enumerating these (well, the strings of 1s and 0s with no adjacent 1s called fibonacci cubes, but obviously the same mathematical structure). I was doing it in a very strict order so that I didn't miss any, and I decided to formalize that order, and having never seen it before, managed to prove your Identity 2 (first proven by Lucas, I believe). I did not do it nearly as intuitively as you show here; it was a real drag through the mud of calculation.
Love the chords! It got me interested in tiling in more dimensions, or with additional/different blocks, and the patterns that may arise from that. Would love a follow-up :)
Hmm.. interesting question. You could potentially try to make some more artificial rule that forces the sequence to start at 2, 1 rather than 1,1, but I can't think of any natural way to do it. This tiling bijection is why IMO the sequence starting with initial conditions F(0) = 1, F(1) = 1 is by far the most natural definition that satisfies the recurrence.
@@ericseverson5608 I can't think of a tiling representation of the Lucas numbers, but here's one involving cell division and fusion. 2 cells fuse into 1 1 cell splits into 3 3 2 cells fuse into 1. 1 cell splits into 3. 4 2 cells fuse into 1. 1 cell splits into 3. 1 cell splits into 3 7 etc etc A split can be drawn as a blob with 3 lines radiating down from it ending in 3 blobs. A fusion is the opposite, 2 blobs with 2 lines converging down from them onto 1 blob, (which always then subsequently splits into 3).
Woah, this is such an underrated channel! As a fellow educational RU-vidr, I understand how much work must have gone into this- amazing job!! Liked and subscribed :)
My thought was different: suppose you want to form a sequence of n tiles then you can either make count the sequence n-1(which you’ll sum (1) tile to) or count the sequence n-2(which you’ll sum (2) tile)
This is cool! I found already knew all these identities and their inductive proofs, but I never thought about looking at them visually. Surprisingly it was quite easy to figure out and the visual proofs are much simpler than the algebraic ones. I took me like maybe 30 seconds of thinking at most for each of the problems, but they were still quite nice and would probably work well as warm up problems. I found the video to be quite well made in general and I especially appreciate the encouragement to pause and ponder throughout the course of the video. I hope you can continue to love math and express it through videos in the future!
This was a topic I've had on my radar, and you've covered it brilliantly! I don't have to be the one to make it anymore :D If I had to add a 4th section, I would've included something on generating functions.
The generating function approach is really hard to make visual, but could have been an interesting addendum. The GF being 1/(1-x-x^2) when we choose these initial conditions is another motivation for this being the most natural form of the sequence. And can definitely relate to some relief that somebody else made some content that you wished existed.
Oh interesting, I haven't heard this before. Do you mean tiling a circle and counting all symmetric rotations as the same? How do you argue the recurrence still holds?
@@ericseverson5608 No, rotations aren’t allowed, or it would definitely give the wrong sequence (hmm but it’s probably already in OEIS too). The only difference with Fibonacci will be that one can have a half of the 2-tile at the “circle start” and the other half at its end, a situation impossible with the segment.
Awesome video on several levels, but the musical accompaniment in particular really stood out to me. Props to Michael Severson on that! (Kind of reminds me of Mr. Rogers's Neighborhood, actually. :P)
52:00 for composing a function with another: A -> M + Y B -> N + Y M + N -> D Y + D -> Ø Y is output if another function consumes Y, you could add a series of reverse-cascade reactions. the 2x function is implemented by: Y -> Z + Z so in this case, to compose it, you would add this rule: D + Z -> C C + Z -> Ø and at that point, the Y + D -> Ø rule is a convenience. this won't work in general, but it would work at least often.
For the sum of squares one, you shouldn't have to limit it to squares. F(n) should thus equal F(k)F(n-k) + F(k-1)F(n-k-1). If k=1, for instance, you recover the original definition.
Your comment about rabbits being used to explain Fibonacci numbers makes me wonder if you've also read The Number Devil. It's a favourite of mine from when I was a kid.
For binary, your shape will have the curve 2^n (or log_2 n depending on orientation) hiding in it. For these Fibonacci tilings, we get a curve phi^n instead.
