Secrets of the Fibonacci Tiles - 3B1B Summer of Math Exposition
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- Опубліковано 28 чер 2024
- A simple problem about tiling will explain multiple patterns hidden in the Fibonacci sequence.
Music by Michael Severson / michaeljseverson .
Animations created using www.manim.community/.
Entry for the 3blue1brown summer of math exposition contest www.3blue1brown.com/blog/some1.
0:00 - Intro
3:39 - Identity 1: Partial Sums
5:19 - Identity 2: Pascal's Triangle
6:52 - Identity 3: Sum of Squares
9:01 - Summary
this whole 3b1b summer of math thing is just godsend. So many talented people
These jazz chords that complexify as the size of the samples increase is delightful.
Delightful, to say the least!
wow, that band-aid concept at 8:20 was so elegant!
commenting before this bows up
@@zihaoooi787 great idea!
omg another carykh sighting in the wild
@@zihaoooi787 lmao
Yes, seeing that proof was how I first heard about this interpretation of Fibonacci numbers, and was the original motivation for this video.
Unfortunately can't remember where the original source I saw it was from.
There's a beautiful moment when you see a pattern click into place. And to have three such moments, each better than the last, in under 10 minutes (and then a 4th moment thanks to your bonus problem), that's a wonderful creation! ❤️
Packing in so many ah-ha moments is really impressive. Choosing the size 8 case really seemed to help make them easy to come up with.
Lovely video, thanks for making it!
Incredible video quality. Audio, animation, pacing, content, clearity - everything is just as perfected as a 3B1B video (or even better). A miracle is required for this not to win the challenge.
Don't forget to like and subscribe.
Thank you so much!
@@ericseverson5608 it really is a brilliant video. As an engineer with background in music theory I appreciate this video on many levels! Marvelous creation. Hope you continue in this style! Earned a subscription from me.
Wow! I think this is the first "3b1b-inspired" video that really manages to match the quality of a 3blue1brown video. And that is the highest praise I can think of for a math video
Amazing video and great visuals that help show what you're explaining. I was worried this would be just another Fibonacci video that didn't really add anything, but this really did surprise me and teach me something I didn't know before. Please make more videos like this, as I will definitely watch them.
Your
You can come
My own new Fibonacci identity: F(k) F(n-k) + F(k-1) F(n-k-1) = F(n)
Great video!
Exactly! Which is immediate to come up with once you see these tiling arguments, but would be harder to notice otherwise.
@@ericseverson5608 I think it is actually much more straightforward to discover from the matrix form because it follows from A^n = A^k A^(n-k)
This way you can also find more involved formulas from A^(m+n+k) = A^m A^n A^k
Let me clarify: I'm not trying to take away from the video at all. The quality is astonishing and the visual combinatorial proof brilliant. Awesome production!
I made the mistake of watching this late at night, and the chimes really made it easy to fall asleep to.
Now having watched it in full, it's really good! The aforementioned sound really does add something to the video too, by being linked to what's on screen.
Great job!
Everything about this is perfect.
- Tying the mathematics to simple approachable structures,
- content pacing,
- prompting the viewer to analyze the situation themselves,
- on point animations,
- coordination with music,
- including chapter markers and subtitles and timestamps in the description,
- ending with open ended proposal for self investigation,
you are a legend
I really want to ask: Roughly how long did you spend making this? I know it is hard to quantify since stuff like prior experience with manim and time thinking about script are hard to stick a number to. The love and care you have poured into this really shows!
Way way too long, and it's pretty hard to give an accurate estimate. Maybe 40-60 hours?
I was glad to have a contest that forced a deadline upon me. I'm also very much still learning Manim, so hoping that I can get a quicker workflow for future projects.
I love the sound in this video. The chimes helped keep me engaged.
Also great content and great animation
I would’ve never drawn the connection between Fibonacci and Pascal’s triangle. That’s why math is so awesome
Oh. My. God. This is a gem. I’ve no idea why I’ve never seen this before, but it’s utterly beautiful. Thank you for showing this to me!
The music and sound design here is amazing!
Great video! As a hardcore math enthusiast myself, I have several comments:
1. The sum of F(n)/2^n over all natural n results in 1 (with an off by 1), and it can be explained with the tiles. Take a sequence of n zeros and ones. there are 2^n of them, but if only count those where there is no pair of consecutive 1's, you get F(n), because each 1 can be paired with the 0 after it to create a domino. Now, take a random infinite sequance of 0's and 1's. What is the probability that it has two consecutive 1's? on one hand, it's 1. on the other hand, It's the sum of all probabilities that until the n-th place, there is no pair of consecutive 1's, and then there is a pair: and that probabily, as we showed, is F(n)/2^n.
2. There are loads, LOADS, more patternces of the fibonacci sequence. There are even more ways to visualize it. Here is a collection of a lot of them, that starts with basics definitions and formulas and then moves on to several differnet areas (such as combinatocis, number theory, generating functions and even trigonometry): drive.google.com/file/d/10k1zDLuCJvotjizy2jrB7KCoA6jOnW2L (I'm sorry it's in hebrew, but google translate can help.)
3. It's a common mistake, but the indices are wrong. The number of ways to tile a strip of length n is F(n+1). It makes way more sense to define f0 = 0, f1 = 1. Then the explicit formula will be simpler, and a lot of number theoretic things will make way more sense. For example, gcd(F(n), F(m)) = F(gcd(m, n)).
3. One reason I like f0 = 1, f1 = 1 as the choice of indexing, beyond it directly being this tiling problem, is that it gives you a simpler generating function f(x) = 1 / (1-x-x^2) = 1 + x + 2x^2 + 3x^3 + 5x^4 + ...
Indexing it the other way gives a generating function f(x) = (1+x) / (1-x-x^2)
1. Ooh that is nice. The indexing is a little messy here, though.
The bijection to domino tilings replacing dominos by substring 10 will miss all the strings that end in a 1. But then that makes the probability easier to count, since the probability of the first 11 appearing in positions (n+1) and (n+2) will be the probability the first n have no 11 and don't end in a 1 (which is f(n) the tiling number, or F(n+1) under your indexing) times 1/4 that the last two characters are 11. So this ends up giving the sum
1/4 + 1/8 + 2/16 + 3/32 + 5/64 +... = 1.
Mind blown. The first part was fairly trivial and known, but these "geometric" proofs of identities are just awesome! Love it!
That's one of the best first video's I've ever seen for any newborn chanel. Absolutely delightful!
This was absolutely beautiful. You made one of the best videos I've seen in this competition
It's so cool how you find the connection to different Fibonacci sequence properties and this very visual problem.
All very nice and elegant!
I have played a lot with stuff like this and have seen many videos, this vid still gives me something new!
Proud to be your 125th subscriber.
Another hidden gem that flourishes up recently. Glad to see this.
VERY few things in life please me more than a good recursion problem. Thank you for introducing this into my life!
I've been getting math videos in my feed and this is the best one. the concept was excellent for a video and the pacing was perfect
That resolution to the tonic at 2:41 is nothing short of magical!
Beautiful! I recently "discovered" the sum of squares pattern myself, and an algebraic proof, but this visual proof just blew my mind! Well done!
the music is such a great addition to this, nice!
mr. rogers meets 3B1B :D :D
Man, The quality of this video is astounding, Please keep making videos
Lovely video! This could be generalized to count the number of ways to cover a 1*n block with blocks of length 1 to length k. This gives the recurrence relation F(n+1) = F(n) + ... + F(n-k+1), where F(1) = 1, F(2) = 2, ... , F(k) = 2^(k-1). For k=3, this gives the Tribonacci-sequence and in general maybe something like a "k-bonacci" sequence?
Correct! You can go even further, adding multiple different colors of the same size tile, which will give you any linear recurrence F(n) = a_1F(n-1) + a_2F(n-2) + ... + a_kF(n-k). The corresponding tiling problem will then have generating function f(x) = 1 / (1 - a_1*x - a_2*x^2 - ... - a_k*x^k).
Can we generalize beyond the k-bonacci sequence by recognizing that x^2 - x^1 = 1 is to the Fibonacci using dominoes and single squares (2 and 1), as x^3 - x^2 - x^1 = 1 is to the tribonacci using trominoes, dominoes, and singles (3, 2, and 1), and for example as x^3 - x^1 = 1 is to Narayana's Cows (OEIS A000930) using trominoes and singles (3,1), with the recurrence relation N(n) = N(n-1) + N(n-3)?
I love that more people are using Manim to create math videos.
this video is a gem. so happy I found this channel
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!
Amazing video! Thanks for the shocking proofs and the well animated visuals.
wow i love the jazzy sound effects you added, its making this math fun
Great video and visuals! The idea is intuitive and simplistic, yet powerful and elegant
This video was incredibly interesting and surprisingly satifying!
Thank you!
Very beautiful proofs, animations and clear explanations.
Well done!
Beautiful video. Structure and music are simple and support the math so well... and reflect the elegance of the math. Thanks!
The music here is just great, thank you for making my ears happy as well as my brain
Absolutely beautiful.
Absolutely amazing video! Subscribed.
This video is beautiful. The music makes my brain happy since it relates objects and pitches
Excellent visual and logical explanation to the problem. Really gave me a wow moment even in the second half of the video.
There's a fourth beautiful identity !
If you take the partial sum of the square of Fibonacci séquence, you can obtain the area of a rectangle made of two consecutive number of the séquence.
Congratulation for thé vidéo !
Very beautiful approach.
J³
This is making me so jealous. :-) I'm teaching math in college, and thanks to the pandemic went to produce explainers for the stuff in my courses, only to discover that I suck at it. I just wished I had your talent, determination, and/or skill, or whatever this is. Wonderful video, thanks for sharing it.
Very nicely done!
You really visualize the cool secrets found within math well! Thanks for the video!
This is beautiful!
While doing my master thesis in the field of mathematical physics, I worked a lot with a type of mathematical object called combinatoric functions, which is the solution of any linear and homogeneous recurrence equation. What does amaze me is that your video is exactly the visual representation of the combinatoric function for the Fibonacci equation. Good job.
This construction also generalizes to any linear homegeneous recurrence relation with integer coefficients (different sizes and colors of tiles)
This is incredible. Thank you.
This is a really great video, well done.
This was fantastic!
Videos that show the visuals of math are soooo helpful in understanding and literally even make it fun.
Fun fact: the Greeks were so good at math because they studied geometry. They used that for math and while it only goes so far, geometric math is basically almost all math, or basically almost all of math can be represented geometrically, which is how they came up with so many formulas and discovered pi
This is beautiful. Please make more videos!
This is so beautiful
So uh... I have no idea who you are or why your video was recommended to me, but it was a good recommendation and you've got yourself a subscriber. More of this please.
It's so cool that someone else came upon this! I actually found it when i was asking the question of how many possible games of Street Fighter there were, and simplified down to the blocks representing the lengths of different moves.
Similar blocks in ancient Indian poetry seem to be the first historical discovery of this
(see twitter.com/stevenstrogatz/status/1080623259593465856?lang=en)
This video is incredible! I wish I had been taught about the Fibonacci sequence this way in school
Fantastic ! Many thanks.
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.
Best Fibonacci video so far
Im speechless, great video
love the music. Great touch
The music is an excellent touch.
I saw these patterns and immediately thought of the fractal pattern of binary numbers - which makes sense, because it can be formed in a very similar fashion.
Cool video, thanks for sharing!
Really beautiful combinatorial proofs!!!
The first problem reminds me of when we discussed in a combinatorics class, except we had 2xn tiles, with either vertical or horizontal dominoes (2x2). I believe that one was also based off the Fibonacci sequence, and was similarly satisfying to work through! Great video, beautifully concise and incredibly well explained!
Yes, there is a pretty straightfoward bijection between both problems. Dominos tiling a 2xn must either be vertical or come in horizontal pairs making a 2x2 block. Once you show this, you can just look at the top 1xn block which then looks like a tiling with singletons and dominoes.
this is beautiful
super duper cool!!
This video is amazing! Please make more videos like that! :)
Great video!
This was super interesting to watch
Amazing!
Amazing math, amazing music
This is a great topic! And those tilings are so cool to prove much more complex identities. Once i figured out how to prove Newton's binomial for Fibonacci numbers with those tilings. Its much harder than just using binomial on Binet's formula, but also more satisfying
Woah, this is such an underrated channel! As a fellow educational UA-camr, I understand how much work must have gone into this- amazing job!! Liked and subscribed :)
I thought the video was going to end at 3:40 but then it kept going 🙏💯💯
Great work
Beautiful
wow! great video
Mindblowing stuff
That was rather lovely and rather restful. Apologies, I thought I was clicking on a 3B1B one, but subbed :)
The way you organized it also kinda looks like a fractal, which is pretty cool
Great video and nice use of music
Those sound effects are soooooo nice : )
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 :)
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.
Lovely Video, and "pause ans ponder" is a wonderful phrasing, I hope you keep going ^-^
That one is borrowed from 3blue1brown, and I agree it's a great phrase.
4:43 was really satisfying. I couldn't figure it out from the equation alone, but the visual made the answer clear.
The jazz chords were just so perfect 😭
Nice video!
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)
Good job.
as the block approaches infinity, notice how there is an immergence of a fractal pattern
the same can be said in a sequence of incrementing binary values stacked on top of each other (e.g. 0001, 0010, 0011, 0100, 0101, etc.)
if you take a 16 bit value, starting at 0, increment up by 1, then place that incremented value directly below the one before it, you can do this until the last value in the subsequent block is all 1s. after which, you can see a similar pattern arise if you replace the 0s and 1s with different colored blocks.
Satisfy!
Are you sure you aren't 3B1B himself, or perhaps his clone? This is great math demonstrated in an exceptionally intuitive way. Just like 3B1B.
I love the music
Wow this is really underrated
good animations good music and great quality of math. Did you come up with the bijections yourself or is from a book? Would love to see more
I saw the sum of squares argument somewhere.. unfortunately can't remember the exact source. The first two I hadn't explicitly seen, but they seem like the straightforward explanations of those identities.
A book that I've heard very good things about if you are looking for more combinatorial proofs is called Proofs that Really Count.
By the way, i love that you display a Penrose tiling in your chanel background.
Yeah, that was a fun project. Cut a bunch of wooden tiles and glued them to bedroom wall.
1, 1, 2, 3, 5, 8, 13, *13 splits into 10 and 3* 21, 34, 55, 89, 144, 233, 377, 610, 987