Secrets of the Fibonacci Tiles - 3B1B Summer of Math Exposition 

These jazz chords that complexify as the size of the samples increase is delightful.

this whole 3b1b summer of math thing is just godsend. So many talented people

wow, that band-aid concept at 8:20 was so elegant!

Lovely video, thanks for making it!

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! ❤️

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.

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.

My own new Fibonacci identity: F(k) F(n-k) + F(k-1) F(n-k-1) = F(n) Great video!

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!

I would’ve never drawn the connection between Fibonacci and Pascal’s triangle. That’s why math is so awesome

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!

I love the sound in this video. The chimes helped keep me engaged. Also great content and great animation

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!

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)).

The music and sound design here is amazing!

Mind blown. The first part was fairly trivial and known, but these "geometric" proofs of identities are just awesome! Love it!

That resolution to the tonic at 2:41 is nothing short of magical!

I love that more people are using Manim to create math videos.

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.

Another hidden gem that flourishes up recently. Glad to see this.

That's one of the best first video's I've ever seen for any newborn chanel. Absolutely delightful!

Beautiful! I recently "discovered" the sum of squares pattern myself, and an algebraic proof, but this visual proof just blew my mind! Well done!

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

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

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.

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?

wow i love the jazzy sound effects you added, its making this math fun

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!

Absolutely beautiful.

this video is a gem. so happy I found this channel

The music here is just great, thank you for making my ears happy as well as my brain

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 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.

This is beautiful!

the music is such a great addition to this, nice! mr. rogers meets 3B1B :D :D

Great video and visuals! The idea is intuitive and simplistic, yet powerful and elegant

This is incredible. Thank you.

Very nicely done!

Fantastic ! Many thanks.

This was fantastic!

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.

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

Beautiful video. Structure and music are simple and support the math so well... and reflect the elegance of the math. Thanks!

Amazing video! Thanks for the shocking proofs and the well animated visuals.

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.

this is beautiful

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!

Absolutely amazing video! Subscribed.

This is so beautiful

Very beautiful proofs, animations and clear explanations. Well done!

Best Fibonacci video so far

This video was incredibly interesting and surprisingly satifying! Thank you!

Im speechless, great video

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 music. Great touch

The music is an excellent touch.

This is beautiful. Please make more videos!

This is a really great video, well done.

You really visualize the cool secrets found within math well! Thanks for the video!

Mindblowing stuff

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

Great work

Great video!

Amazing!

Amazing math, amazing music

Beautiful

This video is incredible! I wish I had been taught about the Fibonacci sequence this way in school

Really beautiful combinatorial proofs!!!

super duper cool!!

wow! great video

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.

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!

The jazz chords were just so perfect 😭

This was super interesting to watch

4:43 was really satisfying. I couldn't figure it out from the equation alone, but the visual made the answer clear.

I thought the video was going to end at 3:40 but then it kept going 🙏💯💯

The way you organized it also kinda looks like a fractal, which is pretty cool

Great video and nice use of music

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 video is amazing! Please make more videos like that! :)

Those sound effects are soooooo nice : )

Satisfy!

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.

Nice video!

Good job.

I love the music

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 :)

1, 1, 2, 3, 5, 8, 13, *13 splits into 10 and 3* 21, 34, 55, 89, 144, 233, 377, 610, 987

I did a very similar exercise in my "Introduction to Mathematical Reasoning" course just today (hadn't seen the video beforehand). The only difference was that we were looking at the tilings of a 2×n board using only dominos, which is equivalent, as vertical dominos get replaced by squares here. Still interesting to see how much this simplifies a lot of phenomena around the Fibonacci sequence...

That was rather lovely and rather restful. Apologies, I thought I was clicking on a 3B1B one, but subbed :)

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)

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.

This was my 3rd paper in my GCSE maths exam (which was always an open ended investigation question). I showed everything up to 3:35, and generalised it to different size tiles (and that it makes 2^n if you allow any size tile), but everything after that is new to me.

Wow this is really underrated

Lovely Video, and "pause ans ponder" is a wonderful phrasing, I hope you keep going ^-^

I never knew until now how much music chords add to math videos