The Thue-Morse Sequence (with visualizations)

[@magicdatabase]

You had me saying "W H A T?" about every other minute watching this video. This stuff is incredible.

[@MathVisualProofs]

Super cool sequence right?

[@benjaminlum5894]

Good to know I'm not the only one XD

[@Adityarm.08]

Very insightful, thank you! Nothing could be more intellectually satisfying for me. I've always been obsessed with symmetry & developed this habit of chewing food according to this sequence when I was very young - trying to keep the load balanced on my left vs right molars. Just typed the first 16 terms on OEIS today & landed here. Glad to have found your channel along the way :)

[@MathVisualProofs]

Thanks for checking it out!

[@searge123searge]

Was working on mathematical stuff for over a year but couldn't find anything similar to it. Today someone told me about the thue-morse sequence and it really is just the same thing. This makes me very happy.

[@MathVisualProofs]

Such cool math right? Have you followed the linked paper then from Shallit and Allouche?

[@searge123searge]

​@@MathVisualProofs Yes, cool math indeed! And yes, I read the paper after watching the video. In addition, there are some weird things going on, specifically when doing operations with numbers inside of the fraction, that I haven't been able to find described by somebody else. Cheers!

[@MathPhysicsEngineering]

I came back to watch this for the second time! You have done an incredible job! This video truley deserves to get over a million views! The fact that this video got only 3.2k views over 7 month only shows how unfair youtube's algorithm. It promotes only channels that are already big! Though I have to say that with your quality contenet you mannaged to get nearly 70k subscribers. You should get more views from subscribers. Good luck, I will be following your channel.

[@MathVisualProofs]

Thanks for the comment! I don't know how the algorithm works - I suppose I really need to improve my thumbnails and title game :) The subscriber growth seems to have come from KZbin Shorts, but those don't translate to views on the wider, high definition videos. Anyway, thank you for your comment and hope you keep up with your content as well!

[@mostly_mental]

Thue-Morse is definitely my favorite sequence, and I learn something new every time I run across it. Great to see it illustrated so well.

[@MathVisualProofs]

Thank you. It is definitely an amazing one.

[@theemeraldprogamer731]

This will make me cry and NOT tears of joy

[@EtienneGracque]

I first heard about the Thue-Morse sequence in a Numberphile video, but it barely scratched the surface. You took that to a whole new level! You should do a follow-up video called "more amazing marvels on the Thue-Morse sequence", I want to learn more!

[@MathVisualProofs]

I haven’t seen the numberphile one. I’ll check it out. @standupmaths has a great video about it. I’ll see if I can follow up with more stuff. Check the linked paper too :)

[@EtienneGracque]

​@@MathVisualProofs My bad, it was Good Old Matt Parker indeed. Sorry about the confusion.

[@MathVisualProofs]

@@EtienneGracque no worries. That is a great video (like all of his).

[@Adityarm.08]

Thanks!

[@MathVisualProofs]

Wow! Glad you liked the video. Thanks for the visit and support!

[@benjaminlum5894]

The most mindboggling thing to me is the end, where people have discovered the limit of P but not Q. Feels like a half proof waiting to be fully solved. Exciting times!

[@MathVisualProofs]

So cool right?

[@youngcwel]

can't believe you are so unpopular, you're really underrated, I hope you succeed

[@MathVisualProofs]

Thanks!

[@horlickminton6534]

I don’t know how this guy does it but he has the best math videos on KZbin. And I love his voice. I would marry this guy 🧠😍

[@MathVisualProofs]

thanks!

[@Bob_the_Jedi]

Found you from my recommendations, perhaps the algorithm has discovered you. Very interesting video.

[@MathVisualProofs]

Thanks for clicking and checking it out !

[@SarahParker-o5k]

To add to your cool list of properties of the Thue-Morse sequence, I've just come across another one: "An Un-mixable Packet of Playing Cards" (relative to virtually every systematic shuffling procedure used today)! While studying the properties of "Cyclic, Mirrored, and AMP structures," Kent Bessey (a professor at BYU, I believe) discovered an infinite number of portions (i.e., sections) of the Thue-Morse sequence that give rise to packet structures of playing cards that are invariant under nearly all of the common systematic shuffling procedures people use to mix cards. In fact, here is a link to the playlist of video presentations that discuss this discovery and some of the applications to mathematical card magic: kzbin.info/aero/PLz_0A1YUkZzhfz3LV7hRjQ96yuTpr6EyL

[@wieneryron]

Interesting! Re the Koch Snowflake: What happens when if you let that sequence go out to infinity? I am guessing the turtle plot eventually loops back around so that the full snowflake is outlined (although maybe not! maybe we are just looking at an infinitesimally small portion of the curve(??)). But if it does loop back around and since there's no repetition (and thus no overlap of the turtle path), then does each loop around add to the chaotic "roughness" of the snowflake? And if so, does this path actually converge to the true snowflake in the limit? Ie, does it become the fractal?

[@MathVisualProofs]

Good questions. I haven't spent a lot of time looking at the details of the results, but the short answer to your questions is "yes". Here is a paper that surveys some of the ideas: arxiv.org/pdf/math/0610791.pdf (I can't find the cited Holdner/Ma paper not behind a paywall, but that would be a good one to check out too).

[@columbus8myhw]

I'm fairly certain it doesn't loop around, it just keeps on growing

[@MathVisualProofs]

@@columbus8myhw yes. I am not sure as well. I haven’t read the papers I mentioned well enough to know why they mention the snowflake (and Parker’s video, linked, mentions same thing)… maybe one day I’ll spend more time. Here I just wanted to see it drawn :)

[@dayhill9855]

Ultimately we can only discern/perceive a pattern based on the current perspective of zooming out, ultimately infinity means the pattern we see could actually be just a small part of the true repeating pattern that could be completely different than the snowflake currently seen, or it could not even have a repeating pattern at all.

[@youtubeuniversity3638]

How would one generalize this to higher bases, like working with 0 1 and 2?

[@dayhill9855]

So just so I know for the combinatorial property of 0w0w0 or 1w1w1 can w be as simple as just 0 or just 1 or does a binary word have to have so many digits to be a binary word?

[@MathVisualProofs]

Even just 0 or 1 works (even empty word). Pretty cool right?

[@dayhill9855]

@@MathVisualProofs so even 111 or 000 will never be there? Interestingly I've always been intrinsically fascinated with this sequence pattern and have severe OCD disorders and upon finding out it actually has a name I notice many people with OCD say the same that this pattern is intrinsically in them, I think there's a link with it and OCD.

[@MathVisualProofs]

@@dayhill9855 Right, those strings don't appear. If one of them did, it wouldn't be the fair sharing sequence anymore because the one person involved would get an extra item at one point and would have more 2 more than the other person for a short moment (whereas the idea is that each person has either the same number of choices at any given point or just one more than the other; and they each spend similar time with the lead of one).

[@bahachicken89]

wow, I used to this and still do this ever since I was 13-14, I do taps with my left and right hands. firs I do LRRL then I imagine L=LRRL and R=RLLR, so what I had done isn't LRRL but is actually just L so I complete the LRRL sequence which is now LRRLRLLRRLLRLRRL so the sequence is now complete, but actually not because now I imagine that L=LRRLRLLRRLLRLRR and R=RLLRLRRLLRRLRLLR so I didn't complete the sequence of LRRL, I only completed step 1. and I go as far as I can before messing up. just realized now that this is actually a thing. wow. I also used to constantly double in my head starting from 1 before I learned about powers, this moment really reminds me that moment.

[@jakobthomsen1595]

Amazing!

[@MathVisualProofs]

Thanks!

[@gulesinsezenklc4583]

...thanks four binary thue Morse.

[@piguy314159]

While I haven't proved it, the "power sums" property still seems to hold if you generalize the Thue-Morse sequence to other bases. Define GTM_b(n) = the base-b digit sum of n, mod b (e.g. GTM_3 would start 012120201120201012...) and let r, k be nonnegative integers; then if you partition the first r*b^(k+1) integers into b groups such that m and n are in the same group iff GTM_b(m) = GTM_b(n), all groups will have the same sum of k-th powers.

[@gulesinsezenklc4583]

now I've got these 0s and 1s allover my FILES!

[@columbus8myhw]

I'm curious about playing "Thue-Morse chess": the player to move is determined by a Thue-Morse sequence (white, then black, then black, then white, then black…) For extra fun, shift the Thue-Morse sequence by a random offset and don't declare whose move it will be in advance.

[@MathVisualProofs]

Check these slides: cs.uwaterloo.ca/~shallit/Talks/green3.pdf. Infinite chess with tm is a thing.

[@abdelmadjidabdelli7781]

Pascal triangle horizontal rows sum are a doubling sequence, in 2018 someone introduce a little change, he added sine wave coefficients (1,+1/2,0,-1/2,-1,-1/2,0,+1/2,+1) to the sum and gets fib sequence !!

[@youngcwel]

7:53 I literally cried there LMAOO😭😭

[@MathVisualProofs]

😀👍

[@fawzibriedj4441]

When I saw the Koch curve, I thought "This must be a joke..."

[@kafkaontheshore9102]

How is this useful? Can't you just make player 2 take 2 on their turn and then 1 therefore?

[@MathVisualProofs]

This shows how you should do multiple rounds. So yes, player 1 goes then player 2 goes twice. Then player 1 goes once, then player 2 goes once, then player 1 goes twice and player 2 goes 1, etc.

[@phiarchitect]

I wonder if Fibonacci is hiding in there somewhere?

[@MathVisualProofs]

I don't think so. Though there is a related Fibonacci word.

[@muqtarjamaegal6071]

I watched 3blue 1brown but I understand here you are hero please calculus

[@MathVisualProofs]

What kinds of calculus would you like to see? Not as easy to work visual proofs in

[@matteoaldegani228]

Q=1/P=(2)^(1/2)

[@MathVisualProofs]

This doesn’t follow. Q is unknown still as far as I can tell.

[@Jaime-mq8lj]

im sorry but the 0w0w0 made me cackle

[@strictmec4139]

Not me thinking of using this in gambling 😂

[@ibrahim_raven1701]

200mg of Modafinil lead me here.

[@MathVisualProofs]

Thanks for checking it out

[@ibrahim_raven1701]

@@MathVisualProofs .-- . .-.. -.-. --- -- .

[@purplenanite]

I got Q=1.628160129718 as to what that is in closed form - not sure

[@MathVisualProofs]

Very cool! Yes, we can definitely approximate it, but the question is if you can find a "nice" form for it like there is for P :)