Math and OCD - My story with the Thue-Morse sequence

Announcement: New Book by Luis Serrano! Grokking Machine Learning. bit.ly/grokkingML
40% discount code: serranoyt
The story of how I once stumbled upon a famous sequence, the Thue-Morse sequence, while walking on the sidewalks as a child,, and the subsequent tale of how this sequence satisfies amazing properties regarding sums of powers of integers.

[@devinjoe3123]

I thought I was the only one. Thue-Morse torments me to this day whenever I accidentally tap my foot.

[@jeremydavie4484]

I have the same idiosyncrasy for breathing in and out (sometimes)

[@nielenventer9747]

Stumbled on this video from your auto-encoder one (which is really great) and couldn't believe what I was seeing. When I was much younger I was completely obsessed with this same pattern and would tap it with my left and right teeth during class (once you start it's hard to stop it). The teachers were convinced I was chewing gum and I'd always get in trouble. Of course they'd never understand when I tried to explain. Thanks for sharing, it's really validating to see not just you but all the others in the comments with the same experience.

[@markmarki9511]

I used to think about this all of the time as a child. I would knock on something usually LRRL RLLR RLLR LRRL and so on. And I could sit all day and knock and think about how long can I continue on this pattern. I was never able to describe this to anyone. Then I randomly googled ABBA BAAB BAAB ABBA. And my mind is blown the fuck away. I never knew anyone else was able to even understand me. Wow.

[@SerranoAcademy]

Thank you for sharing that story, amazing! I used to think I was the only one, and I'm so happy to know that we are connected with the sequence! :)

[@andrewkeeleyyonda]

I'm so glad to know there are others who do this!!! I've been doing this pattern since I was 10 years old, and still do it obsessively in middle age. I see it as a pressure release for my brain--if I am stressed, anxious, or just don't have enough to occupy my brain, it activates this pattern in my shoulders, toes, any symmetrical body parts.

[@ozzyfromspace]

So, from what I’m reading in the comments, other people grew up with this pattern. It’s amazing how connected we all are. I’m from Southern Africa, and through mathematics, I can totally relate ☺️🙌🏽

[@SerranoAcademy]

Float Circuit, thanks for your message, so happy to hear we’re connected through math!

[@henrilaloux4492]

Since I was a child, I've been doing the same thing with a race between two people. One starts before the other, ... I showed this pattern to my family and friends, but no one understood it. I am happy to see that I am not the only one who has noticed this pattern.

[@MrAmalasan]

Thanks Luis. great honest video. this world has the need for so much diversity and you help so so much.

[@SerranoAcademy]

Thank you Glen, that means a lot. :)

[@CoolBreezeAnthony]

The pattern in my mind was the same. I have OCD and of course, the sequence you are explaining was going on in my head as a child that became difficult to stop. It was excessive and I really am glad I have managed to deal with it. I only recently been made aware of the Thue-Morse sequence. It can drive a person to tears. Since I also have ADD I reach a point where I have to let go because I can't continue past a certain point.

[@nameismetatoo4591]

We need someone to do a study on this, because clearly this is not some one-off thing. I have mild OCD and ADD (inattentive type) as well, and as a kid this sequence got stuck in my head all the time. It was my way of making things "even". For example, say I had an itch on my left arm. I'd scratch it, but then I'd have to scratch my right arm. But it still felt uneven, because I ended on my right arm, so I'd scratch my right arm again followed by my left arm. Repeat ad nauseam. There's just something about this sequence that's intuitively "fair" or "even", and it would be interesting to study the relationship between this sequence and the way our brains work. Is it something different about the brains of people with OCD that leads to finding this sequence so satisfying, or is it universal to all humans and is just more likely to be noticed by people with OCD?

[@CoolBreezeAnthony]

@@nameismetatoo4591 I think there should be research on this. I also suffer Bipolar 1 so that only exacerbates my level of anxiety. I also find it necessary to spell all my words correctly before posting. I feel comfortable only when I am listening to complex music that helps take my mind off constant obsessive thoughts that go for days and perhaps months unresolved.

[@mahimanzum]

that was just awesome. Please make more videos and keep up the good work.

[@twoduece]

i genuinely cannot believe this but i do the same stepping pattern on sidewalks, but if i havent stepped on a crack then i walk with my left foot first per every sidewalk square instead of balancing that. ive never heard of anyone in my life that also did anything even close to that.

[@ozzyfromspace]

such a beautiful quiz question and even more beautiful solution 😭😊🎊

[@RichardASalisbury1]

Fun and brilliant!

[@CeleryBruh]

I am 15 and have only known about my OCD for just over a year. I get a lot of symmetry obsessions and always did as a small child. If I would accidentally hit my right hand on something, I would have to match the same on the left. What would bother me is that the right hand always started so it had a priority of sorts in my head. I would subsequently match the pattern by finishing it with a left then another right. 1001. One day I went one 'iteration' further, treating the entire '1001' as a '1' and '0110' as the '0'. 1001 0110 0110 1001. This would go on in my head for hours sometimes. Of course these weren't 1's and 0's in my head, they were usually based on the situation. I got interested in fractals again recently and tried to find correlations between this and shapes, drawing out the patterns in diagonal lines drawing my 1's as rights and 0's as lefts, and I would see many strange coincidences. I only recently decided to look it up extensively and found out about the thue morse sequence with my sister. I even found I had drawn the same recurrence plot diagram for the sequence in an old math book, purely coincidentally. You don't know how relieving it is to find someone who went through the same thing and how nice it is to come to some answers. Something interesting about this pattern I found while sketching it out in 'iterations' is worth mentioning. When on a square grid paper and treating the 1's as right diagonal lines (one square length to the right and one square length up) and 0's as left diagonal lines you get a strange zig-zag looking pattern. When repeating the pattern over the existing one -starting from the same point as before- but doubling the length of your lines, you get rectangles that form from these new lines. Those rectangles also face first to the left, then right, then right, then left, mirroring the initial pattern. Repeating this process a number of times leaves you with a strange looking mess of rectangles, but if you follow your original heirarchy of when you drew them, all rectangles found, no matter the size or what iteration they were drawn in, collectively still follow the pattern but by what direction they are facing. (seemingly matching how long the first ever iteration went for). Also, if you colour code each size of rectangle (which is the same as colour coding them by iteration or generation), each 'generation' of rectangles follows the same pattern by itself, again by which direction they face. Sorry if this wasn't very clear, just something interesting i found while experimenting with this. I might link to a photo of one of these sketches if it helps convey my thoughts. Again, thank you for uploading this, helps me make sense of it a lot more

[@SerranoAcademy]

Thank you so much for your comment, it’s amazing that we went through the same process independently! This rectangle pattern sounds super interesting! Do you have a picture of it? I’d love to take a look and explore more!

[@dvrsflrs]

Fascinating video, and somewhat recognizable :-p

[@Robfnord]

I used to go over this pattern in my head as a kid too!

[@cafeinst]

Magical sequence

[@scherwinn]

Great.

[@darrenstensland5301]

Very cool. Is there a video for the proof yet?

[@stephennehemiah]

Oh man ! Great Video ! I'd like to request you to place give us some coding videos on deep/machine learning :), would be great if you could do that.

[@ozzyfromspace]

Omg I promise I used to do the same thing with walking. If I stepped on cracks, it would drive me nuts that balance was broken even though no one could tell. Funny enough, I’m doing research on identifying operators that are compatible with groups in general and just found a sequence that behaves like the sum of the elements of the Thue-Morse sequence for small values and n=2 i.e. {0,1,3,5,6,...}. I’m having one hell of a night lol Fun video!

[@TheRevAlokSingh]

I do this exact sequence on sidewalks too.

[@SerranoAcademy]

:) we're sequence buddies!

[@olekravchenko]

Hola! Welcome to the FB group facebook.com/groups/afunctions/ about Thue-Morse sequence and math behind.

[@jjakinlade6278]

Yup, I have exactly the same pattern and drive to play it out. I also know one other person with the code.

[@smithingstuff5775]

Wow ive never seen anyone else do this apart from me. I like to visualise it like _--_-__-_--_-__- ...... But i also think about a similar patter aswel that goes; _--__--__--__--__--__--_, like RLLRRLLRRLLRRLLRRLLR....

[@jackwilson4324]

I currently have the same problem

[@BigAsciiHappyStar]

Here is an outline of proof. Unfortunately, the details are somewhat messy and hence omitted. NOTE: this proof works easier if we pretend the sequence starts at 0 not 1. For instance Red = 0,3,5,6 and Blue = 1,2,4,7 Suppose we want to show that sum of cubes works for 16. Let A/B be the sum of red/blue cubes. Every number can be decomposed as a sum of unique powers of 2. For instance 11 = 8 + 2 + 1 and we can expand 11^3 = (8+2+1)^3 using the distributive law. When fully expanded, A and B can thus be represented as a sum of powers of 2 (which may contain repeats). But if, for example, 8*2*2 appears N times in A then it must appear N times in B. To illustrate, (8+2+1)^3 contributes three lots of 8*2*2 to A, but (8+2)^3 contributes three lots of 8*2*2 to B. We can find a 1-1 correspondence between powers of 2 in A and B, hence A=B. The same proof also applies to higher powers, e.g. sum of fourth powers works for 32. Note that for this proof to work, A and B must have a very specific structure. For instance, the colouring at 9:39 is incorrect 😊 The proof can also be modified to show that e.g. the sum of fourth powers does not work for 16.

[@yeepyorp]

just another person commenting SAME although I'm just autistic and probably don't have OCD

[@saurabhiim]

Fantastic Dude ...but why you stop making videos in English ??

[@scherwinn]

Why 1 + 2 + 3 + 4 +... = -1/12 ???