The Ancient Mathematics of the DVD Screensaver

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⬣ LINKS ⬣

⬣ ABOUT ⬣
We've all seen the beloved DVD Screensaver and been excited to see it hit the corner. But is this guaranteed to happen? If it happens once, will it happen again? And long will you have to wait between corner hits? All these questions answered and more via the ancient number theory of Euclid and Diophantine.

Spreadsheet here:

ComputerCow video on the topic:

⬣ TIMESTAMPS ⬣
00:00 - Introduction
03:03 - Simplifying the Problem 1
08:40 - Simplifying the Problem 2
14:14 - Diophantine Equations
21:44 - Euclidean Algorithm
31:15 - Solution
36:21 - Extra Questions

⬣ INVESTIGATORS ⬣
Nothing for you here. Sorry!

⬣ CREDITS ⬣
Music by Danjel Zambo.
"Bionic" by Tobias Voigt.
"Cautious Optimism" by Apex Music.

IMAGE CREDITS
Smartwatch
Dashboard
iPhone 13
Square Lattice
Equilateral Triangle Lattice
Rhombic Lattice
Andrew Wiles
Fermat
Arithmetica
Etienne Bezout

B-ROLL CREDITS
"How LED and LCD TV's Work in Slow Motion [Original]" - Warped Perception
"Happiness comes from the simple things" - SpartaYoshi
"Bouncing DVD Logo Screensaver 4K 60fps - 10 hours NO LOOP" - Raúl Blanco

This video was sponsored by Brilliant.

[PedroPaulo]

That fact that you can remove some many of the variables in the beggining transforming the problem and that the solution is basiclly diophantine Equations is just so beautiful

[not_David]

I don't think I can express how excited I am to see a dvd logo hit a corner (edit: was not disappointed)

[johndray]

Before there were DVDs, there was a similar problem. In cheap versions of breakout (balls bouncing against walls) on early home computers (e.g. ZX81/spectrum) the ball moved by an integer number of pixels left/right and an integer up/down. In these versions the ball bounced off the bat with the same angle of reflection as incidence. This meant that sometimes it was not possible to clear a board as there were some bricks that could never be hit with certain ball angles. Oh the problems of youth!

[RunstarHomer]

I'm so impressed with how this channel has grown since its first SoME video a year or two back. This stuff is seriously higher quality than a lot of very popular math channels, and with deeper and more interesting theory. Thank you for all this!

[ketv]

When I was a little nerdling (3rd form, if I recall correctly), it was in the days before VCR's, to say nothing of DVD's. But I wondered about this exact problem, thinking in terms of a billiard table with m diamonds on one edge and n on another. Bless my mathematics teacher! She showed me the argument involving modular arithmetic, and taught me the _extended_ Euclidean algorithm. I think I managed then to prove most of the results in your video. I'd credit Mrs Smith for the fact that I'm a (semi-retired, applied) mathematician today.

[DustinRodriguez_]

What I find so interesting about this video is the way you use mathematics to tackle this issue. I expected that someone had dumped the firmware code of an old DVD player (I bet you could probably find an old Apex brand DVD player firmware somewhere online, the Apex players were very interesting products that would play basically anything you put on a disc no matter what format it was in, ignoring region locks and other things they were 'supposed to' not do) and you'd be going through their actual algorithm. That's just how I'd tackle it as a software person.

[Zoidle-doo]

Alex doing a 45 minute deep dive into the math of the DVD screensaver? Ohh god yes, I am so here for this.

[computercow]

Great video! Loved the live action visuals. Your explanation of the Euclidean algorithm avoided a lot of the leaps of faith I made in mine. And simplifying to a lattice is a brilliant way to make the numbers easier.
I think your assumption that the logo always hits a wall perfectly doesn't lose any generality, because even if a logo 'crosses' the wall a bit, it will still bounce back on the next frame, making the setup equivalent to a lattice one tile longer. You can confirm this a bit with the simulation I made for my video (which I believe I can't link without my comment being auto-removed by Youtube). If you use the default settings but set the X speed to 7 so that crossing a wall is more obvious, you can see that a screen width of anywhere between 395 and 401 leads to the exact same path!

[txzk]

How tf does this channel only has 43K subscribers? This channel is criminally underrated

[The-Limited-Fun]

You may have already seen it, but Mark Rober released a YT Short a few weeks ago based on this video! It's called "The REAL Truth Behind the DVD Logo".

It seems likely that they used this vid in particular, because they seem to pull the number 58460 out of nowhere, which you calculate at the end by "meticulously" counting frame-by-frame to get x=316, y=185. They also give the same answer of 16m 14s.

Although, they apparently didn't realise that these numbers only apply to one specific simulation uploaded on YouTube and wouldn't work for a normal DVD player! Even though you mention this immediately before giving your calculations. So that's a bit embarrassing.

And of course, they unfortunately don't give credit to this channel, and their video now has over 50, 000, 000 views.

I really love your videos, and this one specifically has helped me recently with a project I'm working on!

[tubebrocoli]

🎶 "Still waiiiting for the DVD looogooo / to hit the corner of my TVvvv" 🎶

[michaelturniansky]

Thanks for this. I've always mentally did this with any kind of rectangular grid (e.g. ceiling tiles), (Hi, ADHD!) but I never thought about starting in an arbitrary place, but always at a corner. I knew it always had to end in a corner, but never knew how to predict which of the other three it would be. Now I do!

[zilvarro]

This video is going to be a guaranteed hit.

[Tysto]

Dude, VHS to DVD was a massive improvement. 1080p to 4k is barely noticeable.

[larperdoodle]

Woah, the editing with the stop motion sticky note was slick.

[writerightmathnation]

@6:41
“Mathematicians have a name for this type of structure, by the way; a lattice.”

Well, lattice theorists don’t restrict themselves to integer points in the plane or even in any other Euclidean space, so you’re not accessing best terminology. This kind of lattice is better referred to as a Gaussian integer lattice because it’s in a plane, and so the elements of the lattice can be viewed as Gaussian integers (complex numbers with integer real and imaginary parts). In fact, you’re locating your grid so that it’ corner is the zero element of the complex plane, and so it’s better described as a finite initial segment of ‘positive’ cone of the Gaussian integers, and the more general notion is that it’s the ‘positive’ cone of a lattice-ordered group. If you analyze the approach, I expect there are some interesting related results in the theory of lattice-ordered groups with similar arguments, and so that this result you demonstrate will be a special case of such results.

Your argument uses periodic extensions that are still contained in the ‘positive’ cone of the Gaussian integers, so that’s where I’d think someone should look if they want to learn the deeper results.




…

[RepChris]

The timing of this video is fantastic considering ive just finished a module on elementary number theory in one of my math courses in university, so this is both a lovely way to see it applied, and a nice refresher of the more fundamental ideas. I really love your method of explaining stuff, and it usually complements the more "university" like explanations I usually have, while still being rigorous and in depth enough to not just be a cursory look into the topic that is more common on YouTube

[guillaume]

Now THAT'S what I've been wanting from this channel !

[ArtArtisian]

I'm so glad this channel exists. Doing the work that needs done.

[MrLuigiBean]

I've been wondering how to go about solving and lo and behold, this video was recommended to me! What's awesome is that I technically already knew about Bezout's Identity, I just never know how to apply it until now! This video was awesome!! 💯