The Ancient Mathematics of the DVD Screensaver 

I had no idea that the DVD logo problem had anything to do with Bezout's identity! (The extended Euclidean algorithm finds your x and y!)

i already have brilliant premium lololol

Amazing video

"Diophantine equations are just integers, they're just basic, right?" Right?

Omg it's Not David I love your videos!

Glad we all agree that the Screensaver is the best tool for agreement

Hi David

How many times did he uhmmmm?

Are you david?

Amazing, wish I'd known that before making the video as I'd have mentioned it for sure!

I'm not sure which cheap versions of breakout you are referring to? But in this ZX81 version you could in fact control the angle of reflection by controlling the exact location the ball hit the bat as you can see in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-ZOH4qAqhXvo.html Which I believe was fairly standard behaviour for such games

Thanks for sharing that story, can't beat a good maths teacher 🙂

exactly, was thinking the whole video about the eea and the linear combination of (s,t) that it gives, which is the solution to the problem

Ahh yes, you're absolutely right! I did all my working out before watching yours, and when you did it as "a bounce is on the first frame when it hits the edge or *passes* the edge" I was jealous that you didn't have to make that assumption! And I think you're right that no generality is lost. Link is in the description to this video for those who want to watch, it's great!

Haha I was thinking the same thing

Jazz Emu immediately entered my head too!

„My eyes are now crrrrrispy to the touch.“ 🎵

I like the direction the video went, just haven't worked out whether it's the right one yet.

​@@KusaneHexakuis that a corner case?

this video is a coprime

dvd to blu ray was way more noticeable imo, maybe because tvs just got better

@@bingxiling9154also because CRT’s have better effective resolution than flat screen displays, so while the jump from VHS to DVD was a bigger jump in quality, the jump from CRT to LCD undid a great deal of that image quality improvement. If you had a top line HD CRT TV, you would have seen a massive improvement when going to DVD from VHS.

i know right? this is some 3blue1brown level stuff

It just got one more!

Exactly. And it is not in general. And if you want it to repeat exactly in a finite time you want nonergodicity and the right initial condition!

This is a nonergodic system in general. And if you want it to repeat exactly in a finite time you want nonergodicity and the right initial condition! This is like saying the Poincare recurrence time is finite! Which requires the resonance between the x and y degrees of freedom you discovered. Occurs in many nonlinear dynamical systems in mathematics, chemistry and physics. Dynamic billiards are well studied: en.wikipedia.org/wiki/Dynamical_billiards (And using the finite difference approach: en.wikipedia.org/wiki/Arithmetic_billiards), blogs.ams.org/visualinsight/2016/11/15/bunimovich-stadium/.

I hope you came back for the good stuff!

Sorry about that -- you won't believe how many times I changed my mind about this! You can see later in the video my spreadsheet has the coordinates running downwards. In the end, I decided that more people were familiar with the Cartesian Plane and how coordinates work there but I spent a long time deciding! Hope it isn't too distracting and enjoy the video!

Nice 👌

I think you're right! I briefly looked into this while planning my video, but decided against bringing it up as I didn't like the idea of introducing yet another problem, especially when I try to make the videos for a broadish audience. Thanks for watching!

no, a-b need to be a multiple of the gcd, not a divisor, which actually means starting positions where the indices differ by 1 in the positive direction will always only hit the corner when every starting position hit the corner. But in a simmilar note when the indices of the starting positions are the same they always hit the corner if going in the positive direction (but might be too obvious if you think about since that means they are literally in the line that connects to (0,0))

No, think about the 12p+8q=2 case... If there were any such p and q, then 4 (the gcd) would divide 2 (your m), and it doesn't.

Actually it's not if one knows how to relate things and substitute. What do I mean by this? We can use the slope-intercept of a line formula y = mx+b where the slope m is defined as rise/run (y2-y1)/(x2-x1) and b is the y-intercept. This is a basic algebraic linear equation. We can the rewrite the slope formula as dy/dx. Now, we can take the basic sine and cosine functions from their right triangle definitions: sine = opp/hyp and cosine = adj / hyp. We can use the relationship of this right triangle in standard form where the angle theta is the angle that is between the line y = mx+b and the +x-axis. With that we can see that within the slope formula dy/dx is exactly equivalent to sin(t)/cos(t). In other words dy = sin(t) and dx = cos(t). Therefore the slope of a linear equation is equivalent to tan(t). With that we can substitute and rewrite the slope-intercept form of a line as y = (sin(t)/cos(t))*x + b or simply y = x*tan(t) + b. This is why we are able to see periodicity within the linear relationships. Also for any number N that is rotated by either +/- 180 degrees or +/- PI radians it is the same exact thing as multiplying it by -1. Why? Well if we multiply 1 by -1 we get -1. What is the dot product of 180 degrees? It is -1. And arccos(-1) is either PI radians or 180 degrees. There's more to it than just that. Just from the simple expression of 1+1=2 which is basic algebraic arithmetic, this simple expression is in fact a linear transformation. This transformation of adding one to itself without realizing it is in fact the unit circle with its center located at the point (1,0). If we take the general definition of an arbitrary circle (X-h)^2 + (Y-k^2) = r^2 where (X,Y) is any point on its circumference, (h,k) is its center point, and r is its radius. We can see that this is a specialized version of the Pythagorean Theorem A^2 + B^2 = C^2. The definition of the unit circle located at the origin is X^2 + Y^2 = 1. The only difference here with the simple expression of 1+1 = 2 is that the unit circle is translated 1 unit to the right with its center at (1,0). If we take this expression and substitute it into the equation of the circle we end up with this: (X-1)^2 + (Y-0)^2 = 1. We can then take this and use algebra to simplify it and to solve for it in terms of Y. (x-1)^2 +y^2 = 1 x^2 - 2x + 1 + y^2 = 1 - 1 -1 x^2 - 2x +y^2 = 0 -x^2 + 2x = -x^2 + 2x y^2 = -x^2 + 2x sqrt(y^2) = sqrt(-x^2 + 2x) Oh wait a minute we have the square root of a negative number... Where did that come from? Yup complex numbers are embedded within 1+1 = 2. Why? Because the multiplication of i is a rotation of 90 degrees or PI/2 radians. Euler's formula and identity shows this in greater detail! In truth, all of these properties aren't just embedded within 1+1 = 2. They are embedded within the numbers themselves. If we take the expression or equation y = x which is an equivalence or identity expression which can be written as the function f(x) = x, this is still a linear equation and there is still a linear relationship of any given number x with itself. This line has a slope of 1, and a y-intercept of 0. A slope of 1 is the same as 1 = tan(t) which is 45 degrees or PI/4 radians. All fields of mathematics are related. It doesn't matter if it's Algebra, Geometry, Trigonometry, Calculus, Probability, Statistics, even Logic or Boolean Algebra. Within Boolean Algebra the OR gate is relative to addition and the AND gate is relative to multiplication. When you get into circuitry with the context or abstraction of logic gates and we use boolean algebra we can simplify things through the use of either The Product of Sums or the Sums of Products. More than just that, but Boolean Algebra and Logic itself has a direct connection with Log2 arithmetic. And Logarithms are Algebraic. Even our understanding of physics, and chemistry is rooted within mathematics. Why and how is everything related? Simply because of motion which is a transformation. Neither Energy or Matter can be created or destroyed they can only be transferred or transformed. There's also a direct connection between these, number and group theory, information theory, and fractal geometry. It may not be evident at first but for every linear transformation (moving along a straight line), there is always an implied rotation. This is something that isn't necessarily taught in schools or universities, however it is an underlying pattern that I've recognized that is always present. When you walk a straight path, your legs are swinging at an angle within the socket of your hip joint. When a bird flies through the sky and their wings are flapping in the air, they are rotating. It is to my understanding that you can not have linear transformations without some form of rotation and vice versa. They are interdependent of each other. Once you have one, you have the other. Otherwise we wouldn't be able to have reflections or symmetry. Just some food for thought! There is periodicity in y = x, never mind a + b = c. If there was no periodicity then there would be no identity. Just wanted to shed some light into the matter because it is quite sound. John 1:1-5, Genesis 1:1-5

@@skilz8098 Euclid’s orchard is viewing a square lattice of rational points from the origin. The angle θ of the slope as described is the inverse tangent since tan θ = opposite/hypothenuse. The rotation is given by exp(iθ) with angle θ in radians. The dot product of two perpendicular vectors is always zero. 1+1 = 2 is a linear transformation but I don’t find its reoccurring mentioning here too relevant. It is not standard to have the origin not at (0,0) when considering coordinates on a plane. It will shift any calculation if it’s not. The square root isn’t additive so this additional shift doesn’t derive it. It is easier to consider the complex numbers on the plane in their exponential form. Integers are a subset of the reals which is a subset of the complex numbers. Complex numbers can be written in exponential form z = r*exp(iθ) which is a succinct and useful way to describe the polar vectors that describe each point of the lattice in question. All mathematical fields are related and there are theorems which describe some of the relationships. Applied mathematics does solve many problems such as finding when exactly the DVD logo will hit the corner. Circular motion does propel most real world things forward. I think there are groups which are understood through symmetry that have faithfully representations as the orthogonal group which describes rotations but I’m not sure if it holds in general. It was something to think about and ponder. Functional equations do equal themselves and this is sometimes called their identity. You are very faithful. I was thinking more of Langlands program which equates periodic functions to algebraic expressions in a general way but was not coming up with anything then. Euler’s formula exp(ix) = cos(x) + i*sin(x), roots of unity, and their Fourier series is more of what I was after.

​@@Jaylooker Nicely stated. Yet it's the purity of math and the relationships of numbers that intrigues me. Especially when we consider how a simple equation or expression can generate an infinite complex structure with amazing beauty such as the Mandelbrot Set, and that's just the tip of the iceberg.

@@skilz8098 Then study math and do some of the practice problems. Studying Fourier analysis and harmonic analysis may interest you because of the periodic functions they describe.

@@Jaylooker I've written my own 3D Graphics / Physics Simulation in C++ using DirectX 10-11 / OpenGL 3.x. I started to learn Vulkan a bit too. I even worked on doing a 6502 NES emulator. I'm quite familiar with the math and physics! I've done a little bit of 3D Audio Processing as well.

Haha I agree, just a lightheaded dig at him really! Thanks for watching 🙂

... because there was no proof included?