Thanks for sharing. I remember reading Introduction to Lambda Calculus many years ago and it was one of those times where I had crazy math dreams and would wonder around SF convinced everything is functions. Like literally everything.
I wonder if it's an age/brain development thing. I also had crazy math and programming dreams in my early 20s. I don't get those any more, though I still spend a lot of waking time (possibly more now than back then) obsessing over math and programming problems.
I don't obsess anymore. I just make observations and move on. For most problems, there are many facets with few distinctions.
Reminds me of a Pynchon novel or my favorite lyrics from The Dismemberment Plan.
"So in the end, whatever, we die, we dissolve / Equations unbalanced, riddles unsolved / And we were never connected or involved / Except for the intersections and crazy mathematics / With no time and no space and no schedule and no place."
I saw a video series long ago where a guy kept playing with shapes algorithmically like this, and his catch phrase was something like "oops, I made a fractal again". Wish I could find it!
The video didn't make this clear, but it is answering real questions. Its used in the study of geometrical optics, celestial navigation, etc. Ie set up a 'toy' problem that gives you insights into rather more complicated dynamical systems.
From Tabachnikov's book Geometry and Billiards:
Classical mechanics and geometrical optics, discussed in this chapter, are intimately related. The configuration trajectories of mechanical systems are extremals of a variational principle, similar to the trajectories of light. In fact, mechanics can be described as akind of geometrical optics; this was Hamilton’s approach to mechanics(see [3] for details). The brachistochrone problem is a good example of this optics-mechanics analogy.
The video contains no equations, but it cites original, ongoing, research that has been going on since the 50s. Sure, you could rip a bong and go "woah, cool colors", or you could follow the links and read, say, Tabachnikov's book.
Personally I found that the videos instantly made some group theory aspects of this stand out. Maybe I'm completely off, and that's not really how this is approached, but now I have a question/idea as a foil which I can use to look a bit deeper. In any case, the animation SW came from a researcher in the field, I don't think they are doing this to generate junk food, but to solve actual problems in their field.
p.s. From Schwartz' paper "Unbounded Orbits for Outer Billiards":
Outer billiards is a basic dymamical system which serves as a toy model for
celestial mechanics....
I tried as hard as possible to write a proof that is independent from Billiard King [the visualization SW], but still the reader of this paper would get a much greater appreciation for what is going on by learning to use Billiard King. A few minutes playing with Billiard King is worth hours of reading the paper.
That’s not entirely true. There are parts of mathematics that have useful real world applications. Sometimes even the parts we thought were esoteric and abstract turn out to have applications that were not conceived of yet at the time of discovery.
I think it might be the mix and the (low) level of voice over background and the stereo image combined with what I am listening on that makes me uncomfortable for some reason.
Delightful video on the lambda calculus and other compution concepts https://www.youtube.com/watch?v=RcVA8Nj6HEo
Reminds me of a Pynchon novel or my favorite lyrics from The Dismemberment Plan.
"So in the end, whatever, we die, we dissolve / Equations unbalanced, riddles unsolved / And we were never connected or involved / Except for the intersections and crazy mathematics / With no time and no space and no schedule and no place."
https://www.youtube.com/watch?v=9HIy5dJE-zQ
I was curious what they used to generate the graphic's and it's their own tool:
https://github.com/2swap/swaptube
0: https://en.wikipedia.org/wiki/Donald_in_Mathmagic_Land
[1] https://youtu.be/9HIy5dJE-zQ
BRB going to go make hyperbolic outer billards art
Could also be the Stars of David (1m46s) and swastikas (2m24s) that end up drawn by the repeating patterns.
From Tabachnikov's book Geometry and Billiards:
Classical mechanics and geometrical optics, discussed in this chapter, are intimately related. The configuration trajectories of mechanical systems are extremals of a variational principle, similar to the trajectories of light. In fact, mechanics can be described as akind of geometrical optics; this was Hamilton’s approach to mechanics(see [3] for details). The brachistochrone problem is a good example of this optics-mechanics analogy.
Personally I found that the videos instantly made some group theory aspects of this stand out. Maybe I'm completely off, and that's not really how this is approached, but now I have a question/idea as a foil which I can use to look a bit deeper. In any case, the animation SW came from a researcher in the field, I don't think they are doing this to generate junk food, but to solve actual problems in their field.
Outer billiards is a basic dymamical system which serves as a toy model for celestial mechanics....
I tried as hard as possible to write a proof that is independent from Billiard King [the visualization SW], but still the reader of this paper would get a much greater appreciation for what is going on by learning to use Billiard King. A few minutes playing with Billiard King is worth hours of reading the paper.
https://arxiv.org/pdf/math/0702073
edit:
ah, and here is a Dogru & Tabachnikov paper where they make the connection between this problem and celestial mechanics clear. Very readable!
https://drive.google.com/file/d/1sm47vPUwDqfcY-IGEKSoKIBqyqr...