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.
This entire youtube channel is a gem btw.
Delightful video on the lambda calculus and other compution concepts https://www.youtube.com/watch?v=RcVA8Nj6HEo
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 miss having the crazy math dreams I used to get when I was in college. If only life could be an endless stream of turning over new stones.
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."
[dead]
s/wonder/wander (although wonder kind of accidentally works too)
I loved this one about periodic double pendulums and their state space https://www.youtube.com/watch?v=dtjb2OhEQcU
This channel manages to be both informative and psychedelic at the same time, I love it. Their video on the quintic is amazing, too.
https://www.youtube.com/watch?v=9HIy5dJE-zQ
I enjoyed this and it warmed my heart a bit to see 4m views for some math animations. Maybe popular on r/trees or something :)
I was curious what they used to generate the graphic's and it's their own tool:
https://github.com/2swap/swaptube
The emergent patterns are... stunning. Especially the ones around 2m35s https://www.youtube.com/watch?v=kL9BTbIGxLg&t=2m35s
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!
It reminds me of the billiards segment in Donald in Mathmagic Land[0]. I absolutely adored it as a kid.
0: https://en.wikipedia.org/wiki/Donald_in_Mathmagic_Land
Wow this is stunningly beautiful and extremely cool. And I learned something I had never heard about.
BRB going to go make hyperbolic outer billards art
I think my favorite 2swap video is still the "solutions to the quintic" video [1]
[1] https://youtu.be/9HIy5dJE-zQ
I will never cease to be amazed by the range of thought experiments that get published as math papers.
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.
Don't know why but coloring and backround sounds of this animation feel very uncomfortable to me.
I could imagine some people getting motion sickness from watching.
Could also be the Stars of David (1m46s) and swastikas (2m24s) that end up drawn by the repeating patterns.
I gotta know if the music is relating to the math or image in a generative way or if its just vibes/style.
The music is by 6884; the track for this video (I assume commissioned specifically for it) is available here: https://6884.bandcamp.com/album/outer-billiards
Very rare to see such visual beauty in math like this. Well done.
Shooting pool with dmt entities right here
Stay down. After your shot, don't stand up right away.
LSD is worse
Youtube quality really let's stuff like this down.
oddly I felt like the compression artefacting during the hyperbolic part enhanced the psychedelic effect
This is math junk food. Pretty though.
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.
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...
I would call it art, not junk food.
How so? I think applied mathematics is the exception.
It made me reconsider what I think about mathematics in general. So, if it was junk food, it was a nutritious one at that.
Entire math is junk food. Real animals don't touch that.
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.
Great, but audio and audio mix is absolutely horrendous and super turbo 2.0 annoying. So much so I had to turn it off and read CC.
I liked it, felt right for the video
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.