Visualizations like this truly highlight how much there is to be gained from viewing the 3D phase space, but also how much richness we miss in >3D!
(I wonder if there are slick ways to visualise the >3D case. Like, we can view 3D cross sections surely.
Or maybe could we follow a Lagrangian particle and have it change colour according to the D (or combination of D) it is traversing? And do this for lots of particles? And plot their distributions to get a feeling for how much of phase space is being traversed?)
This visualization also reminds me of the early debates in the history of statistical mechanics: How Boltzmann, Gibbs, Ehrenfest, Loschmidt and that entire conference of Geniuses must have all grappled with phase space and how macroscopic systems reach equilibrium.
The conclusion I’ve come to from works like Flatland, 4D toys, etc., is that we simply don’t have the neural circuitry to grasp anything beyond three dimensions. We can reason about them, we can make inferences about the whole from partial understanding, but we cannot truly grasp more than three, or perhaps only for an instant of forced conceptualization using heuristics like you mentioned. Even three is a stretch, our minds have adapted to build a three dimensional realm from something like a 2.5 dimensional field of combined visual, tactile, and auditory stimuli. I suspect 3D reasoning itself is a huge adaptive trait compared to most other animals.
I've managed to visualize a Klein bottle in 4d. I easily visualize 3d objects. However I can't really do color - I startled myself recently when I briefly saw red. On that aphantasia test with an apple, I can hold it's 3d shape, but no surface texture or color.
People seem to have surprisingly different internal experiences. I don't know how common 4d visualization is, and I suspect even those capable require exposure to the concepts and practice. However I do think it possible.
The blind French mathematician Bernard Morin is well-known for creating the first visualization of a sphere eversion, a method for turning a sphere inside out without creasing it. His work was based on Stephen Smale's 1958 proof of sphere eversion's existence and on ideas shared by Arnold Shapiro. Morin's method involved constructing a sequence of models, including his "Morin surface," to demonstrate the process.
Your hippocampus has several special clusters of neurons whose members activate and deactivate based on your body's understanding of your position and momentum in a 3D world.
The arrangement of these neurons physically corresponds to reality, and so things are pretty hardwired.
Repurposing these neurons might be possible with advanced training and nootropics, but I'm not sure. You might have better luck engaging other parts of your brain, for example using metaphor or abstraction such as mathematics.
For me, being able to visualize 4D would imply that I can picture four mutually perpendicular axes, something which I find completely impossible for me to do. And I thought it is impossible for any human brain. It would be fascinating if I am wrong.
At least for 4D, would you not consider 3D-over-time as a four dimensional model? Doesn't watching the evolution as seen here allows for building up an intuition ?
Well, what's interesting about 4D is that's not just an extra dimension slapped on top, it's extra rotational degrees of freedom. You can't really get that with time (at least not until you get relativistic, and it still would be hyperbolic rotation, not euclidean).
Can we train our neurons? Like the experiment where human vision adapted to upside down image, could our brains somehow adapt to understanding 4D data from VR headset?
I'm sure some form of training is possible where you get a better understanding of a 4D universe with some limited inference abilities, but with a bad analogy, this would all be "software emulated" with no hardware acceleration - we only have the latter for 3D and we can't update it without a hardware change.
Yes, I believe it's possible to train our brains and learn to perceive better in higher dimensions. There's a great description in the science-fiction book Neverness, where pilots meld their minds with the spaceship computer to visualize and navigate hyperspace.
Good point, why not? Communicating it back to us could be a problem.
Hmm.. what if future ais hide data from us in dimensions we can’t wrap our heads around?
He also did a full video series. "Chemical History of a Candle" was originally a series of lectures given by Faraday; he recreated the lectures and the demonstrations. Definitely worth a watch, especially if you don't have a conceptual understanding of fire
Awesome, thank you! "a series of six lectures on the chemistry and physics of flames given by Michael Faraday at the Royal Institution in 1848, as part of the series of Christmas lectures for young people founded by Faraday in 1825 and still given there every year."
This is interesting, but why does he speak archaically? I understand it fine, but I'll have to stop the video every 5 seconds to translate for my kid. Is it part of his style in general?
Thanks for linking that. I did not know the Engineer Guy had a website. His youtube videos are superb. I need more of them, and more often. I was going to comment that I think he is the Carl Sagan of engineering because he explains things with such clarity, and then I noticed on his website that he actually recently received the Carl Sagan award.
I highly recommend that everyone checkout his videos.
Brilliant and powerful statement. Kudos to Lichess. But this part:
> Imagine if scientists kept the result of every scientific study to themselves. The same work would have to be done over and over again as everyone was forced to reinvent the wheel countless times to do anything at all. Instead, scientists share their work and collaborate which benefits all of us.
raised my eyebrows. If only it were true. Aside from paywalled journals, we don't have a centralized repositories of data in most fields, probably because a lot of it is proprietary (or intends to be) in the first place.
This is brilliant work. I've been playing with this for the last half hour and have found cool papers that I otherwise wouldn't have. To me, this has a great edge over google scholar and lens.
A wishlist:
1) To be able to share maps with fellow researchers, not as pdfs or bib files, but as a url to the map so that they can interact with it. I can see that a "share link" is possible for an article, but not for a custom map. Such a link could also be used to embed such maps on sites like how one can with plots from https://ourworldindata.org/
2) To be able to get more metadata like sponsor, country, etc. like https://www.lens.org does.
3) To be able to upload pdfs and annotate them. I can see why this is dicey, but if possible, would be great to see an integration with something like https://fermatslibrary.com/margins
> It's currently in early access so it's free to use, and you don't need to create an account to get started.
I know software developers need to eat and totally respect your choice in making it a paid service in the future, but I will confess a deep desire for such a thing to be affordable, if not free.
So given that the Super League won't work, what then _will_ work?
It seems to me that innovations can come from two forces:
1) Fairness: Make teams operate fairly. Audit thoroughly, no slap on the wrist
for violating Financial Fair Play, salary cap (I can dream can't I?), etc.
Given how our society favors the capitalist over the regulator, all efforts
on this front have amounted to mere asymptotic jokes.
2) Eliteness: Make the league the creamiest of the cream. Increase the
magnitude of what happens when athletes with incredible talent are
wedded with facilities that enhance and curate that talent. This was
in fact the point of the premier league, but the Super league took it
a step too far. The perception is that the "magic" in a game between
Real Madrid and Manchester United (Goliath and Goliath) is more
watchable than a game between Derby County and Manchester United
(David and Goliath). Except that Derby County can win that game,
have won that game, and the resulting dreams instilled in people (not
just fans of Derby County, but fans of United) comprise the very
source code that makes sports worth watching: a tale of the
impossible that parallels and informs everyday life. Some men with
$$$ in their eyes just don't seem to grasp that, for all their
"finger on the pulse" bragging. Or they did, but felt that the source
code could be rewritten and that's all that matters.
The "Eliteness" front tries to maximize the quality of a sport just short of
steroid use: story of the goose that laid the golden eggs. But what about
fairness? How can we set up incentives for that?
(I wonder if there are slick ways to visualise the >3D case. Like, we can view 3D cross sections surely.
Or maybe could we follow a Lagrangian particle and have it change colour according to the D (or combination of D) it is traversing? And do this for lots of particles? And plot their distributions to get a feeling for how much of phase space is being traversed?)
This visualization also reminds me of the early debates in the history of statistical mechanics: How Boltzmann, Gibbs, Ehrenfest, Loschmidt and that entire conference of Geniuses must have all grappled with phase space and how macroscopic systems reach equilibrium.
Great work Shashank!