Cohomology fractals
ฝัง
- เผยแพร่เมื่อ 18 ต.ค. 2024
- We explain how to make fractal images from cohomology classes in hyperbolic three-manifolds. You can try out the web app for yourself at: henryseg.githu...
Cohomology fractal zoom: • Cohomology fractal zoom
Non-euclidean virtual reality using ray-marching: • Non-euclidean virtual ...
This is joint work with Dave Bachman and Saul Schleimer. This video was filmed during ICERM's Fall 2019 "Illustrating Mathematics" program, which was made possible through the support of
The National Science Foundation (Grant Number DMS-1439786)
The Alfred P. Sloan Foundation (Grant Number G-2019-11406)
A Simons Foundation Targeted Grant to Institutes (Award ID 546235)
Fill the rest of that with clear resin and you've got quite a bowling ball
oh hi robert, internet is a small place after all
@@morkovija Quite.
Legend has it that Saul is still to this day standing by that blackboard and explaining what a cohomology fractal is.
How have none of the comments mentioned the shirts yet? THIS is what a mathematician is supposed to look like.
Yeah where are these shirts from?! :O
green screen
Murch
NOOOOOO!! I want to hear more of the ending lecture!!! It's so damn hard to find sources on math outside an unaffordable university degree -- please, link us to more of his lectures!! I was genuinely fascinated by it!
If you're interested in cohomologies, there are lots of books on algebraic topology :). I recommend Homology Theory: An Introduction to Algebraic Topology by James Vick
Most professors would flattered or, at worst, indifferent if you asked them if you could sit in on their lectures. Don’t be afraid to walk into a classroom and ask if you can spectate! Many professors will even engage you directly in the class discussion and provide you with class materials!
There's a nice symmetry there between the exponential growth as you go outwards of hyperbolic space, and the exponential difficulty in the last part of the video.
Thank you for letting me off the hook by saying so!
I was like, uh-huh, yeah, ok... yup, weird but makes sense... Wait... what? What does that mean again? Ok, now I know I have no idea what's going on.
I came here looking for an intuitve explanation of what cohomology is - and you deliver (8:36): Thanks!
The ending was so mean. I wish I could sit in his class for billions of hours.
this has to be one of the best thumbnails on youtube
The explanation was super interesting, is there a longer talk online somewhere? :)
Unfortunately I think it goes pretty quickly from this level of explanation straight to topology papers.
I think this would be great for a VR application, being able to immerse yourself in these spaces completely.
Hyperbolic spaces look like flat plainting
I've experienced an astrological projection into an infinite geometric area. Columns and other shapes as far as I could see. I still see the patterns in a lot of these videos and ones on sacred geometry.
i of course understand everything you are saying . thank you .
super-nice. plus the ending can be a clip by itself. thank y'all
Pretty cool! I suppose the pictures would be less interesting with spherical or flat manifolds. Have you tried with the other funky Thurston geometries, like Nil or Sol?
Love the effort you put into this presentation!
thank you. I could not really appreciate the other video before this. But i would have really liked an extended explanation
I've struggled to push my math skills further since college. A lot of stuff on the internet is either too basic, or way above my head. It was really cool to see a video on a subject that was accessible to me, but also showing me cool new things. Thanks!
I found that purchasing graduate textbooks and looking through them can be pretty helpful for pushing one’s math skills beyond undergrad.
Hi, Cheyne. I'd guess at this point you've found some of life's big secrets. I'm happy for you!
Please tell me that lecture actually can be watched in full
There isn't much more to Saul's "lecture" at the end - we only went as far as we needed for the joke! But he's on twitter @SaulSchleimer, so you can ask him what he would have said if he continued...
If you want to see Saul (green shirt) talk about this for an hour, see here: icerm.brown.edu/video_archive/?play=2037
Excellent video. Very interesting, informative and worthwhile video. I look forward to more of your videos.
The way you slowly made the lecture less accessible made everyone more interested I wonder if this technique can be used to increase curiosity in kids for teaching methods lol
When do you suppose our computers will be able to instantly render a Mandelbrot zoom at a similar speed? Thank you!
They pretty much can already, though it depends on the resolution and the computer
🤯🤯🤯 If you haven’t read Greg Egan’s “Diaspora” now is the time to do it
Awesome! Thank you for explaining, I followed at least a good chunk of it! Now where can I buy a shirt?
If there were a love button instead of the like I'd leave that in a heartbeat. I aspire to be like ya'll, especially in your community of people who are curious and capable about something important. Q: Are we literally seeing math during a trip? If so then then this must apply to all vision all the time, but somehow what you've shown feels more fundamental than our (albeit beautiful in their own right) neural network abstractions that we interpret out visual inputs with.
It would be nice if you listed the references he talked about in the explanatory segment at the end. And perhaps, if you uploaded/linked to the full explanation.
We will hopefully have more of a full explanation in an upcoming paper.
hahahah that fadeaway
Cohomology fractal drip 🥶🥶🥶🥶🥶
This is quite beautiful thank you just an old cowboy.
And to think, if he had only worn socks in the VR video, another cohort of people would now be learning about Cohomology
How does that cohort compare to the legion of bare feet afficionados now taking their first steps into algebraic topology?
Cool!
Explanation of Dave And Henry Was Quite Easy to Understand
Gerard Washnitzer is a relative of mine just distant enough for me not to know personally and I’m the developer of extensive mathematical models of a nannoreplicator. My nannoreplicator uses fractal imaging to mass produce molecules cells and tissues at low data and uses cohomology to design itself as a simple fractal as well. Knowing this, you’d expect me to be an expert on cohomology, but I built my Pixel Fractometry system from scratch. I would like to know more about the real Monsky Washnitzer cohomology theorem.
Very well done gentlemen. I sincerely hope you keep it up
he was just about to explain the different classes of pattern!!!!
I'm looking for ways to come up with original ideas for (amateur) stained glass projects. This kind of app is really cool for that. :)
More please
"Hey kids, wanna buy some infinity?"
This is awesome!! Didn't understand the construction of the "visual sphere" where he talked about ray from center connecting the pixel (what is a pixel?) and counting how many times it intersects with the lifted surfaces in universal cover of the 3-mfld with sign ( not sure what intersection would be -ve though).
A pixel is one of the dots on your computer screen - each dot gets a different colour in order to display something on the screen. The “visual sphere” is the sphere around you, of directions you can see in. You could also think of it as tangent vectors of a fixed length based at a point. We count intersections of rays with a surface, and since the surface is oriented, we know which side of the surface the ray passes through, which allows us to count with sign.
How did you make that spherical model so well coloured? Seems very hard with 3D printing
This was great!
TH-cam subtitles: "Hi I'm annie. Hi 'm Solomon"
Watch the zoom sequence at the beginning of this video - from 0:10 to 0:45 - but stop playback at 0:35 and keep your eyes on the center of the image.
I seriously miss attending maths lectures
Unfortunately I cannot use the app on my computer (or it runs really slow and i dont want to use it). Are alle the manifolds in your app 3-manifolds? As far as I understand it 2-manifolds should give interesting pictures too? Why do you choose hyperbolic manifolds? Can you get similar pictures for higher cohomology classes by building simplices out of rays? Amazing and fun to watch video thank you
All of the manifolds in the app are 3-manifolds. It would require an extensive rewrite to include 2-manifolds as well, just on the back-end before you get to draw anything. Then the analogous picture for a 2-manifold would be a 1-dimensional picture, which likely wouldn’t look that interesting to us 3D people. We use hyperbolic manifolds for a few reasons. First the data is readily available in the software SnapPy. There isn’t (as far as I know) software to explore manifolds with other geometries. This in turn is because manifolds with the other geometries are far better understood by mathematicians, so there was no need for anyone to write a SnapPea equivalent to understand them. It’s an interesting question though - what would cohomology pictures look like for non hyperbolic manifolds?
The video has an answer for if the manifold is the three-torus: you get a pretty boring half light/ half dark picture. I guess that most 3D geometries would have similarly boring pictures.
I love this educational content :) Thank You
I would love to buy one (or maybe several) of those shirts. They look awesome!! Where can I get them?
We never ended up making them available - the material is polyester rather than cotton, so they aren't that comfortable, and the setup required with the company that made the shirts looked a little too messy for me to have got around to doing it...
@@henryseg hmm... But they're just so perfect for clothing patterns!
@@henryseg Not may t-shirt places that do custom prints support patterns that take up the entire shirt like that. That sucks the material isn't ideal, I'd love an ultra-soft cotton version. What t-shirt company does the polyester prints?
Merry Christmas, Dave, Saul and Henry! :)
I can't figure out my times tables and people are over here making 3D FRACTALS.
It should be noted that that's not quite what the geodesics look like on the torus as you drew it (but rather the geodesics in the uniformly flat metric)
Right, we are being a little loose with the geometry there.
I love this
I got lost at the red loop, blue loop bit at the beginning. Are the loops and the surface of the toroid the entire space you're talking about, or are you talking about them in the sense that the observer is a 3D entity in 3D space, looking at the surface of the toroid around the surface of which the 3D space is curved? If the former is the case, then wouldn't the red and blue loops obscure your line of sight such that you would never really be able to see what they truly are, but rather just red and blue swatches of color filling your entire field of vision, like in Flatlander?
You're doing fuzzy counting of how many surfaces, or adjusting the exposure as per distance from nearest surface or something? Otherwise you'd keep getting jumps in the image when you float though the seem.
Ah, well spotted. We keep track of when the camera goes through the surface, and update the initial count for each ray accordingly. This removes the jumps.
Thanks
Ah... light rays are BENDING! I get it! ... ... wait... I don't get it. Seriously what amazing videos you guys make.
How can the co-homology fractal converge as the radius goes to infinity, if it changes sign infinitely often? Is this like the popular visualization of the Mandelbrot set where the points outside the set are colored by the rate they diverge?
This is an interesting and subtle point, which we gloss over in the video. If you increase the radius of the visual sphere beyond the point at which the image stops being blurry, it starts turning into noise. This is because we only send one ray per pixel - if the rays keep going further, then they end up getting less and less correlated with the neighboring pixel rays. If you take the limit as the radius goes to infinity, you don't get a well defined image.
We think that the correct way to do this is not to think of the limiting object as a function that takes a point on the sphere and returns a number. Instead, it should be a "distribution", that takes some region on the sphere, and integrates over that region to get a number. To draw a picture with some pixel density, you then integrate over each pixel, getting a colour for each.
On coloring points outside the Mandelbrot set: we are coloring by counting, but I think that's about as far as the similarity goes.
@@henryseg Very interesting. I'm supposing you don't mean integrating over the limit object, if it isn't well defined, but integrating over a visual sphere of very large size.
@@MushookieMan A distribution is more general than a function, so that's what we want for the limit object. For a simple example, see en.wikipedia.org/wiki/Dirac_delta_function (which is not actually a function!)
Is there an explanation for when I zoom in on the light dark split I get such odd visuals (including very small patterns that resemble cellular automata?)
Likely floating point noise. When you fly towards the “lighthouses”, your position in the manifold is described by exponentially increasing numbers, and eventually we lose precision.
@@henryseg This explains the noise, but I don't understand how floating point impresicion would lead to the intricate evolving patterns I see when I zoom into the lighthouses.
Nice!
Is there a shop where I can buy the shirts shown in the video?
why did you cut off the end? i didnt understand much but i was still learning things. why cut off i still could have gained stuff
the cohomology rant fadeout was so funny 😂
i think this faithfully represents how my girlfriend must feel on a regular basis
I like the thumbnail
Wait where can i get these shirts
8:03 -- This is where I saw this! Thank you so much gents! ❤ ^.^
Love your videos but I can't actually watch these fractal ones because the zoom is too fast for me. Any possibility of slower zooms in future videos?
John Wehrle Sorry about that! If you want to explore the fractals, you can choose the speed that you move in the controls menu of the app. As for the videos, TH-cam does have an option to watch a video at slower speeds. You could try slowing it down for the parts where we show the fractals.
I don't understand about half of it but still very interesting :3
Could make something physical (even if not hyperbolic) with tainted flat or curved mirrors or series of mirrors to get wrapping to work. :-o
How do you feel after exploring the space in VR?
We have yet to put this in VR, although it shouldn't be that hard to do.
It might be hard to stay standing up. :)
I did a PhD in geometric group theory - if you'd like I can look into Fuchsian v QF v DD to try and keep going "past the end of the video"?
Good content, but treat your cubes with more love 4:27 💕
Is any fractal based on circles, or is there an interesting example where it isn't ?
9:26 okay but isn't it just a function to the integers Z? or is that just in this specific example? is the reason he says "a function from loops to, say, the real numbers" because in general the codomain can be any field?
You’re right. In the way that we explain it you’d get an integer. In the code we use floats because it makes it easier to get consistent contrast levels across different manifolds.
The cohomology fractals appear, to me, to be MORE self similar than mandelbrot fractals. Is there any truth behind this subjective view? Can this difference be quantified?
With cohomology fractals you get back to exactly the same image by zooming in appropriately. Mandelbrot you get something similar but not identical.
What is the universal Cover?
These cohomology fractals appear to be exactly the sort of visual distortions people see on shrooms: I bet this says something important about how the visual cortex is altered by the presence of those active chemicals.
Can Somebody explain to me in Short what the Heck Happened around Minute 5:00-5-30 ?
What geometries of tiles can you tile the universal cover with? Square, triangular, hexagonal, Penrose? What about tiling them like Goldberg polyhedra or geodisic spheres?
If the tiling comes by unwrapping fundamental domains of a three-manifold, then individual domains have to be related to each other by some group of isometries of the universal cover - for the torus the isometries are a Z x Z of translations. So you couldn't do something aperiodic like a Penrose tiling, since it doesn't have a group associated to it. I don't think that something like Goldberg polyhedra would work either, since the individual cells are not isometric.
That said, I think the aperiodic generalization of this would work similarly to the Ammann bars
upload.wikimedia.org/wikipedia/en/5/5c/Ammann_bars.svg
Each tile is decorated by lines (in the 3D case I guess it would be surfaces). For the Ammann bars, you get straight lines. In general, I guess you'd get some sort of curve (in 2D) or surface (in 3D).
maybe these are what 4D would look like to us an infinite moving never able to get any wear for use but if you were able to travel 4d you might find something different ?
How is it a function to R if it counts the number of times it goes through the surface. I would thing that would be a function to Z
A function to Z is also a function to R. In the code, the weights are normalized so that the sum over all faces is 1, so in fact it would be most precise to call it a function to the rationals, Q.
This makes me think of those spherical compass with the globes and my idea would be to put a light in it and make a colidoscope 😊
Fantastic stuff. I was actually following quite well until the dread phrase "Since you've made it this far...".
I see what you did there.
I want to see the observer rendered with some kind of avatar in these spaces. Maybe even add some other objects, with a simple, zero-g physics engine... 🤔
Came here to see if someone else has astounded by the precision of the circle at 3:35
I can't even make a sentence straight.
I wish you hadn't cut him off. I was really interested. 😭
I've got a large proportion of "whoah" microscopic proportion of legwork idea (as coding should always be) involving listening to the visuals in and video/animation if anyone is interested in trying it.
"doubly degenerate surfaces" *looks around at apartment* yep mhm
Your cube example resembles the tesseract.
oh man, that grid made me think of qpu's
When you don't need psicodelics cause you have maths hahahahaha
Why not combine them? :)
see: "hyperbolic geometry of dmt experiences"
What happens if you do a quarter twist when you glue the cube sides together?
If all pairs of sides get glued with a quarter twist then the edge degrees are three, and you get a spherical manifold - the universal cover looks like the tiling of the three-sphere by the faces of the hypercube.
@@henryseg, I'm curious. Do you have a mental image of how the geometry must be spherical or of how it tiles?
@@farissaadat4437 I only know this because I made a sculpture that uses precisely this symmetry: www.3dprintmath.com/figures/3-31
These peeps are cool af, wouldhangwith/10
Anyone who wants more intuition on this sort of stuff should look at everything Jeff Weeks has ever made
(Particularly "Curved Spaces" geometrygames.org/CurvedSpaces/index.html.en)
Henry looks like a 25 year old
and at the same time
+10 years
+ another 10 years
The lotion that works
backwards
and
forward
Spoiler
12:50 you zoom out so I can see the backs of your heads. Cute!
could someone make this vr?
everything starts to melt if you look at this long enough
it's the Gorosei
I've seen this 2:43-... cube/3-torus topology before th-cam.com/video/-gLNlC_hQ3M/w-d-xo.html
Three people clicked the dislike button by accident
What are the implications of These hyperbolic manifold's? Are you explaining the universal laws? Consciousness? Or is this just a fun mathematician hobby?
Just Field Theory here -- th-cam.com/video/cpCtgzWSlTc/w-d-xo.html --
This is so cool, but I am so confused
I love the topic but I can't stand the sound or feeling of chalk. I really wish you had whiteboard or something.
...fortunately my interest in the topic overcame the feeling of the chalk sucking all the moisture of of my fingertips.