Cannon-Thurston maps: naturally occurring space-filling curves

No video

Cannon-Thurston maps: naturally occurring space-filling curves

Рет қаралды 211,728

Күн бұрын

Saul Schleimer and I attempt to explain what a Cannon-Thurston map is.
Thanks to my brother Will Segerman for making the carvings, and to Daniel Piker for making the figure-eight knot animations. I made the animation of the (super crinkly) surface using our app (with Dave Bachman) for cohomology fractals. You can play with the app (on Chrome or Firefox) at henryseg.github.io/cohomology....
Also see:
Cannon and Thurston, Group invariant Peano curves, Geom. Topol., 2007.
Mumford, Series, and Wright, Indra's pearls, Cambridge University Press, 2002.
Some of these curves are available in t-shirt form at www.neatoshop.com/artist/Henr....
00:00 Introduction
00:28 The Hilbert curve
01:00 Approximations to Cannon-Thurston map
01:36 What space do they fill?
02:01 Symmetry of the Hilbert curve
02:34 Symmetry of the Cannon-Thurston map
03:10 The Hilbert curve is artificial
03:38 The complement of the figure-eight knot
04:39 The universal cover
05:20 Unwrapping the surface in the knot complement
05:51 The crinkling
06:50 Thurston's pictures
07:24 Comparing algorithms
08:23 s227
09:18 Carvings

Пікірлер: 484

@CodeParade 2 жыл бұрын

The crinkling of the disk reminds me a lot of how when you try to embed the flat torus in Euclidean space (isometrically), you end up with a fractal of crinkles on the surface.

@burakbalcioglu 2 жыл бұрын

That's exactly what I thought too

@saulschleimer2036 2 жыл бұрын

That is a nice observation!

@teo_lp 2 жыл бұрын

Wow, I just found a picture of that by googling it, does that embedding have a name?

@kaidatong1704 2 жыл бұрын

unrelated, but I remember trying to do smth similar, but less inspired / more forced. drew origami folding diagram for merging two angles into one (there was this hands-on activity for fitting coin of diameter x through index card hole with circumference 2x, that creates way too many tiny folds that I couldn’t do it with that thickness of paper, so had to improvise) it’s a long story. I saw some line replacement fractals, but each iterations gets multiplied by a whole number, so at least x2. I wanted lesser ones like fibonacci or smth, but got lazy and was like heck, just split it 90-10. but some parts being more detailed was annoying to my ocd, so just made queue for where to add detail next, like, start with 1. make .9 and .1, resolve .9 next, make .81 and .09, and so on. which has absolutely nothing to do with the nature of the thing itself

@neopalm2050 2 жыл бұрын

Interestingly, both are somewhat related to Gromov.

@kyrius_gm4 2 жыл бұрын

This is so confusing and i feel like i barely understand anything but i love it! Its so interesting!

@saulschleimer2036 2 жыл бұрын

Glad you like it!

@dawnhansen7886 Жыл бұрын

100% agree ❕️

@w_ldan Жыл бұрын

Yeah

@bellaF Жыл бұрын

glad I'm not the only one who feels like this lol ^^

@dreamsolutions3037 Жыл бұрын

Right? Me clicking on this video: "Ooh fractals!" Afterwards: "oh. Nice."

@mikeebrady 2 жыл бұрын

6:37 Henry: "You could imagine..." Me: "No, no I don't think I could imagine that."

@DavidG2P Жыл бұрын

What a time to be alive where intricate and incredibly complex mathematics can be visualized in such beauty!

@lydianlights 2 жыл бұрын

I definitely don't understand the details but you did a great job of explaining the general idea. I think one of the biggest things is that I don't really understand what a universal cover is.

@saulschleimer2036 2 жыл бұрын

We discuss this a bit more in our cohomology fractals video here: kzfaq.info/get/bejne/nM5yg8uaype4nWQ.html I am happy to answer questions, as well.

@peetiegonzalez1845 2 жыл бұрын

This was extremely interesting, but of course it’s such a complex topic that one ad-hoc video like this doesn’t do it justice. I hope you can make a future video, fully scripted, in which you explain clearly what’s going on. And yes, with more animations, and even some actual maths!

@saulschleimer2036 Жыл бұрын

We have various research level papers on this topic (on the arXiv, with the word "veering triangulations" in the title) that begins to lay out the theory. We have three more papers in various stages that will also appear on the arXiv "real soon now". This video is meant as an "explainer" (without too much maths) and an "invitation" (to a wider audience). At least, that is my feeling - Henry may have a different opinion!

@BILLY-px3hw Жыл бұрын

@@saulschleimer2036 Thanks it helps for us visual thinkers, a lot of time I get caught up trying to understand the math instead of just looking at th visual information. Is this related to any of the work Roger Penrose was doing with his tiling?

@saulschleimer2036 Жыл бұрын

@@BILLY-px3hw That is an interesting question. There is no obvious connection, but I think there may be non-obvious ones. In particular, the CT map approximations obey a "subdivision rule" where the tiles come in a particular order. The Penrose tiling also has a subdivision rule. It would be interesting to see if the tiles appearing in the rule could be ordered. We could then play a "connect the dots" game (as in the Hilbert curve approximations) and produce a plane filling curve from the Penrose tiling. I've not see this in the literature...

@saulschleimer2036 Жыл бұрын

@@BILLY-px3hw And now I have done a quick search, and found a paper titled "Space-filling curves on non-periodic tilings" by Fred Henle. So there is at least some work in this direction. The final paragraphs of that paper mention some "uncomfortable compromises", so perhaps there is a deeper theory to be explored here.

@modolief 2 жыл бұрын

You guys do _such a good job_ of interweaving your commentary. That's a real skill!

@SocksWithSandals Жыл бұрын

The surface on my bedroom floor is an ever growing space-filling fractal, which in the limit becomes the floor when I've had enough and tidy my room

@minerharry 2 жыл бұрын

Just wanted to appreciate how well put together this video was, you two talked very smoothly around each other. Great math as always!

@slug.. Жыл бұрын

I noticed that too they sort of reminded me of twins they kind of finish each other's sentences it works very good video

@michaeldeierhoi4096 Жыл бұрын

The fractal aspect of these images are what caught my eye. Thanks for posting this.

@telotawa 2 жыл бұрын

i don't understand the math for once but i do see that it's an awesome shape and i want one

@simongregory3114 2 жыл бұрын

relatable, except I would replace 'for once' with 'as always'.

@telotawa 2 жыл бұрын

i usually understand math videos, not understanding one feels super weird, this feels like occult incantations

@modolief 2 жыл бұрын

Where do I buy the wallpaper??

@bhante1345 Жыл бұрын

Take acid, you get to live in the curve for 8 hours.

@FtwXXgigady Жыл бұрын

I was just looking at a sliced purple cabbage today and realized it looks like a space filling curve, like the cabbage is trying to maximize surface area within a finite space. I tried to find articles about it, like about why the cabbage grows like that, but couldn't find any. I'm gonna have to congratulate whatever algorithms youtube has because this is more or less what I was thinking about, except this is obviously much more theoretically rigorous.

@saulschleimer2036 Жыл бұрын

Excellent point! I looked at some images of half sliced cabbages on-line and found some with a five-fold "loxodromic" symmetry about their centre. This looks a lot like a Cannon-Thurston map in the compact case? (But of course the non-compact and compact cases look a lot like each other...)

@imconsequetau5275 Жыл бұрын

The degree of folding will increase the surface area of the leaves. The human brain has a similar folding to increase the surface area of white matter. This is how our brains can consume so much more energy in a given volume.

@ianwhittinghill 11 ай бұрын

I’m so glad these guys found each other and get to be friends

@Ab-qv8zc Жыл бұрын

Wow! Something that looks random is actual a complex geometrical pattern. Really cool! It outer border almost looks like a Mandelbrot set. This is a great balance of art and science, and it forces the mind to expand in order to enjoy this mathematical beauty.

@steampunkfox Жыл бұрын

I'm autistic and one of my stims was to tap my fingers in a pattern that is almost exactly the same as the Hilbert Map you showed. Fascinating.

@JacobCanote Жыл бұрын

HI Saul. We did Joseph and the Amazing Technocolor Dreamcoat in 1989. Great to see your awesome space filling curves.

@saulschleimer2036 Жыл бұрын

Glad you liked it! Nice to see you again.

@carel91 2 жыл бұрын

This is amazing. I am glad that I can watch all your work. This is really inspiring. Thank you

@henriquealecrim2497 Жыл бұрын

This video is, for me, one of the highest points of mathematical exposition in this whole site

@saulschleimer2036 Жыл бұрын

Thank you very much!

@ifroad33 2 жыл бұрын

The fact that they tile really surprised me! So cool!

@ISawSomethingOnTheInternet Жыл бұрын

I’m pretty sure these guys discovered the geometric fabric of space-time.

@thedebapriyakar 2 жыл бұрын

Came here to have my mind exploded and elevated. Totally worth it!

@TheAlison1456 Жыл бұрын

Really good video. No mentions of abstract mathematics in a way that repels anyone not "in the know" and who "doesn't study" maths. Nothing wrong with "abstract mathematics", rather the attitude mentioned.

@johneonas6628 Жыл бұрын

Thank you for the video.

@attacg 2 жыл бұрын

Beautiful, crinkly. Great figure 8 bubble animation & explanation

@triberium_ Жыл бұрын

Amazing how much thinking goes into making such abstract ideas come to life

@JamesSpeiser 2 жыл бұрын

Fantastic material, presentation and dynamic between presenters. Bravo

@luke2642 2 жыл бұрын

What a funny duo on 2x speed, finishing each others sentences 😂 Fascinating stuff. It'd be helpful if all the other videos linked were also listed in the description!

@dudewaldo4 Жыл бұрын

We are NOT stuck with names! Rename them to whatever you want! Future generations will be grateful

@xgozulx Жыл бұрын

I didn't really understand it, but at the same time I was in awe the whole video... I loved it and im confused

@incription 2 жыл бұрын

Does anyone else have an urge to print fractals in nanometer resolution? I know it's impossible but they would look so cool

@yayforfood100 2 жыл бұрын

it's not entirely impossible. the semiconductor manufacturing industry regularly prints near-nanometer scale images

@incription 2 жыл бұрын

@@yayforfood100 Anyone got 10 billion to spare?

@whatelseison8970 2 жыл бұрын

@@incription Certain fractals are actually used as antennas by etching them on pcb's. There's also current research into using nanoscale rectennas for energy harvesting so who knows, maybe someone is already doing that in a lab somewhere.

@RobertSzasz 2 жыл бұрын

Not impossible, just really expensive.

@alexhudspeth1213 Жыл бұрын

@@whatelseison8970 ssshhh you promised not to tell

@cirecrux Жыл бұрын

The animations are beautiful

@bloomp7999 2 жыл бұрын

great conversation man ! we need more of these converstions on KZfaq

@alexhudspeth1213 Жыл бұрын

It's like we're looking behind the simulation. Fascinating.

@040_faraz9 2 жыл бұрын

Only if someone could make such lucid videos of algebraic things, schemes, varieties, etale cohomology and all things Grothendiecky!

@jacobhawthorne1997 2 жыл бұрын

Outstanding as always!

@chadschaefer5084 2 жыл бұрын

That wooshing noise you heard was my head exploding as your explanation went over it right around the point of mobius transformations and the knot.

@andrettibark Жыл бұрын

I don't even know what classes I would need to take to understand this video.

@saulschleimer2036 Жыл бұрын

Topology. And hyperbolic geometry. Both taught at Warwick Uni. :)

@wacomtexas Жыл бұрын

Beautiful. As soon as you mentioned crinkling I knew Daniel Piker would be involved somehow. Incidentally, I bought some delicious but very flat kale from Borough Market the other day..

@kaleygoode1681 Жыл бұрын

I for one would love tiles like that! Wonderful how they plug together... And great how you overlaid your result over the previous work. Time to put your names on that figure 8 result!

@Perplaxus 2 жыл бұрын

This is a good video for those who kinda understand the subject

@seedmole Жыл бұрын

Very fun, I spent an hour or two reading about Hilbert's Grand Hotel and such--good job pushing this video to me, algorithm.

@brianmcquain3384 Жыл бұрын

very cool I am enjoying this

@mmomus5863 2 жыл бұрын

Already love it. What a great premiere to catch.

@MonkeySimius 2 жыл бұрын

Neat designs. I didn't understand where they come from or what they represent in the slightest.

@jonroland2702 2 жыл бұрын

This must be the formula they use at amusement parks for the lines

@maibster 2 жыл бұрын

Great video, impressive animations

@tisanne 2 жыл бұрын

not entirely certain how i got here but it's very interesting indeed!

@danielhmorgan 2 жыл бұрын

phenomenal

@TheBookDoctor 2 жыл бұрын

That's wildly cool.

@VJFranzK Жыл бұрын

it's one of the most "hand drawn" looking geometric shapes!

@ourladymelody Жыл бұрын

I love the phrase "the compliment of the knot"--that which is Not the kNot.

@FarranLee Жыл бұрын

7:33 I don't know how to express the idea in my mind but this image is the best representation I've ever seen of it. Basically, something like, different levels / planes / realms / fields of reality are functioning in their own ways, but they coincide at points along their paths of activity, and those co-incidences are what draws the actual into the real. Not sure if this idea is remotely relevant to what you're demonstrating, but the last video I watched was about the existence of quantum particles and the quantum fields etc, so this follows on nicely.

@KaliFissure Жыл бұрын

Great video. 👍3D L system is my fidget toy. Space fillingness seems to be emergent in many cases. The surface is space itself and as we are seeing from JWST the distortion gets greater and greater and at limit a single photon is smeared across the entire sphere.The cmb.

@telotawa 2 жыл бұрын

i'd love to have this as a wallpaper, can you publish the code? or make a renderer for this version?

@henryseg 2 жыл бұрын

The code is up at github.com/henryseg/Veering, although I’m not sure how user friendly it is. You’ll need sage with pyx installed for drawing the graphics. The relevant file is “scripts/draw_continent.py”, with some usage examples at the bottom of “draw_continent_hack.py”. The code is not in a very clean state at the moment… the hack was to get one of the animations for this video!

@cyancoyote7366 2 жыл бұрын

Can't promise anything as I'm a but busy these days, but if I have some time today I'll try to get a render for your somehow using this code if I can manage it. If I'm successful I'll upload it somewhere here as an uncompressed image. :)

@leif1075 2 жыл бұрын

@@henryseg Thanks for sharing Henry. I hope you can respond to my email or other message when you can. Thanks very much.

@MySelfMyCeliumMyCell Жыл бұрын

crinkly discs are amazing

@ts4gv Жыл бұрын

1:23 you got me, that’s exactly what i was thinking 😂

@alberthmartinez1750 Жыл бұрын

Awesome!

@ytrebiLeurT Жыл бұрын

It's a fractal, something that resembles itself, self-similarity is everywhere...

@tomtomatron8625 Жыл бұрын

Thank you.

@eyedl Жыл бұрын

whoa, and that was only 10 minutes, awesome

@hughjanus3591 2 жыл бұрын

Very cool video! I did not understand any of it

@saulschleimer2036 2 жыл бұрын

We are happy to answer questions!

@hughjanus3591 2 жыл бұрын

@@saulschleimer2036 so The way canon thurston maps fill the plane are analogous to the way Hilbert curves fill a square but you started bringing in Euclidean and non Euclidean geometry, and wires with bubble film and popping sections and that’s when I started getting confused. A specific question would be what exactly is non Euclidean geometry and how do canon thurston maps interact with it. A little bit of context for me is I start college algebra in a few weeks, but I have used hilbert curves to plot x,y coordinates on a 1d line

@saulschleimer2036 2 жыл бұрын

@@hughjanus3591 "what exactly is non Euclidean geometry" by this we mean "hyperbolic geometry". I can recommend books by James Cannon (Two-Dimensional Spaces, Volumes 1, 2, and 3), by Mumford-Series-Wright (Indra's Pearls), and by Jessica Purcell (Hyperbolic Knot Theory).

@saulschleimer2036 2 жыл бұрын

"how do canon thurston maps interact with it" - well, this is much harder. Hyperbolic space has a "boundary at infinity" which is a sphere. This is already difficult. The spanning surface in the knot complement gives a disk in hyperbolic space, and so gives a "curve" in the boundary at infinity. That is the Cannon-Thurston map. The details of the construction are subtle, and we don't give them in the video.

@kyrius_gm4 2 жыл бұрын

I dont understand much either but i think a great set of videos to understand hyperbolic space is to watch CodeParades videos on his hyperbolic game he made called Hyperbolica :)

@TheStarBlack Жыл бұрын

Before I clicked, I didn't know I didn't know about any of this. Now I know I don't know about it.

@johnnyswatts Жыл бұрын

Interesting content, confusing and chaotic presentation.

@alixsprallix Жыл бұрын

great video

@theman13532 Жыл бұрын

4:08 this would make a CRAZY mario kart battle mode track

@dane5624 Жыл бұрын

As brilliant as these fine gentlemen are, I thought they could explain this in a less chaotic way. It's all good. I get it.

@saulschleimer2036 Жыл бұрын

We are just excited to be alive. :)

@uelssom 11 ай бұрын

amazing

@Life_42 2 жыл бұрын

6:20 beautiful!

@remingtonleeman Жыл бұрын

I feel like I met these guys at a diner and asked what they do for a living

@Jason-bb9vi Жыл бұрын

I noticed someone was cool enough to comment with , "thank you." And I gotta say , that thank you was not without value , but rich and effective in it's purpose. Or so it would seem .

@RIXRADvidz Жыл бұрын

When that last bong hit suddenly hits and your brain opens to the Universal Constant. Then you see everything as it is and are able to translate it down into alternate dimensions.

@jjaapp18 2 жыл бұрын

This is why most KZfaqrs set up a script to follow, so you're not rambling

@henryseg 2 жыл бұрын

Usually we would have, but it was a last minute decision to make this video. We had half a day of prep and one day to film.

@saulschleimer2036 2 жыл бұрын

I tried to write a script on the train-ride over; but it was bad and also much too long... We did try!

@DavidRutten 2 жыл бұрын

This is a good video, or rather, this is an approximation of a good video. I'm sorry, that sounded like I'm dissing you...

@titaniumtomato7247 2 жыл бұрын

Cannon-Thursten maps: its morbin time

@aepokkvulpex 2 жыл бұрын

When they were tiled I could kinda see a hexagonal pattern

@saulschleimer2036 2 жыл бұрын

Good eye. The dihedral angles of the "hyperbolic ideal tetrahedra" are all 60 degrees, which leads to lots of "broken six-fold symmetries" in CT map.

@withnosensetv Жыл бұрын

@@saulschleimer2036 I love how actively you guys are engaging in the comments. Keep up the great work, this was super interesting

@MushookieMan 2 жыл бұрын

I more or less understand what a universal cover is from flying inside manifolds, are there any animations of what the heck the universal cover of a knot complement is?

@saulschleimer2036 2 жыл бұрын

I recommend the video "Not knot" by the geometry centre. kzfaq.info/get/bejne/asd-admJnNe8gIU.html

@henryseg 2 жыл бұрын

The video "Not Knot" from the Geometry Center goes into this. A knot complement doesn't act very differently from any other manifold when you take the universal cover. The main issue I think is understanding where the knot goes. Since the knot isn't inside of the knot complement, it "vanishes off to infinity"...

@leif1075 Жыл бұрын

@@henryseg Wjst exsctly does fibered mean for the knkt co.plement..since it is a continuous smooth surface not sure what this means if you could clarify..

@leif1075 Жыл бұрын

@@henryseg At the end did you mean the second shape is made of MULTIPLE ofnthise figures as opposed to just one? If I understand correctly..thanks for sharing..

@saulschleimer2036 Жыл бұрын

@@leif1075 "Fibered" means "locally of the form a plane cross an interval". Consider three-space - it is fibered by planes parallel to the xy-plane. This is the "local model" of a fibered three-manifold.

@makegrowlabrepeat Жыл бұрын

Is this the secret to a high score in Snake?

@whoatemywendys 2 жыл бұрын

One could say a cannon-thurston map is a frac-tile

@saulschleimer2036 2 жыл бұрын

Nooooooooo! The pain…

@ga5712 Жыл бұрын

Wow. Hard to get your head around this.. but watching it makes me wonder if the universe is a knot

@saulschleimer2036 Жыл бұрын

There are papers that discuss the "global topology" of the three-dimensional universe. In particular Jeff Weeks has papers on this, and he raises the possibility that the universe is "hyperbolic". He discusses the "circles in the sky" technique (that is, patterns of correlations in the cosmic background radiation) for determining the "global topology". Unfortunately, it seems that these circles are absent...

@simialogue 2 жыл бұрын

I am reduced to being existentially nonplussed. It's not that I don't apprehend possible understanding - I mean it's just over there, lurking in the corner - I do. It's just... I seem to be in a round room. Perhaps through fascination and a lot of head scratching, I'll be able to join it.

@stevenmayhew3944 2 жыл бұрын

They look similar to the Julia sets, cousins of the Mandelbrot set.

@saulschleimer2036 2 жыл бұрын

There is a "dictionary" of kinds between holomorphic dynamics and Kleinian groups. One of the main proponents of this is Dennis Sullivan. There is a nice blog post on this by Yankl Mazor. At a more advanced level, there are many research papers, including early ones by Sullivan.

@alden1132 Жыл бұрын

Ooooh, I have visceral dislike of the squiggly patterns. They look VERY similar to the *pattern* of the aura I see right before I get a migraine. If they were opalescent, and flashed like TV static, they'd be identical.

@NonTwinBrothers 2 жыл бұрын

How long did it take the CNC machine to make those posters?? Seems like a very long operation if it has to go through all that path

@henryseg 2 жыл бұрын

Each one ran overnight, taking around 16-20 hours.

@kriterer 2 жыл бұрын

This might be a dumb question, but, since they should contain all of the same points when represented as sets, how do we/can we actually distinguish between a randomly filled plane and a complete, filled tiling of Cannon-Thurston maps? Is there even a point to differentiating between the two planes?

@henryseg 2 жыл бұрын

The difference is that there is an order in which the map visits each point of the plane. You can see this most clearly with the Hilbert curve example: in every approximation to the Hilbert curve you see the polygonal curve goes from lower left to upper left, then upper right then lower right. So in the limit, the Hilbert curve visits the quarters of the square in that order.

@jounik Жыл бұрын

@@henryseg So if I understand correctly, the map is a set with extra structure (i.e. a strict ordering) and to represent it as a plain old set loses that salient bit of the mapping.

@henryseg Жыл бұрын

@@jounik Right. The "mapping" is a map (a continuous function) from the circle to the plane (actually the sphere).

@elijahmitchell-hopmeier182 2 жыл бұрын

I'm not smart enough for this. Still great video. It was really pretty to watch

@poobertop Жыл бұрын

It's crazy to think how DNA imbeds such space filling calculations.

@retematic2351 Жыл бұрын

I want that as floor, looks trippy af

@jacobcowan3599 Жыл бұрын

To the uninitiated eye, a slice of this approximation around one of those focal points looks quite a bit like cabbage

@Kroggnagch Жыл бұрын

I don’t know the geometry well enough to understand half the stuff you guys say. I have much to learn yet...

@MrDaraghkinch Жыл бұрын

Me: "Ooo, fractals, I get this." These guys: "Lift the knot compliment to the universal cover". Me: I am a fucking baby.

@Jason-bb9vi Жыл бұрын

Wow you just dashed a bit of false humility into a bowl full of modesty and did so without competition . By denying competing , you have implicated complementation which is equivalent to bettering the best . Which is to say, growth has occured in your ontological efforts , which is to say , stay the course, the same shall stay forever young. .

@MrDaraghkinch Жыл бұрын

@@Jason-bb9vi I understand, thanks Jason.

@Jason-bb9vi Жыл бұрын

@@MrDaraghkinchthank you. Seriously, you have an amusing way with descriptive words. Thanks for making this even more entertaining . It really was fairly mind blowing to see someone rock this particular field of study to this extent so .really could relate , certainly felt the same way.for the most part

@Mesra73 Жыл бұрын

I like your funny words, magic men

@Veptis Жыл бұрын

space filling curves can also be generalized to different ways of traversing high dimensional latent space of any kind of auto encoder network.

@neopalm2050 2 жыл бұрын

How does the "big crinkly disk in euclidean 3-space" relate to the images you show? Did you convert the sphere of points at infinity (marked by the crinkly plane) into the euclidean 2D images you showed with a conformal transformation or something? Also, are we supposed to think of _anything_ here as riemannian or are we supposed to see everything as only having a conformal structure?

@henryseg 2 жыл бұрын

Ah yes we didn’t really explain that. The boundary of the crinkly disk is the space-filling curve. (In fact, sphere-filling - it fills the boundary of the big ball.) The pattern on the sphere repeats in many different ways. One of these ways is that if you stereographically project the pattern on the sphere to the plane, with the projection point being chosen carefully, then the pattern tiles the plane in the way we showed. The projection point has to be one of the “cusps”. Everything here is not only riemannian but constant negative curvature one: it’s all hyperbolic geometry. There are lots of conformal-structure-preserving maps because hyperbolic geometry has Möbius transformations for isometries.

@neopalm2050 2 жыл бұрын

@@henryseg Is the sphere of points at infinity also curvature -1? What about the 2D plane tiled with a Cannon-Thurston map? Or were you just referring to the 3D space in which the crinkled disk lies? As far as I know, topological spheres must have positive curvature by gauss-bonnet, and the way these tiles fill space doesn't look like a hyperbolic symmetry group to me.

@saulschleimer2036 2 жыл бұрын

@@neopalm2050 The sphere at infinity is "off at infinity". It is not constant curvature -1. The two-plane tiled by the CT map is the sphere at infinity, minus a point (and then stereographically projected). The three-dimensional space (the open "big ball") that contains the crinkled disk is hyperbolic three-space. You are right that spheres "want" to have positive curvature. The connection between hyperbolic geometry in dimension three and conformal geometry in the two-sphere is subtle. Here is perhaps the first link: en.wikipedia.org/wiki/Gromov_boundary and the graduate level book "Metric spaces of non-positive curvature" by Bridson and Haefliger.

@gregsLyrics Жыл бұрын

Fascinating. I think you have inadvertently described the mitochondria's cristea in biological nature. Makes me wonder about curve fitting what nature has made for all humans.

@GiftFromGod Жыл бұрын

came here out of curiosity about the image(s) and here at the end I can confidently say that I barely understood anything xD very interesting stuff though!