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.
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
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.
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!
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!
@@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?
@@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...
@@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.
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.
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...)
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.
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.
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!
@@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.
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..
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!
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.
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. .
@@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
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 .
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!
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. :)
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.
@@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
@@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).
"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.
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 :)
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.
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.
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...
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.
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?
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"...
@@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..
@@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..
@@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.
anyone else enjoy watching videos they do not at all have the baseline knowledge to understand? i just be sitting there like "hmmm yes very interesting (no clue whats going on)"
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!
Yes, very true. Henry and I discussed this, and we agree that you could "hook together" the Hilbert curves to get a plane-filling curve. However, this would involve making more choices...
Space filling shapes are like how we are made up of cells and those cells are made up of organelles and those organelles can be crinkly for more surface area. Whats the surface area so important for? More connections to space? We need a crossover of physics and biology! The final relation of math, energy, and consciousness.
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.
I wouldn’t say they’re approximations. If you zoom in on a Hilburt curve forever, you’lol see the original lines. The cannon-thruston maps you have are steps to the final curve
Great question. There is no two-sphere filling a three-sphere which is invariant under a uniform kleinian group. (This is an "easy" Euler characteristic computation.) The existence a one-sphere filling a three-sphere, invariant under a uniform kleinian group, is open. However expert opinion seems to be that such things do _not_ exist.
Here are mathoverflow references - for the first we have 142621 (Hyperbolic Manifolds which fiber over the circle) and for the second 66000 (F→E→B bundle with B,E,F hyperbolic: possible?)
Every 2D is just on a single dimension so if there was a 3D, the look of it would intrigue me. How X Rays can show a person from only a 2D piece at a time, that is what I want to see with this. It would be like uncovering a 4D art piece
@@Xyabra I agree it would be cool. I don't think that there are examples (in 3D) similar to what the video is about. However, there is some amazing (and related) work of Vladimir Bulatov on three-dimensional "kleinian limit sets".
Okay. I'm not a math genius. Of course, this is still extremely interesting. But I had an idea about laser cutting puzzle pieces which might be used to build a map, but not in just one way. Many possible maps. A design method that would let me have some organic looking variations of interlocking pieces that could form a fantasy world map that could be a game in itself to assemble (think Mappa Imperium) I'm really having a hard time finding out where to begin. Two mathematical concepts that appear useful to me are Voronoi maps and now this. Anyone got ideas about how can I put this together? Leverage Math and Computers to design this puzzle?
Nervous system (n-e-r-v-o-u-s.com) make beautiful laser cut jigsaw puzzles with a very organic feel. They use various biologically inspired iterative methods to generate their piece boundaries. Saul and I have thought about jigsaw puzzles made from C-T maps, unfortunately we get very pointy pieces that won't work well as a physical puzzle.
I suggest the book "Indra's Pearls" by Mumford, Series, and Wright. It starts with the basics and then works through the hyperbolic geometry needed to understand the first examples. After that, I suggest reading "Hyperbolic Knot Theory" by Purcell. And then you will be equipped to read our research papers that actually explain what is going on. (Or you could skip to the end and reach back to the beginning and middle as needed - whatever works for you.)
@@saulschleimer2036- Thanks for the recommendation of Jessica Purcell's book. Really fascinating! Coincidentally, the famous composer Henry Purcell wrote a suite for string orchestra called "The Gordian Knot Unty'd". He published the piece back in 1691! I wonder if she knows about that piece.
This the first time that I've taken proper notice that the true Hilbert curve involves a limit. So, if I understand correctly, the true Hilbert curve occupies all the points in a plane. Now I'm trying to imagine a 3D Hilbert curve, which seems possible. What about N dimensions? Oh, I just noticed that 3Blue1Brown has a vid on the Hilbert curve. Watching that now.
Yep - they exist in all finite dimensions. And in some infinite dimensions. Check out the mathoverflow post "Are there space filling curves for the Hilbert cube?"!
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.
That's exactly what I thought too
That is a nice observation!
Wow, I just found a picture of that by googling it, does that embedding have a name?
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
Interestingly, both are somewhat related to Gromov.
This is so confusing and i feel like i barely understand anything but i love it! Its so interesting!
Glad you like it!
100% agree ❕️
Yeah
glad I'm not the only one who feels like this lol ^^
Right? Me clicking on this video: "Ooh fractals!"
Afterwards: "oh. Nice."
What a time to be alive where intricate and incredibly complex mathematics can be visualized in such beauty!
6:37 Henry: "You could imagine..."
Me: "No, no I don't think I could imagine that."
I’m so glad these guys found each other and get to be friends
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.
We discuss this a bit more in our cohomology fractals video here: th-cam.com/video/fhBPhie1Tm0/w-d-xo.html
I am happy to answer questions, as well.
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
You guys do _such a good job_ of interweaving your commentary. That's a real skill!
Just wanted to appreciate how well put together this video was, you two talked very smoothly around each other. Great math as always!
I noticed that too they sort of reminded me of twins they kind of finish each other's sentences it works very good video
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!
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!
@@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?
@@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...
@@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.
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.
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...)
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.
i don't understand the math for once but i do see that it's an awesome shape and i want one
relatable, except I would replace 'for once' with 'as always'.
i usually understand math videos, not understanding one feels super weird, this feels like occult incantations
Where do I buy the wallpaper??
Take acid, you get to live in the curve for 8 hours.
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.
The fractal aspect of these images are what caught my eye. Thanks for posting this.
I don't even know what classes I would need to take to understand this video.
Topology. And hyperbolic geometry. Both taught at Warwick Uni. :)
HI Saul. We did Joseph and the Amazing Technocolor Dreamcoat in 1989. Great to see your awesome space filling curves.
Glad you liked it! Nice to see you again.
We are NOT stuck with names! Rename them to whatever you want! Future generations will be grateful
This video is, for me, one of the highest points of mathematical exposition in this whole site
Thank you very much!
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!
Does anyone else have an urge to print fractals in nanometer resolution? I know it's impossible but they would look so cool
it's not entirely impossible. the semiconductor manufacturing industry regularly prints near-nanometer scale images
@@yayforfood100 Anyone got 10 billion to spare?
@@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.
Not impossible, just really expensive.
@@whatelseison8970 ssshhh you promised not to tell
Beautiful, crinkly. Great figure 8 bubble animation & explanation
The animations are beautiful
Amazing how much thinking goes into making such abstract ideas come to life
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..
Only if someone could make such lucid videos of algebraic things, schemes, varieties, etale cohomology and all things Grothendiecky!
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.
great conversation man ! we need more of these converstions on TH-cam
Fantastic material, presentation and dynamic between presenters. Bravo
This must be the formula they use at amusement parks for the lines
Very fun, I spent an hour or two reading about Hilbert's Grand Hotel and such--good job pushing this video to me, algorithm.
Neat designs. I didn't understand where they come from or what they represent in the slightest.
This is a good video for those who kinda understand the subject
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!
it's one of the most "hand drawn" looking geometric shapes!
very cool I am enjoying this
not entirely certain how i got here but it's very interesting indeed!
Before I clicked, I didn't know I didn't know about any of this. Now I know I don't know about it.
crinkly discs are amazing
It's a fractal, something that resembles itself, self-similarity is everywhere...
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.
We are just excited to be alive. :)
Outstanding as always!
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.
Cannon-Thursten maps: its morbin time
That's wildly cool.
Me: "Ooo, fractals, I get this."
These guys: "Lift the knot compliment to the universal cover".
Me: I am a fucking baby.
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. .
@@Jason-bb9vi I understand, thanks Jason.
@@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
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 .
i'd love to have this as a wallpaper, can you publish the code? or make a renderer for this version?
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!
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. :)
@@henryseg Thanks for sharing Henry. I hope you can respond to my email or other message when you can. Thanks very much.
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.
whoa, and that was only 10 minutes, awesome
Very cool video! I did not understand any of it
We are happy to answer questions!
@@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
@@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).
"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.
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 :)
Great video, impressive animations
To the uninitiated eye, a slice of this approximation around one of those focal points looks quite a bit like cabbage
1:23 you got me, that’s exactly what i was thinking 😂
They look similar to the Julia sets, cousins of the Mandelbrot set.
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.
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.
Thank you.
6:20 beautiful!
Wow. Hard to get your head around this.. but watching it makes me wonder if the universe is a knot
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...
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.
I have absolutely no idea what I'm looking at :D
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?
I recommend the video "Not knot" by the geometry centre. th-cam.com/video/4aN6vX7qXPQ/w-d-xo.html
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"...
@@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..
@@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..
@@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.
I'm not smart enough for this. Still great video. It was really pretty to watch
Awesome!
great video
Ha, Escher's mind was gorgeous.
It's crazy to think how DNA imbeds such space filling calculations.
I don’t know the geometry well enough to understand half the stuff you guys say. I have much to learn yet...
amazing
That looks like stress diffraction in transparent solids.
anyone else enjoy watching videos they do not at all have the baseline knowledge to understand? i just be sitting there like "hmmm yes very interesting (no clue whats going on)"
pretty pictures.. and lots of words too.. ;9)
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!
I want that as floor, looks trippy af
If only M.C. Escher was still alive!
1:58 - A square can tile also.
Yes, very true. Henry and I discussed this, and we agree that you could "hook together" the Hilbert curves to get a plane-filling curve. However, this would involve making more choices...
I'd like to experience the crinkly spaces in VR
Space filling shapes are like how we are made up of cells and those cells are made up of organelles and those organelles can be crinkly for more surface area. Whats the surface area so important for? More connections to space? We need a crossover of physics and biology! The final relation of math, energy, and consciousness.
I had a job making bits out of pieces, we'd make wrinkles , advertise them as creases.
The Crinkling... dun dun DUNN
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.
I wouldn’t say they’re approximations. If you zoom in on a Hilburt curve forever, you’lol see the original lines. The cannon-thruston maps you have are steps to the final curve
I understood none of this video. but i do know that cool shape is cool shaped
I wonder what a
3D one
would look like
Great question. There is no two-sphere filling a three-sphere which is invariant under a uniform kleinian group. (This is an "easy" Euler characteristic computation.) The existence a one-sphere filling a three-sphere, invariant under a uniform kleinian group, is open. However expert opinion seems to be that such things do _not_ exist.
Here are mathoverflow references - for the first we have 142621 (Hyperbolic Manifolds which fiber over the circle) and for the second 66000 (F→E→B bundle with B,E,F hyperbolic: possible?)
Every 2D is just on a single dimension so if there was a 3D, the look of it would intrigue me.
How X Rays can show a person from only a 2D piece at a time, that is what I want to see with this.
It would be like uncovering a 4D art piece
@@Xyabra I agree it would be cool. I don't think that there are examples (in 3D) similar to what the video is about. However, there is some amazing (and related) work of Vladimir Bulatov on three-dimensional "kleinian limit sets".
Okay. I'm not a math genius. Of course, this is still extremely interesting. But I had an idea about laser cutting puzzle pieces which might be used to build a map, but not in just one way. Many possible maps. A design method that would let me have some organic looking variations of interlocking pieces that could form a fantasy world map that could be a game in itself to assemble (think Mappa Imperium)
I'm really having a hard time finding out where to begin. Two mathematical concepts that appear useful to me are Voronoi maps and now this.
Anyone got ideas about how can I put this together? Leverage Math and Computers to design this puzzle?
Nervous system (n-e-r-v-o-u-s.com) make beautiful laser cut jigsaw puzzles with a very organic feel. They use various biologically inspired iterative methods to generate their piece boundaries.
Saul and I have thought about jigsaw puzzles made from C-T maps, unfortunately we get very pointy pieces that won't work well as a physical puzzle.
Whatever this is, I enjoyed it. Something about...... shapes? 😂😊👍
space filling curves can also be generalized to different ways of traversing high dimensional latent space of any kind of auto encoder network.
Nice shirts.
This was so awesome I fucking love math
Likewise. :)
I so want to do more math. I made it as far as Group Theory. What is the most direct route from there to what you're exploring?
I suggest the book "Indra's Pearls" by Mumford, Series, and Wright. It starts with the basics and then works through the hyperbolic geometry needed to understand the first examples. After that, I suggest reading "Hyperbolic Knot Theory" by Purcell. And then you will be equipped to read our research papers that actually explain what is going on. (Or you could skip to the end and reach back to the beginning and middle as needed - whatever works for you.)
@@saulschleimer2036- Thanks for the recommendation of Jessica Purcell's book. Really fascinating!
Coincidentally, the famous composer Henry Purcell wrote a suite for string orchestra called "The Gordian Knot Unty'd". He published the piece back in 1691! I wonder if she knows about that piece.
The carvings look like red cabbage cross-sections!
MA132/138 forever lads!😆🤣
Mostly incomprehensible to a non-specialist, but the tilings are very interesting.
These would be great for the BACKROOMS maps...
I kinda wanna tile a floor with these
This the first time that I've taken proper notice that the true Hilbert curve involves a limit. So, if I understand correctly, the true Hilbert curve occupies all the points in a plane. Now I'm trying to imagine a 3D Hilbert curve, which seems possible. What about N dimensions? Oh, I just noticed that 3Blue1Brown has a vid on the Hilbert curve. Watching that now.
Yep - they exist in all finite dimensions. And in some infinite dimensions. Check out the mathoverflow post "Are there space filling curves for the Hilbert cube?"!
@@saulschleimer2036 Bang. Hilbert cubes. Thank you so much. Will, search immediately.
I got completely lost after 3:38, lol. never heard of the figure-8 knot.
I can't believe that I am one of the first people to see this!
Any links to fashion designs made with these?
www.neatoshop.com/artist/Henry-Segerman
6:17 this was nice to look at. The world could use more fractal renders with depth/3d movement
I guess I should go play CodeParade’s game
Or, you could play with our app! Henry links to it in the video description.
So the boundary of the cross section (disk) covers the entire ball's surface?
Yes, exactly correct.