Illuminating hyperbolic geometry

Joint work with Saul Schleimer. In this short video we show how various models of hyperbolic geometry can be obtained from the hemisphere model via stereographic and orthogonal projection.
2D figure credits: 4:09 Cannon, Floyd, Kenyon, Parry. 0:49, 1:20, 1:31, 2:12, Roice Nelson. We thank Gazi University, Ankara, Turkey, for its hospitality during filming.
(2,3,7) triangle tiling: shpws.me/Dq6q
Geodesic saddle ({3,7} radius 3, subdivided 4): shpws.me/ES7b
Pseudosphere: shpws.me/GIgq
(2,3,5) triangle tiling (stereographic projection): shpws.me/CKVv
Thanks to M.Y. Zhang for writing CC subtitles for this video.

[@Mathologer]

Among your videos this is my favourite so far.

[@henryseg]

+Mathologer The ones I make with +Saul Schleimer always seem to have higher production values!

[@TheRealBigBash]

If you are interested in hyperbolic geometry. I would recommend universal hyperbolic geometry.

[@TheRealBigBash]

If you are interested in hyperbolic geometry. I would recommend universal hyperbolic geometry.

[@TheRealBigBash]

If you are interested in hyperbolic geometry. I would recommend universal hyperbolic geometry.

[@pissqueendanniella4688]

MATHOLOGER?! Is this a cross over episode??

[@Kram1032]

using light to show off those various projections is really cool

[@AlvinBalvin321]

I honestly want to use one of those as a lamp

[@jerome8314]

Those would sell

[@sheepkillindog]

Same

[@ang3metal]

I want it as a strainer xd

[@petermarsh4578]

This is well above my knowledge level, but i'd love to understand properly one day

[@dogchaser520]

Yes. This is why academics aren't necessarily teachers. The most brilliant minds in a field are often very poor teachers; they train entirely in their field and no pedagogy, so they're unfamiliar with the science of the transmission of knowledge from one mind to the next. Couple that trend with the modern disdain for those who pass on knowledge professionally and you have a hot soupy mess of people saying interesting things that only those with highly specific knowledge in their field can understand. In other words, nearly useless.

[@2b-coeur]

@@dogchaser520 Amen. See "The Sense of Style" by Steven Pinker for a writer's perspective on that issue..

[@dogchaser520]

​@@2b-coeur Looked it up, and it actually sounds fascinating! I'll take a look. "De-academifying" (obfuscated) language our schools is an important step in leveling the playing field and becoming more inclusive. Right now, academics are essentially gatekeeping intellectual status and all that rests upon it through intentionally muddied language. If you're interested in race theory, in the West this practice is essentially White supremacist, sadly.

[@isodoublet]

@@dogchaser520 "they train entirely in their field and no pedagogy, so they're unfamiliar with the science of the transmission of knowledge from one mind to the next" Well, there really isn't a science of transmission of knowledge, and it seems doubtful training has much to do with it in practice (that is, beyond the context of artificially controlled situations with dubious generalizability). A lot of it is due to talent, and teaching talent doesn't seem to discriminate between genius and mediocrity.

[@MindYourDecisions]

Great video! I would love to see more schools use your models to teach these concepts.

[@hamsterdam1942]

but we don't want to get our scholars clickbaited

[@jerzysielicki-baryka9652]

hey this is very interesting and all but why is the outro so creepy

[@shabazahmed]

Reminds me of the two kids from The Shining

[@worshipwormking2327]

New aesthetic

[@Action2me]

These guys are mathematicians not entertainers.

[@Alorand]

The Lovecraftian aesthetic comes from thinking too deeply about the wrong angles of the non-euclidean space.

[@yuudaemones2624]

you like it, be honest.

[@rarebeeph1783]

one of my favorite projections is taking a euclidean plane, pulling back a gnomonic projection to the half-sphere, and parallel projecting to a disk in the plane. it maps lines in the plane to half-ellipses tangent to the boundary of the disk at two opposing points, which makes it very well suited to conceptualizing projective geometry. of course, you still have to implicitly equate opposing points on the boundary. (edit: i suppose you could stereographically project the half-sphere to the disk instead; that would map lines to circular arcs which intersect the boundary at opposing points) you can even model this projection in a graphing calculator like desmos, which means you can graph proper functions and see how they behave. i suppose it shouldn't be surprising that the point at which they intersect the circle at infinity is closely related to the limit of the slope as x goes to infinity (if it exists), so most common functions (polynomials, exponentials) intersect at the top/bottom of the circle (i.e. straight vertical from the origin). as a final example, sin and cos do not have limiting slopes, but they are bounded between two horizontal lines, and so must intersect the point at infinity where those lines do: the horizontal point, corresponding to 0 slope lines.

[@Posesso]

This is so good. It's helpful to explain because it is the real deal. No 'imagine a ray that blabla', here you just show it and say. And now we show you how to make it on paper. Thanks!

[@eptothefrep]

I watched this to understand my absurdly confusing dreams. it helped a bit and the switching voice and outro are fittingly eerie

[@neild4609]

Hyperbolic geometry in dreams? That reminds of me of a talk I watched on youtube called "The Hyperbolic Geometry of DMT Experiences" It seems like it could be relevant, as it's hypothesized that endogenous DMT is released while dreaming.

[@alexandrogarces8384]

Here from CodeParade. Great video!

[@ethos8863]

The idea that different projections are the result of light sources in different positions and directions is rather striking

[@maciej.ratajczak]

DMT (the smoked / vaped form of dimethyltryptamine, a psychedelic extract from certain Amazonian plants) flash allows one to experience this living geometry in real time and enter an apparently other dimension. Mind-bendingly mind-blowing. It's like looking out from the inside of your brain/mind, or maybe vice-versa, looking in from the outside of your brain/mind, and seeing a hyperbolic projection of the world/reality.

[@soulflightclctv1247]

Have you read the Qualia Computing article on DMT and hyperbolic geometry? I think you'd find it up your alley if you haven't read it already. Super interesting

[@maciej.ratajczak]

@@soulflightclctv1247 Thanks. I actually came here from a video I had watched on that exact subject: The Hyperbolic Geometry of DMT Experiences (@Harvard Science of Psychedelics Club) at kzbin.info/www/bejne/oqCmc6mXn5mYick

[@neild4609]

Yes, it's mind-blowing, and incredibly beautiful

[@thomas.thomas]

i kinda have understood it but actually i haven't

[@fm56001]

same

[@happyface-]

Him: *talking about math and science stuff* Me: “Oooo pretty patterns”

[@SimulationSeries]

Love how eccentric you guys are, and excellent visual portrayal of the material, new sub & fan, much love, thank you! :)

[@otaku-chan4888]

I like your funny words, magic man~ this is way above my grade 12-level knowledge of euclidian/non-euclidian planes, but I can tell this is cool stuff!

[@apurbabiswas7218]

Great video. I found it a little too fast for someone uninitiated. I loved the 3D printed models and the demonstrations - pretty cool.

[@geraldsnodd]

These projections are so satisfying.

[@AppallingScholar]

Underated channel

[@happmacdonald]

Hey cool! I love hyperbolic geometry. While I have already been introduced to these three projection types before (at least the ones on the flat plane, the hemisphere model is new to me even though it's such a great exchange medium between the other three!), this is the first time I've seen them compared to one another and their most important shape conservation properties discussed in full. Thanks guys! :D

[@Steve-uu7yx]

this video was very illuminating

[@clockworkkirlia7475]

Sweet! I'd love that as a lighting feature.

[@raihansk5971]

Gems are always rare... but they are always precious. Just like some of these KZbin channels.

[@Amr-Ibrahim-AI]

Thank you. Projecting using light sources is simple and clear

[@GregoryTheGr8ster]

Alternating between 2 narrators is intriguing.

[@abdullahh283]

These are really some thought inducing videos! Great work gentlemen!

[@mikediv8788]

This channel is underrated

[@alansmithee419]

A weird effect of this plane is shown in a game called hyper rogue. You can find the entrance to an area and easily walk around the whole area. But when you enter said area, it opens up to an infinite scale and contains its own areas all of which act in this same way. I really want to see a first person rendering of this sort of thing. Edit: huh, I'm wrong. For details, look at this galoomba: (Second reply to this comment)

[@fishyeverything8530]

CodeParade coded this kzbin.info/www/bejne/oXalYmSGhp54pZo

[@galoomba5559]

The "walk around the whole area" is an illusion created by the projection. Actually the wall is an infinite straight line.

[@alansmithee419]

@@galoomba5559 I literally just reinstalled the game for the sole purpose of proving you wrong. You weren't. I'm an idiot.

[@williamrumping4008]

This guy deserves more subscribers

[@liesalllies]

These projections are beautiful. Anywhere selling prints or posters of them?

[@kenziekline2231]

This is so cool, thank you for taking the time to explain all of this!

[@priolo22]

could you make some outdoor lamps that cast a tennis court (through the shadows)?

[@HeyKevinYT]

astounding demonstration

You, good Sir, have broken my mind

[@weebPaste]

idk how i got here but i enjoyed it

[@petergambier]

Thanks fella's, nicely explained, I sort of get it, but then I don't, but the main goal is helping to make better and better eye candy I should think. We all look for patterns in words and music so we can visually get it but could not fully describe it.

[@farukben]

Stereo sound was pretty dope

[@theclosedcurve]

when he says "we're gonna use the sun" 😂

[@Boldy112]

1 of the better visual explanations, thx for sharing this

[@spotellis]

I have no clue what’s happening💀 But I like how the flashlight acts like a camera and the shadow looks like whats being recorded. thats pretty trippy

[@_casg]

Omgosh I did a project like this for my 3d class, with a light bulb but I did was that I made two Mandelbrot geometric spheres with alternative concentric angles with their geometry one would rotate inside the other sphere and when both spheres rotate in opposite polarities the shadows would start interchanging like crazyyy, It was the wildest experience manipulating the shadows

[@TheJacklwilliams]

Thank you gentlemen. Realizing this particular video is 7 years old, and I’m just learning the subject, I have to ask, how does this, translate to real world work. Aka, in geometry speak, where are these models applied and used? Thanks again.

[@mordechailevinson8646]

...Thank you so much for posting this......

[@6AxisSage]

You would understand my work! Its hard to explain to people that incoming light on the retina maps onto half a geodesic dome of roughly hexagonal (6 triangles) tiling, which is then rescaled by the thalamus onto the V1 Gyri, like an image. As far as the brain is concerned the gyrii bump is geometrically perfect. Same for all the other sulcii and gyrii, as long as each cortical column has wired correctly with its neighbors, then tasks can be performed such as read/write operations, graphical construction, pixel encoding, moire pattern animations.

[@vipulpetkar]

two people talking from different direction is really making this confusing.

[@OMN1N]

This is what appears in your recommendations after watching way too many geometry dash videos

[@rshtg2019]

i like how i ask a question and its answered immediately

[@SunroseStudios]

i came to this video wrongly expecting something about simulating illumination (that is to say, lighting) in a 2D hyperbolic plane... i think i misinterpreted the thumbnail... but i do appreciate the explanation of hyperbolic projections! we've only ever barely understood how hyperbolic geometry works, so it's nice to have some light brought to the subject, even if not as literally as i was hoping. that said, i imagine geodesic-based "raytracing" *could* be used to simulate lighting in the hyperbolic plane...

[@henryseg]

Here's raytracing geodesics in three-dimensional hyperbolic space: kzbin.info/www/bejne/n6ereGd3hNCbj6M. If I recall correctly we are cheating with the lighting - physically correct light intensity drops exponentially with distance, which makes everything far too dark!

[@SunroseStudios]

@@henryseg ohh, i see! interesting

[@Razzer0693]

im just here for the pretty shadows

[@sonuyadav1182]

Thank you for explaining

[@reizinhodojogo3956]

fun fact: normal non-euclidean spaces without hyperbolic or something are easy to imagine a visualization with a brain video(imaginary video of a cube that changes whats inside depending on the angle or another thing), but 4d visualizations are very hard to visualize(at least to me)

[@SpriteGuard]

Nice overview. The half-plane projection seems like the bottom curves upward, is that just a limitation of the materials?

[@henryseg]

+Sprite Guard Alpha The bottom of the region in which I've cut holes out curves up, because of limitations in the material - the holes corresponding to triangles below the curve would have to be too small to print properly.

[@ValkyRiver]

@@henryseg Also 0:19 beautiful crystal ball

[@Yu-Gi-Oh36508]

all I see are cool shapes

[@xXrimjobXx]

i neeeeeed moreeeee

[@alansmithee419]

Who made me the genius I am today? The mathematician that others all quote? Who's the professor who made me that way? The greatest to ever get chalk on his coat?

[@hongkonger885]

Outro: when you have one minute left to do your homework:

[@Guitareben]

This is fantastic!

[@dimitrioskaragiannis1169]

Amazing work sir !🎆 I love your videos ☺

[@TaiFerret]

I would like to see a hyperbolic analog of the Mercator projection.

[@ZenoRogue]

The hyperbolic analog of the Mercator projection is called the band model. The Mercator projection renders the equator isometrically as a straight line, and the rest is mapped conformally. The same is true for the band model -- it renders a chosen hyperbolic straight line isometrically as an Euclidean straight line, and the rest is mapped conformally. While the equator is finite, a hyperbolic straight line is not -- you get an infinitely long band (of finite width, though), and hence the name "band model". You can see it in action in Bulatov's presentation ""Conformal models of hyperbolic geometry", and also in our game HyperRogue -- where it is used as a great presentation of the surprising fact that the path taken by the player during the game is very close to a straight line (the guiding line is taken to be the one which connects the initial and final position here).

[@ishtar0077]

Great video

[@slehar]

Wow! Awesome!

[@rod3134]

It reminds me of fractal geometry...

[@dalegillman5287]

Very cool video!

[@MikeM8891]

Are there any books you would recommend for learning about non-euclidian geometry?

[@michaeldeierhoi4096]

I always find your presentations interesting and informative. And they are delivered in a concise and professional manner. What I found odd in this particular episode was when Henry was holding the hemispheric model above the huge white board which was being supported at one end by an assistant. I thought for sure that you would then move the model away and see how the pattern changed on the white board?! But that didn't happen so my question is why not show us how the image changed with the model higher above the board? Otherwise why use such a big board at all?

[@saulschleimer2036]

Are you referring to the scene starting at kzbin.info/www/bejne/m3iogpKLqranj9U? We are using a giant white board because 1) it is very flat and 2) we could tilt it to be perpendicular to the angle of the sun. (And also, it was what we had!) Since the rays of the sun come in in parallel, moving the model away from the white board will not change the shadow.

[@michaeldeierhoi4096]

@@saulschleimer2036 Oh, I see because the light rays are coming in parallel the image would not be enlarged by moving the model away. Thanks for the explanation.

[@noahgilbertson7530]

illuminating!

[@timber2lease]

if there would be an index like 1/amount of videos with similiar topics like the indexed video (each one of this channel), im sure this channel would have a lot of videos in the top 100. (sry no math expert here, but i think you know what i mean :)) and the channel itselft would be in the top 3. respect.

[@tingwu_]

May I ask why a hemisphere model could represent hyperbolic space? Shouldn't it have a negative curvature?

[@taicanium]

In a way, it does. Spherical space's curvature is positive, so when projected to Euclidean space as a shell, it has a "center". The hemisphere works as a projection of hyperbolic space insofar as...well, imagine that the "center" of the hemisphere is now on its edge instead of at its center. So the lines of hyperbolic space sort of "come out" of the edge of the hemisphere and follow the edge off and away. God I am bad at this.

[@Sundoze_com]

cool video! thanks

[@azsastic69auspi24]

Thank You!!

[@yeno6492]

Why am I watching this even though I'm not a major in geometry or math. But great job I kinda understand

[@luizamorim8265]

Amazing vídeo.

[@tristarperfecta1061]

It's like I am hearing another language but I feel smarter.

[@jamesjosephides]

That's super neat! I love this part at 2:52 Is there any way to get a hold of that 3D model? :D

[@FixedFace]

perfect for people designing lamps

[@W1z3k]

What are those statues at 3:19 ? Which country and city are they in?

[@patrickcrabb6212]

That last frame made it seemed like they are part of some kind of math cult.....but then I'm making the assumption math isn't one massive cult with subsections.

[@fractalgenesis3474]

for some reason the audio was auto panning thru my speaker system. still very cool.

[@andrewwhite6]

Cool!

[@chufa72]

Reading about hyperbolic geometry is denser than the Silmarillion. "Compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms... the Hurwitz surface of lowest possible genus" WTF

[@gianmarcogianni4052]

Hi @Henry Segerman, those shadows are amazing? what kind of light are you using? :O do you have a link? thank you!

[@henryseg]

I use this flashlight: maglite.com/products/mini-maglite-led-2aaa-flashlight

[@gianmarcogianni4052]

@@henryseg thank you very much! very appreciated. I'm looking for a bright point-like source of light. I found very interesting your exhibition at the Summerhall in Edinburgh. There I've seen that you use an Led plugged into the socket with a driver. Did you build it or do you have reference for it? Sorry for my questions but I'm working on a similar exhibition for the Alma Mater in Bologna (Italy). Thank you

[@henryseg]

@@gianmarcogianni4052 I don't know the details of how to do this, but apparently it is not too difficult to modify a battery powered device to run on mains power with an appropriate transformer.

[@clawer626]

yo this shit is lit and your video really helped me understand this subject for my class. thanks dude

[@nicholasheilig3694]

It's clunky but great, I love it!

[@NSHTrollingPebs]

is there a model where lengths aren't distorted (regardless of angles)

[@JamesSpeiser]

bravo

[@shaunteaches]

Amazing video!!Could you share how you made some of these beautiful models? I would love to print some of these for my high school class!

[@henryseg]

Many of my models are available on printables.com, eg www.printables.com/model/167453-732-triangle-tiling. Remaking them from scratch would not be easy…

[@monocore]

3:15 CUZ IM JUST A TEENAGE DIRTBAG BABY

[@NonTwinBrothers]

The other guys voice sounds familiar... Did he by any chance voice "Chaos" by Jos Leys?

[@saulschleimer2036]

Yep! Very well spotted! (Or rather, heard!)

[@NonTwinBrothers]

@@saulschleimer2036 Oh my 😳

[@ianj2895]

using a sphere and hemisphere to demonstrate the projections - is this just to show the effect of the type of projection in representing the geodesics and angles on a euclidian plane? This being the same effect for both hyperbolic and spherical planes? The actual hyperbolic plane isn't the same shape as the spherical plane?

[@henryseg]

The different projections are like different kinds of maps of the Earth. You can use the Mercator projection, or an equirectangular projection, or any of dozens of more possible ways to map the true geometry of a sphere onto the euclidean plane. Likewise, there are many many different ways to map the true geometry of the hyperbolic plane onto the euclidean plane so we can see it. None of these projections are perfect, they all distort in one way or another. And yes, there is no way to perfectly map the hyperbolic plane to the sphere - they are different.

[@calunsagrenejr]

Who kept replaying 2:52? Just me? K.

[@swedhgemoni8092]

The city of R'lyeh is said to have non-Euclidian geometry. Theoretically, how would a model of it look?

[@eatfruitsalad345]

perfectly odd and informative video!

[@deleted-something]

Interesting

[@glitchy9613]

Is there any way to get the gans model using the hemisphere model and a light source?

[@henryseg]

I don't think so. You can do it with two projections, first casting a shadow onto a hyperboloid and then from the hyperboloid to the euclidean plane. But I don't think there's a way to go direct to the plane.

[@swright2134]

cool video guys ty

[@Alorand]

The Lovecraftian aesthetic at the end is due to thinking too deeply about the wrong angles of the non-euclidean space.

[@ubergeraldine]

But aren't light rays in straight lines themselves an optical illusion? Light travels in waves - or even fields since light is an emanation of electricity and electricity seems to travel in spirals?

[@lev7509]

Yes, practically they have a certain volume, and they may display wave- and field-like properties under some conditions, but their *net* travel direction is a straight line.

[@soup2634]

This is. So Cool. Thnks!

[@NafenX]

This is all well and good but what does hyperbolic mean

[@Lycon721995]

I need to know is this a form of a 4th dimensional object or space? Does this have anything to with with 4th dimensional geometry?

[@henryseg]

Not directly, no.