Geometry with a Strange Name

This is a video about the last Thurston geometry we have not previously explained in our videos, "the universal cover of the 2x2 special linear group over reals". Why such a name? An exciting travel through spaces of motion, product, and twisted product geometries!
Uses fragments of our earlier videos:
Non-Euclidean Third Dimension in Games: • Non-Euclidean Third Di...
Nil Geometry Explained!: • Nil geometry explained!
Portals to Non-Euclidean Geometries: • Portals to Non-Euclide...
Made with RogueViz, our non-Euclidean geometry engine. To learn more, watch these videos, play HyperRogue, or visit our discord: / discord
Narrated by / @tehorarogue
Music from the HyperRogue soundtrack: (CC-BY-SA)
E2xS1 : Graveyard (Shawn Parrotte)
twisted S2xS1: Ivory Tower (Will Savino)
twisted H2xS1: Ocean (Will Savino)
twisted H2xR: Lost Mountain (Lincoln Domina)
Starring (in the order of appearance):
Characters from HyperRogue:
Princess, Vizier, Handsome Gardener, Blue Raider, Desert Man, Ratling Avenger, Rogue, Viking, (unidentified) Knight, Rusałka, Green Raider, Yeti, Ranger, Narcissist, Golem, Necromancer, Cultist, Fire Fairy,
Flail Guard, Pirate, Fat Guard, Yendorian Researcher, (unidentified) Bird
Maxwell Cat: sketchfab.com/... (CC-BY 4.0)
Spaceship and Asteroids from Relative Hell
zenorogue.itch...

[@tcaDNAp]

It's gonna be fun when HyperRogue characters start showing up in math textbooks

[@tcaDNAp]

Speaking of text, great work on the captions and transcript! It really helped me parse all the blackboard bold 😂

[@an_asp]

I like the way that, at 8:55, having an unobstructed straight line here means that you're able to make a nice circular motion with no collisions. It's very cool to take these more abstract dimensions like rotation and represent them as spatial dimensions. Some of my favorite examples of this come from visualizing the phase space or parameter space of control problems, where a complicated process suddenly becomes very visually intuitive to navigate.

[@josh34578]

Now I can finally know how many of the 8 geometries give me motion sickness.

[@seansipos8791]

It is so nice to see good clean videos and visualizations on each of the Thurston geometries. Your videos are the only ones that helped me get anything like an "intuitive" and playful sense of nil or solv, so to see you cover the universal cover of the 2x2 special linear group is a real treat.

[@hallucinogender]

Thank you for fixing the audio. These videos are always fascinating. I tend to have difficulty getting my head around these geometries (though I somehow developed an extremely natural intuition for Solv not long after first learning about it) but it's always fun to learn about and try to warp my mental model of geometry into new and exciting configurations.

[@FinnPlanetballs]

both my ears enjoyed this video :D

[@considerthehumbleworm]

only one of my ears enjoyed this video, but that’s just because the other one isn’t especially interested in math

[@cheeseburgermonkey7104]

11:31 This slide explains everything. Following the determinant formula for the 2x2 matrix shown in that slide, simplifying, and setting it equal to 1 (as described by ~SL(2, R)) gives the exact same equation as the unit-split quaternions equation in the previous slide, which [as said] forms a hyperboloid, which is a model of [2D] hyperbolic space! With ZenoRogue's reply on the original, unlisted video: _Yes, this is correct, except one thing: the hyperboloid which is a model of 3D hyperbolic space has formula w^2-x^2-y^2-z^2=1 (1 plus sign, 3 minus sign), this would be SL(2,R) (or anti-de Sitter spacetime)._

[@The8thOpening]

It's so nice to hear a good explanation of SL2 geometry; I've been unable to find one anywhere else. I would love to learn more about the connections of the other thurston geometries connections to group theory.

[@maxnewdf]

only now i realize the audio was wrong thank you

[@toimine8930]

6:42 omg it's maxwell

[@KingJAB_]

This is complicated and hurts my brain. I love it.

[@music-zv6je]

I don't really understand too much because I'm not that smart, but it's very very exciting and entertaining to watch, even for a regular viewer! I feel like the visualisations would go so well with xenharmonic (microtonal) music for the background (the biggest dream of mine) :D It breaks out of the 12-note system and explores completely different harmonic worlds, much like the Non-Euclidean geometries step out of the traditional intuitions into completely novel experiences; would love to see you collab with Sevish or any other microtonalist! If time and energy allows that, of course...

[@SimplyWondering]

It’s not about being smart it’s just about having a pre existing knowledge base with time and a bit of elbow grease you can get there

[@Utesfan100]

The seifert-weber space would be interesting, and the poincare homology sphere.

[@ZenoRogue]

If you want to see what flying through these looks like, you can do so in the "Curved Space" software by Jeff Weeks, or in HyperRogue (special modes -> experiment with geometry -> geometry -> interesting quotient spaces -> ...)

[@snacku7]

When the world needed her most, she returned

[@Wackylemon64]

I never realized hyperrogue characters were 3d models, huh

[@ZenoRogue]

They were always vector graphics, and when HyperRogue started using 3D for walls, the layers also became shifted, thus becoming 3D models. Although such shifting is not good enough for true 3D mode where they are seen from the side, so the models like the "bird" used here had to be "converted".

[@FireyDeath4]

I understood maybe about two thirds of what you were talking about. I don't think those were the important third

[@nin10dorox]

"Holonomy rotates us by the angle equal to the area inside the loop." That just blew my mind! Does anyone know an intuitive reason why that is?

[@drdca8263]

I think it is probably because it is the integral of (something) over the region bounded by the loop? It seems to make sense that it should be additive over regions. Suppose you have a loop, and you split it in half into two loops. (Suppose your base point is at one of the points that are in all 3 of [the original loop, one of the two loops it was split into, the other of the two loops it was split into], I.e. one of the corners/junctions made when splitting it.) If you start at your base point, go around one loop, and then around the other, both in a CCW direction, the edge where you originally split the loop in two will be traversed twice in opposite directions. Walking along some path and then immediately walking backwards along the same path, should undo whatever that path contributed to. So, the holonomy associated with loops around two adjacent regions, should combine to produce the holonomy for the loop around the combination of the two regions. And, walking in a tiny loop should produce only a tiny bit of holonomy. And, we can split the region bounded by any loop into lots of tiny regions, and the holonomy for the big loop should be the combination of what it would be around each of the tiny loops bounding each of those tiny regions. This seems a lot like an integral. And, the geometry of a sphere is very symmetric, no point looking different from any other. So, the contributions from each of the tiny regions, should be the same (at least if they are the same shape, but we can subdivide them to make them pretty much the same shape) So, it should be proportional to the area.

[@nin10dorox]

@@drdca8263 I suppose you could "polygonize" the loop, and then make a triangle from each segment, so that all the triangles have a vertex at the starting point. Then the problem for a general curve reduces to the problem for triangles.

[@drdca8263]

@@nin10dorox I was thinking a bunch of approximately equilateral triangles for the inside (subdividing each into 4, triforce style, when making them smaller and closer to equilateral), and some little bit extra at the edge to connect the polygonalized approximation to the actual curve you want.

[@ZenoRogue]

Yeah, it blew my mind too when I first heard about it! Let us call the excess of a shape A the amount of holonomy (i.e., the angle by which we are rotated) when going around A. (1) For a spherical triangle T, the sum of internal angles is 180° plus the excess. You can prove that the area of T equals the excess by splitting the sphere into lunes (quite easy, but hard to explain without a picture -- you can google "deriving the surface area of a spherical triangle"). (2) It is easy to see that if the shape A can be split into two shapes B and C (for example, a square split into two triangles) then the excess of A is the sum of the excess of B and the excess of C. So it follows that the fact also must be true for other polygons, because you can triangulate them. (In fact, if you think about (2), it should be quite intuitive that the excess must be proportional to the area, so it is enough to check one example to see what the factor is -- as @drdca8263 already explained.)

[@KaliFissure]

Surface(cos(u/2)cos(v/2),cos(u/2)sin(v/2),sin(u)/2),u,0,2pi,v,0,4pi Radially symmetric Klein bottle? Single sided closed surface?

[@AnotherTowerDev]

1:15 Is that a chess set that works in isometric top-down view and in orthographic, next to Maxwell?

[@ZenoRogue]

Yes. (This scene is from our video "Non-Euclidean Third Dimension in Games", and the original chess set is by Polyfjord -- the link is in the description of that video)

[@artexwolf]

Awesome as always! Been meaning to ask though: what engine do you use to make these?

[@ZenoRogue]

RogueViz (the non-Euclidean engine created for HyperRogue)

[@kadavr314]

Double cover = spinors !!

[@denis_smusev]

I like your channel

[@zeotex2851]

Please consider making 3D KZbin videos, they could be watched in VR which would be great for intuition

[@ZenoRogue]

We do have some VR videos on our channel. But not many people have the VR hardware so it seems better to concentrate on those who do not. (VR video seems better for videos which are mostly 3D visualizations, like "Portals to Non-Euclidean Geometries" which has a 360° VR version -- here we have lots of elements which seem better shown on flatscreen)

[@mathematicskid]

I still don't get why there are any loops in PSL(2,R), either way great video!

[@Kaiveran]

It's the equivalent of rotating 360° in the base hyperbolic plane, you come back to where you started. The universal cover simply "undoes" this loop.

[@Kaiveran]

Yay finally 🥳

[@DanielLCarrier]

Where did you explain solv geometry? I can't find it.

[@ZenoRogue]

In "Non-Euclidean Third Dimension in Games"

[@lailoutherand]

Why did my dumahh mind think of "geometry dash"

[@Kaiveran]

Non- euclidean 3D gdash would probably be the ultimate kaizo/troll game.

[@Trismhmm]

Amazinggg, Glory be to Yeshua. Thank You for the Succinct and interesting discussions!!

[@user-xu1ue4oi2h]

Maxwell 7:17

[@antoninduda9078]

Babe, wake up! Zenorogue just posted.

[@antirogue825]

wait, the minimap in the bottom right corner is using actual graphics instead of ascii symbols? is this part of the next hyperrogue update or just a one off for this video?

[@ZenoRogue]

If you activate "current x E" or "twisted current x E" in "experiment with geometry" in HyperRogue, you can also activate "view the underlying geometry". This feature is used in this video. It is in HyperRogue for a long time, but it does a rather different thing than the minimap.

[@maxbrown1990]

What about a 4-dimensional Euclidian geometry?

[@ZenoRogue]

We are showing our visualization of E4 (very different than typical ones) in "Higher-Dimensional Spaces using Hyperbolic Geometry".

[@maxbrown1990]

@@ZenoRogue You mean, only non-Euclidian geometry only?

[@ZenoRogue]

I mean that we use a different method of visualization of E4 which is based on hyperbolic geometry. Typical visualizations are based on slicing or perspective.

[@abellematheux7632]

High quality vulgarisation

[@MushookieMan]

Mmmm... burger sphere

[@SamPuckettOfficial]

nil rider when

[@ZenoRogue]

Nil Rider is available for some time on itch.io... but we are improving it :)

[@music-zv6je]

first

[@SlashCrash_Studios]

Blue shell

[@hiccupwarrior89]

both my ears enjoyed this video :D