@@rocc9944 the cat that maxwell was based on is uni, the actual maxwell model is called "goober" or something similar i dont quite remember and then the internet renamed this model to maxwell
Visualizations of the weirder parts of topology/math are literally my passion and my single favorite thing to mess around with. I am so grateful that you've been making videos like these -- it ALWAYS makes my entire week to see a new post!! I can't wait for the next one, this was great!! I would love to see more details about horospheres like you mentioned here, or the math behind your solvmanifold! I've been going back through your nil posts lately and seeing if I can find the various subgroup symmetries of the Heisenberg group (lower affine space, the non-abelian upper-right term in matrix multiplication leading to the "twist" etc), your videos make SUCH a huge difference for math visualization and communication!!
I love stumbling across your stuff when I'm half-asleep and it's like you're yanking me into math land to break my brain with weird geometries. Never stop 💜
Ok so they've got all these ways of rendering non-euclidean spaces, which is extremely difficult in itself, but they've also got ways of smoothly transitioning between one type of non-euclidean geometry and any other? WTF!? Do these in-between states represent their own geometries? Is there a measure of how close an inbetween state is to the two its made up of? I assume there are likely problems with these geometries, but if they can be rendered they exist right?
Thanks! That is a good question. If you look closely, we never transition directly from non-Euclidean geometry A to non-Euclidean geometry B in this video -- there is always Euclidean geometry in the middle of transition. Every geometry becomes closer and closer to Euclidean geometry as we zoom in. (Just like larger and larger spheres are locally closer and closer to flat.) So we use the following trick: at, say 20% of the transition from Euclidean geometry to A, we do the same embedding as we would do for full A, but we do it to the scene scaled 20%. So it is the same geometry as full A, but with smaller curvature, equivalently, larger absolute unit (relative to the sizes of objects). In some cases a clear and nice direct transition is possible. For example, H3 geometry has 2:1 scaling per Z level in X and 2:1 scaling per Z level in Y, while Solv geometry has 2:1 scaling per Z level in X and 1:2 scaling per Z level in Y. So we could of course transition directly: inbetween geometry has 2:1 scaling per Z level in X and has t scaling per Z level in Y, where t is transitioning between 2 and 1/2 (for t=1 we get H2xR geometry, not Euclidean, since scaling in X is always 2:1). So we get new geometries (which are not Thurston geometries except the three special values, since there are no closed manifolds with these geometries). However, the algorithm we use in this video requires precomputed tables for each such geometry, so that would be difficult to render and we transition through Euclidean instead. (The RogueViz engine has precomputed tables for 2:1, 3:1, 1:1, 1:2 and 1:3; in some videos we are using a ray-based algorithm, and we could render such a direct transition more easily using that algorithm.)
At least it's actual Non-Euclidean geometry rather than just using it as a buzzword like some people I've seen do on other games. It irritates me slightly when they do that.
@@julianemery718 yeah? if you have portals you can draw two parallel lines that converge or diverge, which, among other things isn't allowed in euclidean space.
Summary: ZenoRogue has talked before about how this is better viewed as a manifold because these games tend to not be showing con/div-erging parallel lines really
Summary: ZenoRogue has talked before about how this is better viewed as a manifold because these games tend to not be showing con/div-erging parallel lines really
Desperately begging the next dev who makes a Lovecraft videogame to set the climax in R'lyeh and have real non-Euclidean geometry like this. Imagine if instead of nice chessboards and low-poly animals with bright lighting it was horror themed; stygian darkness with alien imagery and unnatural hues.
Questions. So the horosphere is a flat torus (with 0 curvature) represented in a space with negative curvature and as a result has to look spherical (positive curvature) to compensate, and the clifford torus is a flat torus represented in a space with positive curvature and as a result has to look like a negatively curved shape to compensate?
Yes, that is a good way to put it. The formula is: in three-dimensional space of constant curvature K1, a surface with principal curvatures k1 and k2 will have Gaussian curvature K1+k1*k2. (If k1 and k2 have different signs (and thus the product is negative), it looks like a saddle, and if they have same signs (and thus the product is positive), it looks like a hill. So we get what you say.
Due to more significant coreolis-effects, projectile physics within a rotating-cylinder/ring-habitat would realistically be VERY hard to predict, making realistic "halo" shooter games or any sports-event-game unnecessarily difficult. Meanwhile on earth, some places have barely-significantly lower-surface-gravity-levels, making international sport competitions less comparable between any 2 locations. I am still waiting for "halo tennis with realistic physics" if only as a game-jam-joke, tennis with much stronger coreolis-effets, that mostly depends on latitude+longitude within the rotating ring habitat, besides waiting the space-sim-game that includes realistic time-dilation and length contraction and redshifts, using the opensource ["a slower speed of light"] engine (that sadly only runs in a very old unityengine version without MANY code-fixes to update it). You may "jump into a non-euclidean space that has a higher-speed-of_light" for faster-than-light travel, BUT PLEASE make it look better than a tunnel of white stripes, and maybe make it a noneuclidean-space-nightmare-to-get-lost-in like the warhammer40k books describe "ftl-travel to be a dangerous nightmare hell-realm".
Thanks for the amazing explanation! Non-euclidean geometry may not have much practical use but it is very good for the brain. I want to ask you, will the iOS version of hyperRogue ever be updated beyond v11.3h?
Thanks! Good for the brain is practical use, and these structures are used for modeling various real-world things :) The last time I have tried to update HyperRogue on iOS, it seemed impossible without buying a new macbook... it seems this platform is not very friendly for experimental software, I have seen some other cool-looking non-Euclidean games that were iOS only and they are long lost. :(
The horosphere returns! What a neat application of it! So you're essentially embedding a Euclidean world inside of a non-Euclidean space? Do you use ray tracing or ray marching or something similar to render this (I guess it may be difficult to use OpenGL's ordinary rendering pipeline to render anisotropic spaces), and what frames rate do you obtain? Are you able to render this in realtime (I would consider that to be at least 60 frames per second)?
All this video is rendered using OpenGL's ordinary pipeline (I mean, triangles, custom non-Euclidean vertex shaders are used to compute where triangle vertices are rendered). (Contrary to e.g. "Portals to non-Euclidean Geometries" which is all ray-based and fast enough for VR.) Rendering of some scenes, e.g. "inside horosphere", was very slow (
@@ZenoRogue Oh! I guess if you use OpenGL's ordinary pipeline, you must project each point (triangle vertex) in the non-Euclidean space to Euclidean space somehow. For hyperbolic space, you can do that by using the Beltrami-Klein model, since that is azimuthal and preserves straight lines. For spherical geometry, you can use a gnomonic projection instead, which also has those properties. Is there a corresponding projection for the anisotropic hyperbolic space? I have no idea how you would go about finding such a projection in practice. Would it be some kind of mix between the Beltrami-Klein model and a gnomonic projection? Also, if you're able to use OpenGL's ordinary pipeline, how come you get < 1 fps? Sure, 200,000 is much more than 500, but I still don't think it seems that much if you have a comparably large number of triangles.
This is called the inverse exponential map in Riemannian geometry (denoted "log"). For a point x, log(x) is a vector whose direction is the direction of geodesic reaching x, and length is the length of that geodesic. (I also like to call this "azimuthal equidistant projection".) While log(x) returns the exact length, what we actually need is that the ordering is correct (the points further away return longer vectors). So Beltrami-Klein/gnomonic is different than azimuthal equidistant, but it still works. Actually, Beltrami-Klein/gnomonic works better, because it also happens to give correctly ordered on interpolation (in other words, depth testing is guaranteed to work correctly not only on vertices, but also on the whole triangles). But I do not see any glitches caused by incorrect interpolation for azimuthal equidistant in anisotropic. (I think we also adjust the distances to make it more Klein-like.) There is also another challenge: that log(x) is multivalued (not in Euclidean/hyperbolic/NIH, tame in spherical, but a problem in Nil and even wilder in Solv). Turns out just returning one value works fine enough, though. (See our paper for more details: "Real-Time Visualization in Anisotropic Geometries") Apparently the chessboard model has about 2e5 triangles. So the number of triangles in chessboards rendered per frame is roughly 2e5 * 2e5 / 4e2 = 1e8 on chessboards alone. (Other things have less.) Also I get ~1 fps after disabling the "gldraw" function, so lots of time is spend in various functions on the CPU side, probably various things done in HyperRogue, just in case if there are interesting things on a tile, bookkeeping to prevent the same tile being drawn again in the same position, etc. I have not profiled this.
Do you have an explanation somewhere on your channel for how geodesics work in Nil geometry, similar to the one here about Solv? I've always found Solv to be surprisingly intuitive, but Nil confuses me.
Yes, see the video "Nil geometry explained", in which we also used these birds. What's funny, actually some people on that video wanted a similar explanation of Solv :)
Love your channel! The visual and the content itself is solid but a better recording equipment will be appreciated. When the rest of the stuff is so good, small issues like these stand out.
@ZenoRogue : Since you have implemented non-commutative coordinates (Spherical geometry is more or less Quaternionic and Hyperbolic geometry is more or less Split-Quaternionic, even though PSL geometry is also connected to the Split-Quaternions, similar things are true for other Non-Euclidean Thurston geometries, if we allow Dual Quaternions etc), have you considered implementing geometries with non-associative coordinates like those of Octonions and Split-Octonions? These would definitely be interesting since their primary symmetry structures are alternative non-associative Moufang Loops rather than associative non-commutative Groups, although they can always be expanded into multiplication groups as secondary symmetry structures, plus they are very closely related to Groups with Triality (a la Glauberman and Doro)! For Split-Octonions there is a very nice implementation using real Zorn Vector Matrices; with a bit of work this is possible also for regular Octonions using complex Zorn Vector Matrices with some restriction - Skew-Hermitian i think.. Just like Groups can act on geometric sets/topological spaces through left and right Group Actions (or both combined, which always commute with each other), so can Moufang's Loops act on geometric sets or topological spaces through left and right Moufang Loop Actions (or both combined, which do NOT commute usually). For Groups and Group Actions we have: f*(g*h) = (f*g)*h = f*g*h 1*f = f*1 = f f'*f = f*f' = 1 ~ f(g(x)) = (f*g)(x) f'(f(x)) = f(f'(x)) = 1(x) = x ((x)f)g = (x)(f*g) ((x)f)f' = ((x)f')f = (x)1 = x (f(x))g = f((x)g) = f(x)g f'(x)f =/= x generally, instead this is conjugation of x by f, similarly with f(x)f' =/= x For Moufang Loops and Moufang Loop Actions we have: f^m*(g*f^n) = (f^m*g)*f^n = f^m*g*f^n f*(f^n*g) = f^n*(f*g) = (f^(n+1))*g (g*f^n)*f = (g*f)*f^n = g*(f^(n+1) f*(g*(f*h)) = (f*g*f)*h ((g*f)*h)*f = g*(f*h*f) (f*g)*(h*f) = f*(g*h)*f 1*f = f*1 = f f'*(f*g) = f*(f'*g) = (g*f)*f' = (g*f')*f = g ~ f(g(f(x))) = (f*g*f)(x) f'(f(x)) = f(f'(x)) = 1(x) = x (((x)f)g)f = (x)(f*g*f) ((x)f)f' = ((x)f')f = (x)1 = x (f(x))f = f((x)f) = f(x)f f((x)(f*g)) = (f(x)f)g ((g*f)(x))f = g(f(x)f) (f*g)((x)f) = f(g(x))f (f(x))(g*f) = f((x)g)f f'(x)f =/= x generally, conjugation (f*g)'(f(g(x))) =/= x generally, another conjugation, similarly with f(g((f*g)'(x))) =/= x (((x)f)g)(f*g)' =/= x generally, yet another conjugation, similarly with (((x)(f*g)')f)g Just like Group Action Laws are derived from Group Laws, so Moufang Loop Action Laws are derived from Moufang Loop Laws. The smallest possible non-associative Moufang Loop has 12 elements (just like the smallest possible non-commutative Group has 6 elements), it consists of 1 unity, 9 reflections and 2 order-3 rotations. This M2(Sym(3)) is most probably embeddable into the Split-Octonions, just like Sym(3) is embeddable into the Split-Quaternions.
random personnal fact : the first time i've learned how devs trick playesr into believing their square grid is mappen on a sphere was with the game Glaxy55. Never heard of it? Yeah sadly it died in alpha, with the final nail on the coffin being the departure of the main dev of this project. Is there any form of link with this video? Well yes, after that he released his own games ..... notably the one at 2:49 :)
Flateartheners: See? There’s no difference between flat surface and spherical one. It’s just one way of perspectives, camera settings more like, not the actual structure itself. Scientists: Well ughm I dunno
In what direction/shape does light travel in solv geometry? (It’s a straight line for Euclidean but it’s a helix in Nil, so I’m wondering what it is for Solv)
The path drawn about 9:30 repeats. So it is like up, north, down, east, up, north, down, east, up, north, down, east, ... which is kind of helical. Of course the inhabitants might just call it a straight line :)
Combining multiple space-bending ideas (portals, hyperbolic geometry, 4D, etc.) in one game is probably too mind-bending. And it is not clear if there is any coolness that could be gained from that rather than doing them separately.
Uniform usually means tiled by regular polyhedra, and the concept of a regular polyhedron does not work in these anisotropic geometries. But you can have vertex-transitive, tile-transitive tilings (but not edge-transitive, I think).
so can you make an sandbox that lets you modify the geometry like in the video? like this gui _______________________________________ | X multiplier : 0.5 +------- 1 | | Y multiplier : 0.5 +------- 1 | | Fold Y? : [ON] - [OFF] | | Fold X? : [ON] - [OFF] | | _Custom X/Y formula? : [ON] - [OFF] _| because im curious if i can make spherical geometry and more :D
In what sense? All these visuals are available in HyperRogue, although they do not affect the gameplay, which is still designed primarily for 2D top-down. We would love to inspire other game developers and other artists (writers, painters, movie makers) to try some new things.
@@ZenoRogue The level in the video has nice details and 3D (or more D?) objects, which makes it much more interesting to explore. From what i've seen in videos about HyperRogue, it's levels have not many features, and objects are represented by 2D sprites. So, if the program in the video is not a future game, then what is that? Just an engine to do fun stuff and make videos? I'm genuinely interested. Is it available in some form?
@@kras_mazov HyperRogue gameplay is designed for top-down, so most videos are in 2D. In the 2D scenes in this video, you can see some 2D sprites in the HyperRogue style. The world is indeed somewhat featureless, on the other hand, hyperbolic geometry makes it bigger than any Euclidean game -- the game is about exploring this huge world. The lack of decoration makes the gameplay more clear (in the traditional roguelike style), and also is meaningful for some navigation quests. However, it also has special modes to display the world in 3D, including the styles shown in this video. Then the 2D sprites are replaced by 3D models (unfortunately very poor, as it does not seem worth it to create professional ones). However, the engine has no problem with rendering more complex 3D models or richer environments. So we use the HyperRogue engine with a visually richer scene. You can play this scene yourself in RogueViz (see the link and info in description), although it has no actual gameplay.
MINECRAFT producers planning to make minigame inspired from MINECRAFT but like TEMPLE RUN named NOOB RUN NONEUCLIDIAN whick will uses the non-euclidian geometry like "horosphere" and "soft geometry" and "clifford Taurus", with graphic like in MINECRAFT where the Noob after was stolen a forbidden idol is chased allways by Huggy Wuggy in this non-euclidian space, then this Noob must avoid any obstacles for to not been caught by Huggy Wuggy.
Well, really there are different kinds of 4-dimensional spaces; "4D" just means that any position in that space is described by four numbers. Usually when people say 4D they mean regular old euclidean space with an extra dimension, where most of our 3D geometric principles still apply (e.g. parallel lines never converge or diverge) You could easily have a hyperbolic 4D space though, where for example you can have five hypervolumes meeting at each face (rather than only four), with the dihedral angle between adjacent hypervolumes being 90 degrees. Non-Euclidean 4D spaces are very difficult to reason about though -- you pretty much have to give up on visualizing it and just use the relevant mathematics. (though I wouldn't be too surprised if the ZenoRogue team have developed some ability to imagine that kind of geometry!)
@@trainjumper Well, I don't know... Since trying to imagine the 4D space is so hard with the strict '90 degrees between the axises' rule, then maybe those axises are folded down a bit and they are only getting unfolded from our perspective? Because the sunlight normally goes straight, but near large objects it curves in space... 😁
@@maxbrown1990 Yeah, space in our universe is technically non euclidean since it curves towards massive objects (general relativity). You are definitely also free to imagine a 4D coordinate system with flattened axes; mathematically it's technically equivalent to one where all axes are at 90 degrees, as long as your axes are linearly independent (e.g. you shouldn't be able to create the vector -W-> by any combination of -X-> -Y-> and -Z->, since in that case the extra dimension wouldn't be adding any new information). Sorry for the long response, basically dimensions are a flexible mathematical tool and can be whatever you need them to be
![[#2024MAMA] ROSÉ (로제), Bruno Mars - APT. | Mnet 241122 방송](http://i.ytimg.com/vi/dgGqD28J6aQ/mqdefault.jpg)
I appreciate they took the time to put Maxwell and not just any cat in the scene
I appreciate they took the time to put purr whenever camera near Maxwell
the cat's name is uni
@@rocc9944 HIS NAME IS MAXWELL.
@@flamegame5877 ok so i did some research and apparently maxwell's real name is jess
so stfu
@@rocc9944 the cat that maxwell was based on is uni, the actual maxwell model is called "goober" or something similar i dont quite remember and then the internet renamed this model to maxwell
the reveal that the cat was maxwell was the best part
I'm fully satisfied with the chosen player model 🐾
the light ray graph was key to helping me understand the Nil geometry, i am very happy you used it again to explain solv!
very good, from the explanation to the models, and the storytelling, its a complete video
Visualizations of the weirder parts of topology/math are literally my passion and my single favorite thing to mess around with. I am so grateful that you've been making videos like these -- it ALWAYS makes my entire week to see a new post!! I can't wait for the next one, this was great!! I would love to see more details about horospheres like you mentioned here, or the math behind your solvmanifold! I've been going back through your nil posts lately and seeing if I can find the various subgroup symmetries of the Heisenberg group (lower affine space, the non-abelian upper-right term in matrix multiplication leading to the "twist" etc), your videos make SUCH a huge difference for math visualization and communication!!
I love stumbling across your stuff when I'm half-asleep and it's like you're yanking me into math land to break my brain with weird geometries.
Never stop 💜
And the cat is amazing loo
Thanks to your channel I've discovered Hyper Rogue. You have kickstarted a fascination with non euclidean geometry. I appreciate all of your content!!
The Catloaf 💀
The cat cameo was lovely!
You do a great job explaining so many cool concepts!
Also, hyperrogue is so cool
Always a treat to watch your videos and explanations/explorations.
is the cat purring at 2:30?
Ok so they've got all these ways of rendering non-euclidean spaces, which is extremely difficult in itself, but they've also got ways of smoothly transitioning between one type of non-euclidean geometry and any other? WTF!?
Do these in-between states represent their own geometries? Is there a measure of how close an inbetween state is to the two its made up of? I assume there are likely problems with these geometries, but if they can be rendered they exist right?
Thanks! That is a good question.
If you look closely, we never transition directly from non-Euclidean geometry A to non-Euclidean geometry B in this video -- there is always Euclidean geometry in the middle of transition.
Every geometry becomes closer and closer to Euclidean geometry as we zoom in. (Just like larger and larger spheres are locally closer and closer to flat.) So we use the following trick: at, say 20% of the transition from Euclidean geometry to A, we do the same embedding as we would do for full A, but we do it to the scene scaled 20%. So it is the same geometry as full A, but with smaller curvature, equivalently, larger absolute unit (relative to the sizes of objects).
In some cases a clear and nice direct transition is possible. For example, H3 geometry has 2:1 scaling per Z level in X and 2:1 scaling per Z level in Y, while Solv geometry has 2:1 scaling per Z level in X and 1:2 scaling per Z level in Y. So we could of course transition directly: inbetween geometry has 2:1 scaling per Z level in X and has t scaling per Z level in Y, where t is transitioning between 2 and 1/2 (for t=1 we get H2xR geometry, not Euclidean, since scaling in X is always 2:1). So we get new geometries (which are not Thurston geometries except the three special values, since there are no closed manifolds with these geometries). However, the algorithm we use in this video requires precomputed tables for each such geometry, so that would be difficult to render and we transition through Euclidean instead. (The RogueViz engine has precomputed tables for 2:1, 3:1, 1:1, 1:2 and 1:3; in some videos we are using a ray-based algorithm, and we could render such a direct transition more easily using that algorithm.)
At least it's actual Non-Euclidean geometry rather than just using it as a buzzword like some people I've seen do on other games.
It irritates me slightly when they do that.
I mean some of those games *are* actually non-euclidean, but just the boring kind of non-euclidean (portals). these are much more interesting
@@thezipcreator
So, you saying portals are a type of non-euclidian stuff or...?
@@julianemery718 yeah? if you have portals you can draw two parallel lines that converge or diverge, which, among other things isn't allowed in euclidean space.
Summary: ZenoRogue has talked before about how this is better viewed as a
manifold because these games tend to not be showing con/div-erging
parallel lines really
Summary: ZenoRogue has talked before about how this is better viewed as a
manifold because these games tend to not be showing con/div-erging
parallel lines really
Desperately begging the next dev who makes a Lovecraft videogame to set the climax in R'lyeh and have real non-Euclidean geometry like this. Imagine if instead of nice chessboards and low-poly animals with bright lighting it was horror themed; stygian darkness with alien imagery and unnatural hues.
You guys are the most genius mathematicians I've ever seen work on video games.
Yeah but @CodeParade is a close match!
@@henrikljungstrand2036 Shoutout to CodeParade too!
I very much agree 😄🤓
Cheers to all fellow mathematicians!
@@MetaBuddhaPlease check out @kayturs as well with his Hyperblock game prototype!
@@MetaBuddhaAnd @pato32rs (Patrick Owen) with a hyperbolic space rendering demo.
@@MetaBuddha It is interesting with the spherical space building game Metrica by @aaronkriegman8684 also.
Hello Maxwell!!
Questions. So the horosphere is a flat torus (with 0 curvature) represented in a space with negative curvature and as a result has to look spherical (positive curvature) to compensate, and the clifford torus is a flat torus represented in a space with positive curvature and as a result has to look like a negatively curved shape to compensate?
Yes, that is a good way to put it. The formula is: in three-dimensional space of constant curvature K1, a surface with principal curvatures k1 and k2 will have Gaussian curvature K1+k1*k2. (If k1 and k2 have different signs (and thus the product is negative), it looks like a saddle, and if they have same signs (and thus the product is positive), it looks like a hill. So we get what you say.
2:28 I love how the cat just purrs if you're close to him
two minutes ago zenorougue video??? wow!
I love the narration on these
Due to more significant coreolis-effects, projectile physics within a rotating-cylinder/ring-habitat would realistically be VERY hard to predict, making realistic "halo" shooter games or any sports-event-game unnecessarily difficult.
Meanwhile on earth, some places have barely-significantly lower-surface-gravity-levels, making international sport competitions less comparable between any 2 locations.
I am still waiting for "halo tennis with realistic physics" if only as a game-jam-joke, tennis with much stronger coreolis-effets, that mostly depends on latitude+longitude within the rotating ring habitat, besides waiting the space-sim-game that includes realistic time-dilation and length contraction and redshifts, using the opensource ["a slower speed of light"] engine (that sadly only runs in a very old unityengine version without MANY code-fixes to update it).
You may "jump into a non-euclidean space that has a higher-speed-of_light" for faster-than-light travel, BUT PLEASE make it look better than a tunnel of white stripes, and maybe make it a noneuclidean-space-nightmare-to-get-lost-in like the warhammer40k books describe "ftl-travel to be a dangerous nightmare hell-realm".
The cats purrs
wait is that the 2d to 3d chess set from that one video
У меня мозг начал вытекать из ушей, пока я пытался сориентироваться в этих пространствах -_-
Maxwell
Thanks for the amazing explanation! Non-euclidean geometry may not have much practical use but it is very good for the brain. I want to ask you, will the iOS version of hyperRogue ever be updated beyond v11.3h?
Thanks! Good for the brain is practical use, and these structures are used for modeling various real-world things :) The last time I have tried to update HyperRogue on iOS, it seemed impossible without buying a new macbook... it seems this platform is not very friendly for experimental software, I have seen some other cool-looking non-Euclidean games that were iOS only and they are long lost. :(
The horosphere returns! What a neat application of it! So you're essentially embedding a Euclidean world inside of a non-Euclidean space? Do you use ray tracing or ray marching or something similar to render this (I guess it may be difficult to use OpenGL's ordinary rendering pipeline to render anisotropic spaces), and what frames rate do you obtain? Are you able to render this in realtime (I would consider that to be at least 60 frames per second)?
All this video is rendered using OpenGL's ordinary pipeline (I mean, triangles, custom non-Euclidean vertex shaders are used to compute where triangle vertices are rendered). (Contrary to e.g. "Portals to non-Euclidean Geometries" which is all ray-based and fast enough for VR.)
Rendering of some scenes, e.g. "inside horosphere", was very slow (
@@ZenoRogue Oh! I guess if you use OpenGL's ordinary pipeline, you must project each point (triangle vertex) in the non-Euclidean space to Euclidean space somehow. For hyperbolic space, you can do that by using the Beltrami-Klein model, since that is azimuthal and preserves straight lines. For spherical geometry, you can use a gnomonic projection instead, which also has those properties. Is there a corresponding projection for the anisotropic hyperbolic space? I have no idea how you would go about finding such a projection in practice. Would it be some kind of mix between the Beltrami-Klein model and a gnomonic projection?
Also, if you're able to use OpenGL's ordinary pipeline, how come you get < 1 fps? Sure, 200,000 is much more than 500, but I still don't think it seems that much if you have a comparably large number of triangles.
This is called the inverse exponential map in Riemannian geometry (denoted "log"). For a point x, log(x) is a vector whose direction is the direction of geodesic reaching x, and length is the length of that geodesic. (I also like to call this "azimuthal equidistant projection".)
While log(x) returns the exact length, what we actually need is that the ordering is correct (the points further away return longer vectors). So Beltrami-Klein/gnomonic is different than azimuthal equidistant, but it still works. Actually, Beltrami-Klein/gnomonic works better, because it also happens to give correctly ordered on interpolation (in other words, depth testing is guaranteed to work correctly not only on vertices, but also on the whole triangles). But I do not see any glitches caused by incorrect interpolation for azimuthal equidistant in anisotropic. (I think we also adjust the distances to make it more Klein-like.)
There is also another challenge: that log(x) is multivalued (not in Euclidean/hyperbolic/NIH, tame in spherical, but a problem in Nil and even wilder in Solv). Turns out just returning one value works fine enough, though. (See our paper for more details: "Real-Time Visualization in Anisotropic Geometries")
Apparently the chessboard model has about 2e5 triangles. So the number of triangles in chessboards rendered per frame is roughly 2e5 * 2e5 / 4e2 = 1e8 on chessboards alone. (Other things have less.) Also I get ~1 fps after disabling the "gldraw" function, so lots of time is spend in various functions on the CPU side, probably various things done in HyperRogue, just in case if there are interesting things on a tile, bookkeeping to prevent the same tile being drawn again in the same position, etc. I have not profiled this.
Do you have an explanation somewhere on your channel for how geodesics work in Nil geometry, similar to the one here about Solv? I've always found Solv to be surprisingly intuitive, but Nil confuses me.
Yes, see the video "Nil geometry explained", in which we also used these birds. What's funny, actually some people on that video wanted a similar explanation of Solv :)
Beautiful cheese holes
Really cool. I especially appreciate mr. Rat
Love your channel! The visual and the content itself is solid but a better recording equipment will be appreciated. When the rest of the stuff is so good, small issues like these stand out.
@ZenoRogue :
Since you have implemented non-commutative coordinates (Spherical geometry is more or less Quaternionic and Hyperbolic geometry is more or less Split-Quaternionic, even though PSL geometry is also connected to the Split-Quaternions, similar things are true for other Non-Euclidean Thurston geometries, if we allow Dual Quaternions etc), have you considered implementing geometries with non-associative coordinates like those of Octonions and Split-Octonions?
These would definitely be interesting since their primary symmetry structures are alternative non-associative Moufang Loops rather than associative non-commutative Groups, although they can always be expanded into multiplication groups as secondary symmetry structures, plus they are very closely related to Groups with Triality (a la Glauberman and Doro)!
For Split-Octonions there is a very nice implementation using real Zorn Vector Matrices; with a bit of work this is possible also for regular Octonions using complex Zorn Vector Matrices with some restriction - Skew-Hermitian i think..
Just like Groups can act on geometric sets/topological spaces through left and right Group Actions (or both combined, which always commute with each other), so can Moufang's Loops act on geometric sets or topological spaces through left and right Moufang Loop Actions (or both combined, which do NOT commute usually).
For Groups and Group Actions we have:
f*(g*h) = (f*g)*h = f*g*h
1*f = f*1 = f
f'*f = f*f' = 1
~
f(g(x)) = (f*g)(x)
f'(f(x)) = f(f'(x)) = 1(x) = x
((x)f)g = (x)(f*g)
((x)f)f' = ((x)f')f = (x)1 = x
(f(x))g = f((x)g) = f(x)g
f'(x)f =/= x generally, instead this is conjugation of x by f, similarly with f(x)f' =/= x
For Moufang Loops and Moufang Loop Actions we have:
f^m*(g*f^n) = (f^m*g)*f^n = f^m*g*f^n
f*(f^n*g) = f^n*(f*g) = (f^(n+1))*g
(g*f^n)*f = (g*f)*f^n = g*(f^(n+1)
f*(g*(f*h)) = (f*g*f)*h
((g*f)*h)*f = g*(f*h*f)
(f*g)*(h*f) = f*(g*h)*f
1*f = f*1 = f
f'*(f*g) = f*(f'*g) = (g*f)*f' = (g*f')*f = g
~
f(g(f(x))) = (f*g*f)(x)
f'(f(x)) = f(f'(x)) = 1(x) = x
(((x)f)g)f = (x)(f*g*f)
((x)f)f' = ((x)f')f = (x)1 = x
(f(x))f = f((x)f) = f(x)f
f((x)(f*g)) = (f(x)f)g
((g*f)(x))f = g(f(x)f)
(f*g)((x)f) = f(g(x))f
(f(x))(g*f) = f((x)g)f
f'(x)f =/= x generally, conjugation
(f*g)'(f(g(x))) =/= x generally, another conjugation, similarly with f(g((f*g)'(x))) =/= x
(((x)f)g)(f*g)' =/= x generally, yet another conjugation, similarly with (((x)(f*g)')f)g
Just like Group Action Laws are derived from Group Laws, so Moufang Loop Action Laws are derived from Moufang Loop Laws.
The smallest possible non-associative Moufang Loop has 12 elements (just like the smallest possible non-commutative Group has 6 elements), it consists of 1 unity, 9 reflections and 2 order-3 rotations. This M2(Sym(3)) is most probably embeddable into the Split-Octonions, just like Sym(3) is embeddable into the Split-Quaternions.
MAXWELL
Fabulous, thank you
This is so incredibly cool!
Amazing work
Omg, the sequel
Cool! So... They are here from equilateral triangles and octahedrons but the perspective makes it look like squares!? Wow
Maxwell❗❗❗❗❗
Great video
8:25 Solv
Nice cameo with Maxwell cat.
Just found your channel, these videos are mind blowing! How can I contact you about getting rights to use some footage?
You can write an e-mail to zeno (at) attnam.com.
great video!
Nice Job B-Mo
random personnal fact : the first time i've learned how devs trick playesr into believing their square grid is mappen on a sphere was with the game Glaxy55.
Never heard of it? Yeah sadly it died in alpha, with the final nail on the coffin being the departure of the main dev of this project.
Is there any form of link with this video? Well yes, after that he released his own games ..... notably the one at 2:49 :)
These demonstrations always fascinate me. Any chance RogueViz will someday be available for Mac?
It should not be a big problem for us to compile a Mac version; on the other hand, it should probably work via Wine without problems.
@@ZenoRogue it doesn't wine :(
Yeah, I tried wine. No luck.
I love these videos
Flateartheners: See? There’s no difference between flat surface and spherical one. It’s just one way of perspectives, camera settings more like, not the actual structure itself.
Scientists: Well ughm I dunno
When the sky becomes a line, it's a different kind of skyline.
Yeah? What do y’all want?
4:28 trippy 😮😮😮
5:50
It's... It's the Smith Chart.
Your world is a Smith Chart.
I miss ur videos ❤❤❤❤
Horosphere? More like horror sphere with that creepy-ass cat staring at me!
In what direction/shape does light travel in solv geometry? (It’s a straight line for Euclidean but it’s a helix in Nil, so I’m wondering what it is for Solv)
The path drawn about 9:30 repeats. So it is like up, north, down, east, up, north, down, east, up, north, down, east, ... which is kind of helical. Of course the inhabitants might just call it a straight line :)
imagine: Portal™ but hyperbolic
Combining multiple space-bending ideas (portals, hyperbolic geometry, 4D, etc.) in one game is probably too mind-bending. And it is not clear if there is any coolness that could be gained from that rather than doing them separately.
I am on a plane in a plane. Hum.. how very plain.
Toast, pimento cheese, spinach, bacon, jalapeños, and honey. 😋
4:57 It is hyperbolic?
Yes, the geometry of the 3D space in 4:57 is hyperbolic.
oohe i got motion sick
Can Nil or Solv be uniformly tiled?
Uniform usually means tiled by regular polyhedra, and the concept of a regular polyhedron does not work in these anisotropic geometries. But you can have vertex-transitive, tile-transitive tilings (but not edge-transitive, I think).
so can you make an sandbox that lets you modify the geometry like in the video?
like this gui
_______________________________________
| X multiplier : 0.5 +------- 1 |
| Y multiplier : 0.5 +------- 1 |
| Fold Y? : [ON] - [OFF] |
| Fold X? : [ON] - [OFF] |
| _Custom X/Y formula? : [ON] - [OFF] _|
because im curious if i can make spherical geometry and more :D
The problem is that (in the current implementation) modifying the geometry this way needs recomputing some stuff, which takes some time.
@@ZenoRogue oh okay
Oh I can tile a sphere with squares. It's called a cube :)
This was in my recommended. where am I. why
zeno, can you read my other suggestion comment?
Which one? (try joining the HyperRogue Discord server, that is better for conversation)
Are you going to make this into a game?
In what sense? All these visuals are available in HyperRogue, although they do not affect the gameplay, which is still designed primarily for 2D top-down. We would love to inspire other game developers and other artists (writers, painters, movie makers) to try some new things.
@@ZenoRogue The level in the video has nice details and 3D (or more D?) objects, which makes it much more interesting to explore. From what i've seen in videos about HyperRogue, it's levels have not many features, and objects are represented by 2D sprites.
So, if the program in the video is not a future game, then what is that? Just an engine to do fun stuff and make videos? I'm genuinely interested. Is it available in some form?
@@kras_mazov HyperRogue gameplay is designed for top-down, so most videos are in 2D. In the 2D scenes in this video, you can see some 2D sprites in the HyperRogue style. The world is indeed somewhat featureless, on the other hand, hyperbolic geometry makes it bigger than any Euclidean game -- the game is about exploring this huge world. The lack of decoration makes the gameplay more clear (in the traditional roguelike style), and also is meaningful for some navigation quests.
However, it also has special modes to display the world in 3D, including the styles shown in this video. Then the 2D sprites are replaced by 3D models (unfortunately very poor, as it does not seem worth it to create professional ones).
However, the engine has no problem with rendering more complex 3D models or richer environments. So we use the HyperRogue engine with a visually richer scene. You can play this scene yourself in RogueViz (see the link and info in description), although it has no actual gameplay.
@@ZenoRogue Thank you.
what is even this video? is it for teaching? I just found this and i'm high as fuck
length → span; weird → uncanny; less → lesser → fewer
MINECRAFT producers planning to make minigame inspired from MINECRAFT but like TEMPLE RUN named NOOB RUN NONEUCLIDIAN whick will uses the non-euclidian geometry like "horosphere" and "soft geometry" and "clifford Taurus", with graphic like in MINECRAFT where the Noob after was stolen a forbidden idol is chased allways by Huggy Wuggy in this non-euclidian space, then this Noob must avoid any obstacles for to not been caught by Huggy Wuggy.
🤍
It’s “what” our world would look like. Not “how it would look like.”
What if the 4D space is non-Euclidian? 😆
That’s not how it works
@@farrankhawaja9856 How comes?
Well, really there are different kinds of 4-dimensional spaces; "4D" just means that any position in that space is described by four numbers.
Usually when people say 4D they mean regular old euclidean space with an extra dimension, where most of our 3D geometric principles still apply (e.g. parallel lines never converge or diverge)
You could easily have a hyperbolic 4D space though, where for example you can have five hypervolumes meeting at each face (rather than only four), with the dihedral angle between adjacent hypervolumes being 90 degrees.
Non-Euclidean 4D spaces are very difficult to reason about though -- you pretty much have to give up on visualizing it and just use the relevant mathematics.
(though I wouldn't be too surprised if the ZenoRogue team have developed some ability to imagine that kind of geometry!)
@@trainjumper Well, I don't know... Since trying to imagine the 4D space is so hard with the strict '90 degrees between the axises' rule, then maybe those axises are folded down a bit and they are only getting unfolded from our perspective? Because the sunlight normally goes straight, but near large objects it curves in space... 😁
@@maxbrown1990 Yeah, space in our universe is technically non euclidean since it curves towards massive objects (general relativity).
You are definitely also free to imagine a 4D coordinate system with flattened axes; mathematically it's technically equivalent to one where all axes are at 90 degrees, as long as your axes are linearly independent (e.g. you shouldn't be able to create the vector -W-> by any combination of -X-> -Y-> and -Z->, since in that case the extra dimension wouldn't be adding any new information).
Sorry for the long response, basically dimensions are a flexible mathematical tool and can be whatever you need them to be
maxwell
maxwell
maxwell