Portals to Non-Euclidean Geometries
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- Опубликовано: 1 май 2024
- On this tour, portals will take us to various non-Euclidean geometries. This is not Minecraft!
A cool holonomy effect happened during this tour, but it was not explained by our guide Tehora! Have you found it? Please tell us in the comments!
Tehora Rogue's channel: / @tehorarogue786
Visited geometries:
* Three-dimensional Euclidean space 𝔼³ (cyan floors, music: Icy Land by Shawn Parrotee)
* Product space ℍ²×ℝ (green floors, music: Living Caves by Shawn Parrotee)
* Three-dimensional hyperbolic space ℍ³ (yellow floors, music: R'Lyeh by Shawn Parrotte)
* Product space 𝕊²×ℝ (blue floors, music: Ocean by Will Savino)
* Three-dimensional spherical space 𝕊³ (purple floors, music: Land of Eternal Motion by Shawn Parrotte)
* Solv (brownish floors, music: Lost Mountain by Lincoln Domina)
HyperRogue soundtrack under the Creative Commons BY-SA 3.0 license
An interactive version of this scene should be added to RogueViz soon. We need to make it work in VR first!
EDIT: available in RogueViz 12.0L zenorogue.itch.io/rogueviz (RogueViz demos ⟶ portal collection ⟶ [press Enter multiple times or 9] ⟶ inter-geometric portals ⟶ run)
Play HyperRogue to have more fun with non-Euclidean geometry! To learn more about non-Euclidean geometry, play HyperRogue or visit our discord: / discord 
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I feel like I'm being taught a preschool class by a 5th dimensional entity
You summed it up perfectly. This video has such a unique vibe
That is exactly what’s happening.
Maybe we are. Maybe life is one big preschool class on learning space-time before we ascend into the 5th dimension
@@bxnny0374 unique vibe... or trip back to the mid 90's... because this feels very mid 90's to me :P
Man what is it with condescending women and trippy geometry
This video is the definition of 'Never let them know your next move'. She could say "In the next geometry, all squares are your mom" and I'd believe her
we need that
Your mom's so geometric, she uh... idfk. Becomes a non-Euclidean reality?
That's not how that saying works but ok.
@@itsgonnabeanaurfromme well that's not how his mom works obviously either but like.
Lmfao! That sums it up perfectly! I can, kind of, get a sense of the math behind what she's saying, but as far as actually comprehending it in any meaningful way, I might as well be trying to figure out Proust in Mandarin translated through a Swahili to Swedish interpreter program.
This is like the tutorial level I should not skip before getting into multiverse
😂
“For your convenience we have equip you with jet packs” 😂😂😂💀
This simulation would make a terrifying horror game
Your future
It is indeed a big thing in the lovecraft universe
All you need is a backrooms monster chasing you while you try to find the exit 💀
I want dungeons with geometries like these in a zelda game
I'd like to see any plane of Oblivion done like that. And maybe daedric shrines are built with "all those weird angles and stuff" in an attempt to depict Oblivion's geometry on simple euclidean Nirn.
It felt so good to be in Euclidian geometry again after those crazy pesky pentagons
Pentagons work in euclidean geometry. This is not non-euclidean.
Hahaha
Home sweet 180 degree home
Really? I felt the opposite. While Euclidean geometry is preferred by most, I find it more enjoyable to be in non-Euclidean space.
I wish I was trippin rn 😭😭
The best part of this video is every zone having its own music like mario
I'm actually amazed at the fact that I can more or less understand what is happening. Demos like these not only move science forward, but move societies forward.
Can I just say how much I appreciate you saying "okay enough, you might need some rest in the euclidian world" at the exact right moment? That's very considerate.
1k likes and no replies? let me contribute absolutely nothing to society by replying👍
@@Oscribusyou did contribute my friend
It made me feel ashamed of being a human
its amazing how the nonchalant music turns a bad trip into a nice afternoon stroll
this edible aint sh
2:40 that pattern in the back is literally lsd
Bruh
I think this is the music from a game with a non euclidean world, I forget the name, something rogue, pretty cool game
@infanatosuks4880 um, you might be mistaken. All the music is in the description...
I can finally understand why Lovecraft was so unnerved by architecture using non-Euclidean geometry now. Phenomenal stuff!
Most people tend to just think of extra dimensions being added when speaking of non-euclidean geometry, I'm glad to see someone take us through some 3D geometries that are non-euclidean. It's strange and fascinating how much you can change the rules of nature and still have a functioning system.
Nothing is really being changed though, just how space is perceived. For example, to a photon travelling around a planet, it's just going in a straight line, it's space that is bending. Warped space is still technically flat, and the curve of space only really matters in determining the position/distance of something in relation to something else. It doesn't actually change anything else about how physics itself operates. Without a 4 dimensional material, a 3 dimensional object in 4 dimensional space, would still be just 3 dimensional. And that's the rub. We always hear about 4 dimensional shapes, but never about 4 dimensional matter itself, because that's where the theory essentially ends, because they haven't a clue of what 4 dimensional matter would even look like, especially when again, 4 dimensional is generally related to perspective and shapes, with zero consideration for the atomic level.
And since 3 dimensional matter would still be only 3 dimensional in 4 dimensional space, it most certainly couldn't be matter at all as we know it, the entire makeup of atoms would have to change in an unfathomable way. But let's say a 3 dimensional object was somehow altered by 4 dimensional space, and like a klein bottle, shapes of matter actually looped in on themselves. Well that again wouldn't change much, because since it's 4 dimensional space, nothing is actually intersecting, so at least you get rid of the impossible problem of having 2 atoms occupying the same space. But that still means that nothing really changes about how the matter interacts. It would only really allow for some odd behaviors like if you had a long rectangular magnet with repulsive sides on each end, anything you put on either end would behave as if 2 ends are pushing from both sides, even though from a 3 dimensional perspective it would look like an extra magical force is coming from nowhere. So while bending space in such a way could be useful (and likely impossible), it would mostly be useful for technology.
Another problem with 4 dimensional space, which likely doesn't exist at all and is more of a thought experiment, is that even if it did exist, nothing would really be able to form in it because of how drastically space is altering how things interact, it would be too chaotic. For example, imagine if the solar system was suddenly in 4 dimensional space. You might have the earth be on one side of its orbit, but getting gravity pull like it's closer to planets on the other side of the orbit. What this ultimately means is a stable orbit wouldn't be possible. Now imagine these space changes down to the atomic level. You could have electrons essentially leaving the orbit of their atom. Oops, no more atom. Self annihilation. Which means ironically, 4 dimensional space would probably be just a soup of particles that can't really coalesce into anything, with atoms essentially blinking in and out of existence as electrons come and leave from their electrons.
This would also mean interactions between atoms at all would be intermittent and then broken again. It could also result in other behaviors, like particles becoming anti particles at seemingly random, and that's the ultimate proof that a 4 dimensional universe would actually be empty, because the whole thing would just self annihilate over time, leaving behind just energy and a soup of elementary particles.
I'd love to see a feature where you can throw a ball. The inertia and speed could always be consistent so we could check how the trajectory and visual matches our expectations.
I HAD THE SAME THOUGHT! Just with a stone but, It would be so interesting to see and then also make sense of it all
I like this idea.
I don't know if you can do it. You're talking about a ball under free fall, then you'd have to modelate a gravity field. Since actual game engines manage the gravity regarding to an euclidian geometry you'd probably have to program it yourself. Then rises the question up where the hell would you put the gravity center or gravity line and set another reference in the frame
@@Edo_Aelio putting it that way got me thinkin now fuck
@@x_angel87 Yeah. I am not sure how someone would program this to work. My best guess have the player be attacted to the nearest surface. I am thinking like super Mario galaxy.
For anyone who didn't know, the purple pillar in the solv geometry (6:43) is the ceiling.
fascinating
what is solv geometry?
@@unneccry2222
Confusing visually, but I've read a comment in other of ZenoRouge's videos (the Nil geometry game trailer I think) where the explanation is actually fairly straight forward.
*huh??*
Is two toruses.
Take the last geometry world in this video, add zooming in and out fractals on every surface, speed it up, and you get what I experienced in the peak of DMT.
I would love to watch a VR version of this where we could look around ourselves cause some of those shapes was something else
This does work in VR (see the description).
My brain exploded when she showed us that all the pentagons had right angles
dead
same. o-o also my eyes hurt.
Makes sense its the most replayed.
Mine healed. I realized its just reality stretched out another 20% and it all makes sense. In my mind, you'd just add one more cardinal direction and to turn around you just turn 10% further
she?
I once read a Warhammer 40k book were they talked about seeing shapes with angles that didn't add up, and I could never really visualize what that meant.
I get it now.
Webway can be confusing. Warp even more so. :)
Now try to imagine the non Euclidian geometries and architectural buildings in the Drukari world called Commorrage lol
@@pws354_8 DO NOT VISIT COMMORAGH WORST MISTAKE OF MY LIFE
@@Polmax2312 Was neither. Weird 5-legged aliens instead. (One of the eisenhorn books IIRC)
Whoa. That's the wildest thing I've seen all month. Thanks!
This is amazing. The presentation was very well done. I liked how you explained/toured the geometries, and the fact that each “area” for lack of better term had its own music was a nice touch
one of the most high quality videos ever, it makes everything really clear and easy to understand, even the confusing parts made sense when she finished speaking, 11/10
The idea of getting lost/stuck in one of these rooms irl is terrifying lol
I thought of it like half way in, I started getting a little sick at the thought of going through multiple portals and not remembering the way back
@@MrDannyloco If there was a way to survive and interact with other humans I think you'd be suprised by how swiftly your brain would learn to adapt to these viewpoints in order to increase your chances e of survival
This is pretty much how it would look like if you use psychedelic substances, so yeah you pretty much stuck in these "room" when you use those substances
That is very easy to do in the hyperbolic ones
That's what psychedelics do
This video felt like it was 3 hours long and 15 seconds long at the same time.
Yes
"Time to go back to your Euclidean world!"
"NOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO"
You murdered me, I'm dead. Thanks for this unique experience!
There needs to be a VR laser tag game made with this. That'd be trippy!
It wonder how fast it will be almost unpopulated, because it made people throw up.
My brain would literally self destruct
everyone would be sick within 5 seconds
It’s called dmt lol You see a lot of this with closed eyes visuals but it’s brighter and moving faster
Well, a VR visit of such place would be a start. Time to search that side of VRchat again...
This is like a guided meditation except instead of trying to clear your mind the goal is to get a migraine
Wow, these Daggerfall mods are getting more ambitious by the day
In all seriousness, awesome program and awesome video, it’s mind bending to see these portals to worlds unlike our own, but also helpful to visualize otherwise pretty abstract 3D non-euclidean geometries :)
I miss Bill Thurston. Thanks for showing us what he may have seen.
This got me thinking about H P Lovecraft and his descriptions of strange geometry's but he never had the chance to see anything like this.
Adding monsters also obeying weird geometry would be terrifying.
Bro, that's such a very good horror game concept.
@@cellphoneguy5698 Lost In R'lyeh would be a great name. Running from Cthulhu and Deep Ones!
This is the sole reason I clicked on this video.
play hyperrogue. it's what you just said, and made by the person who made this video. it even has rylyeh and cthulhu! it's a roguelike and not a horror though
Wouldn't a normal person look like a weird monster with these geometries?
Cthulhu isn't a squid faced dragon, He's just a guy called Steve who was born with the wrong geometry
The holonomy effect mentioned in the description is on 1:43, the camera just moves up, down, left, and right but after returning to the first angle the surface that was a floor becomes a wall
cool
Interesting, when I first watched it I thought a corner had disappeared when they zoomed in. It wasn’t till I went back cause of this comment and counted that I realized it didn’t.
@@pogolaugh wow
@@pogolaugh interesting lol
I knew what happened, but I didn't know the effect was called holonomy.
i honestly just absolutely love how solv works, its so.... hard to conceptualise, its easy to understand but actually processing any of it is impossible for me, it might be my favourite geometry i've ever seen
Interesting how each new world made me initially seasick until I my mind could get its bearings on how motion effected what you see, and in what ways it was and wasn't consistent.
Rare I can feel the process of brain learning haha very cool!
You can just tell that the code behind this is extremely elegant.
Suddenly, Lovecraft's description of cities like R'lyeh feel much more treacherous to read about as characters try to move through them.
😰
Right? Imagine walking along a warping path, only to realize just in time that in two more steps you're going off a 90 degree angle cliff, but gravity is pulling on your mass in a way that makes it seem like you'll be fine if you try. The only way forward is going back the way you came, then up becomes down, so don't go up those twisted stairs; you'll fall. Jump in the hole instead to get to the top floor.
@@kilderok Going back the way you came might not be an option to you anymore as you may have shifted your angle ever so slightly midway through your realization and that was enough to put you off your trail by 90 miles.
one of the pieces of background music is called R'lyeh
@@shartbimpson 😲
@@Dazreiello-old Imagine getting ambushed by Cthulhu in his home city.
If I ever get lost in spherical geometry or saddle geometry world, I'm never coming back
FINALLY I managed to stumble across a useful demonatration of this! omg ty so much, this was lovely.
I'm a programmer myself, but I can not imagine how hard it must be coding this...
What does the data structure look like to store non-euclidean geometry? 💀
Graphics engines are basically math function plotters, give it the right function and parameters, and you get euclidian geometry (ie. Draw things that are far away small) change the functions and parameters and you get all of this craziness. What was puzzling me is how they designed the "level" within the rooms. They either did it by code (coordenates) or they have also wrote an editor for varying geometries.
@@elkikex Yes, that's exactly what I mean... I am programming games, so I know a lot about game engines... But those "levels", or rather their geometry, needs to be stored somehow to be "plottet"... But there literally is no datastructure I could think of to store non-euclidean coordinates, right?
@@triplezgames3882 codeparade talks about some of these problems for his non-euclidian game
@@uhrguhrguhrg Yeah, right after my comments I watched all of his devlogs on Hyperbolica. From what I understood, the magic behind the transformation is done using gyrovectors. Also he had to use square tiles
Some Indie developer needs to pick this up to make an absolutely mind breaking puzzle game with non-euclidean geometry.
The RogueViz engine this video is made with was originally made for HyperRogue, and is also used in a few other smaller games (Nil Rider, Bringris, Relative Hell). Do you need anything more? :) Of course the more developers do absolutely mind breaking puzzle games with non-euclidean geometry., the better!
Try the game Hyperbolica. It's not exactly a puzzle game but it has worlds with non Euclidean geometry.
@@halcyonacoustic7366CODE PARADE!!!!
@@ZenoRogue is one of those games a VR game?
@@doppelhelixes They all support VR, but not necessarily are designed for it. HyperRogue is designed to have great gameplay when played top-down, but it you want to see what the character would actually see, it has a VR mode too. Nil Rider should be cool in VR. Bringris works in VR too. Relative Hell has no VR. Hyperbolica is probably the best if you want specifically VR.
Very cool! This could be useful for building up solid intuition for these geometries! Great work!
the amount of intelligence and work that must've went into making and polishing this must have been off the fuckign charts
I love how every geometry type has its own theme. Very creative, loved it!
Pretty sure the music comes from the game hyperrouge, which is also based in hyperbolic geometry. Each biome has its own music.
@@ragingfirefrog I can confirm that at least some of the music (maybe all, I am not super great with memory) comes from HyperRogue, as I have played it a lot. It is one of my favorite games!
@@spookyblush-speedruns yeah it all does. Zeno has a habit of reusing music in his videos. I still like how he used different songs for different geometries in this
You acted exactly as a spirit guide through a shamanic vision. I am very impressed with the presentation and my awareness is expanded.
Oddly enough this video feels like what a spiritual encounter feels like
Definitely reminds me of my most profound mushroom and LSD experiences.
Finally, an accurate visual of my dreams
Wow, incredible! Thank you, have a great rest of your day too!
You've done great work visualizing the non-Euclidean geometries.
If I might make a suggestion, it might be helpful to see Euclidean space morph into the other geometries, especially the really odd looking ones like Nil and Solv. Maybe start with fog obscuring the long lines of sight so that it still looks Euclidean because it's all local and then gradually lengthen the fog visibility length, thus giving a sense of how long the various lines-of-sight are. Or start with most of the map in darkness except near the viewpoint ("near" according to some underlying metric) and gradually widen the lighted area, giving a sense of when we're seeing the same thing multiple times along different geodesics because the multiple copies all get lit up at the same time.
But it's just an idea. Good work!
There's an INFINITE number of geometries, formed by a combination of different basic geometries. Euclidean at x-y plane combined with hyperbolic at y-z plane is just one example, seen in this video. These can also go to higher dimensions.
@@r.a.6459 How does that relate to the comment your're replying to... ?
If I night make a suggestion, turn this into a Call of Duty map and teach an entire generation of kids to understand non Euclidean Geometry.
@@emersonsnyder369 I don't think Call of Duty's engine renders non-euclidean geometry
@@r.a.6459 IRRELEVANT SHUT UP
Such a cool video! I remember reading stories by H.P. Lovecraft in which characters found themselves in alien cities where there was something wrong about the geometric layout of the buildings and such. It struck me as a fascinating idea, and yet, I couldn't picture what such a place might look like. This video has provided me with some possibilities. Thank you!
Same! I just read The Call of Cthulu and I was wondering what exactly would "non-euclidean geometry" look like, it's a great horror trope to use the incomprehensible and now I have a better understanding!
Thanks for making HyperRogue, it's really good
I love this video.
I come back to it ever so often.
Helps me expand my thinking.
visual non-euclidian geometry feels like a nausiating fever dream, thank you
Now that it's well-known how to do these things, I can't wait for proper black holes and wormholes in space games
Regarding wormholes, there is a cool video "Project Manifold: Non-Euclidean Universe with Wormholes", but I do not know whether they are doing anything more with their engine.
LOVE THIS! …. It’s like giving a navigation compass, an axis system for weird experiences like dreams!
I had no idea what I was looking at 90% of the time, but I enjoyed the journey.
This sums up this terrifying dream I keep having of getting lost in some inter dimensional gate system
R'yleh?
@@dubuyajay9964 Uh-oh.
VR tours through some really crazy spaces like this is such an insane concept to me.
This was actually quite engaging, I might show this to my friend's kids.
Can't wait to take a toke and watch this in full. Very cool.
I love how other-worldly it feels.
This is kind of how DMT feels but faster and way more complex, with shifting geometry like the buildings in Dr Strange.
Tbh any faster and I would throw up.
@@travislowrider6623 yeah… I almost threw up….
Note to self: Never do DMT
@@cubedtothex It's only for like 15 minutes, you could handle it
@@Nae_Ayy lmfao “don’t worry you’ll only feel like you’re dying and directly talking to your subconscious mind for 15 minutes.”
Thanks for the excellent visualizations of various alternative geometries. Gives me ways to imagine those alternative geometries while reading Lovecraftian Cthulhu stories.
this is beautiful o.o Thank you
I hope that something like this exists in hyperrogue at some point. Maybe it won't be sound gameplay-wise, but it would still be very cool to explore/play around with.
Ah, apparently it's going to be on rogueviz. Good enough for me!
Check out the game Hyperbolica, coming out in March (on pi day).
@@atimholt it's out now
I didn't think Hyperbolica had much of a game to it, it was just walking. I think we need a raytrace shooter in hyperbolic space.
@@chaotickreg7024 I agree. It's a walking simulator in hyperbolic geometry.
The music in the full hyperbolic world is intense. I need more of it.
It is the music in the R'lyeh land in Hyperrogue
This is so dope! i fell in love with hyperbolic spaces! Infinite space with itself is just mind blowing.
Amazing! This is the way I would try to imagine different dimensions! Terrifying!
Mesmerizing, I would love to see more of this stuff implemented in films and shows, these would fit perfectly with the likes of the 2001 Star Gate/Interstellar Black hole/wormhole imagery, or Doctor Strange's visions.
or even a fantasy game. this would be a great fantasy world
@@second_second_ Oh absolutely! Bringing in bizarre geometry or physics concepts into a fantasy world would be phenomenal I think, something to rationalize the bizarre in those realms.
Games. Vr games
@@viktorvondoom9119 Maybe, but I hear people get motion-sick as-is in VR, I don't know what something like this would do to them
@@jmaster2855 It would very much be like 2001. Transpose human-form
Oh man... I would love a technical explanation as to how you programmed all of this!
Yeah. If I had to implement a renderer for that, I'd probably start directly with raytracing / raycasting as it is a (relatively) straightforward way to deal with curved space but it would be interesting to know how they actually did it.
Its easy as fuck
@@fackynaxicht8603 thank you very helpful
@@Lessen0 geometry is not your grandma's teaparty. You either got it or not. Its not about "being helpful", its about knowledge. And why would i waste it huh?
@@fackynaxicht8603 crab in a bucket dickhead mentality.
Thank you for making this video.
Thanks for the tour BMO.
Thoughts:
The calm narrative made me think of Powers Of Ten. I can imagine a similar documentary where we travel to progressively stranger geometries. Is there a canonically meaningful ordering of them? Altering one constraint at a time?
Is it meaningful to transition slowly from one geometry to another? Can there be intermediate states as with fractal geometry? Is there a mapping that can enforce a different spacial geometry onto a fixed terrain?
I would have liked to see the camera pitch and roll.
And clearly someone needs to implement an FPS in other geometries. I like the idea of chasing someone from one geometry to another. Shooting would be confusing enough, but aiming at a target through a portal would be... interesting.
And this obviously leads to a sequel called GeoPortal where your portal gun has selection buttons for what geometry you will be in after passing through. Lots of puzzle solving opportunities. What happens to conservation of momentum when transitioning? The gravity vector?
Now you’re thinking with portals
Now you’re thinking with portals
Literally like DMT visuals.
It makes me happy to see so many Lovecraft references in the comments. Cool video!
thanks for the tour bmo
This mimics many of my trips on salvia divinorum. I'm kind of floored. Who knew the subconscious was non-euclidian?
I genuinely cannot tell what tries to communicate, but I can for certain say this is incredible
Not gonna lie. I wound up here a few weeks ago because I realized that I only had a faint idea of non-Euclidean space. I was good at Geometry and Trigonometry in HS, but it made my head hurt the whole time I was getting good at it. My head hurts again, but in a good way. Watching this feels the same way that looking at pictures of stars and galaxies when I was a kid felt like. It annoys me that there is such a huge hole in my understanding that I was unaware of. I'm glad to finally take the time to find out, and this is a joy to encounter.
I love this! I’m subbing
I can go on forever in these without a break, it doesn't hurt to look at, it's fascinating, so interesting how geometry works in non-euclidean spaces, I can go all day
You must be a 5th dimensional being if this doesnt give you vertigo
@@leokm9586 a
@@leokm9586 yeah it doesnt bother me either, ive always considered myself a bit eldrich
Same, in fact it's a kink to me
@@bottlebeard Wait wut-
Mesmerizing. I've sometimes felt a bit lost in your previous videos even if I enjoyed the geometries and visuals, but I was fascinated by this 8-minutes journey, always eager to see what geometry the next portal would lead to ! The commentary is a really nice addition
Stoned and loving this ❤
I'm guessing that very last world, the 'sky' of that world was the purple panels: the black in the distance was just the space between the sky and ground.
It was oddly relaxing, though the Sphere world was anxiety-inducing a bit.
I'd LOVE someone to make a Lovecraftian game using these geometry perspective shifts. Lol
Thanks for bringing us back to euclidian geometry in the end
Great stuff! Is it possible to explain Sol to non-mathematicians? I'd love some kind of lecture or demonstration.
For Solv, the three dimensions of space don't work the same. As you travel up the vertical dimension, one horizontal dimension shrinks exponentially and the other grows exponentially.
This means that the shortest path between two places at the same height can involve going up or down to get a shortcut, which is why light follows such paths and makes the floor appear to curve up/down in different directions.
@@SimonClarkstone Thank you! Good grief, that is absolutely bonkers.
@@kateorman The practical effect: if you build an elevated bridge in Solv in one direction, say, north-south, going along the bridge will significantly decrease the distance you need to go -- the higher the bridge the shorter the trip. But if you'd build a bridge in the perpendicular direction, east-west, the distance will *increase*. For east-west travels, you need a tunnel burrowed under the surface, not a bridge.
Here's the explanation by Zeno himself:
This geometry has interesting features not exhibited by any 2D geometry, and is much weirder than all the geometries we have seen so far. It is also quite impressive visually. (It is kind of sad that the more interesting geometries have such boring names.)
Imagine a plane, tessellate it with squares, and put a 1x1 cubes on each square. Then, put a 2x2 cube on each four cubes on the first level, a 4x4 cube on each four cubes on the second level, and so on. Also do the same in the direction below the plane (0.5x0.5 cubes, and so on). Consider this a map in R3 of a manifold, where the size of the cube corresponds to the metric (our model distorts the distances, all the cubes are actually the same size in the actual manifold) -- so, for example, we can get from the cube (0,0,0) to (1024,0,0) in just 21 steps (move 10 cubes upwards, one cube to the right (which corresponds to 1024 steps on the 0 level), and 10 cubes downwards). We already know this manifold -- this is the hyperbolic geometry H3, viewed in the Poincaré half-space model, with its "{4,4} on horospheres" honeycomb, already described.
To obtain the Solv geometry, we also start with 1x1 cubes arranged in a plane, but on top of these 1x1 cubes, we put (1/2)x2 cuboids instead. On top of them, we put (1/4)x4 cuboids, and so on. For example, We can reach the cube at (1024,1024,0) from (0,0,0) in just 42 steps -- first, we go 10 levels upwards (in the Z direction), make one step in the Y direction, and 10 levels downwards. We are now in (0,1024,0). Now, go 10 levels downwards (as previously, we stack 2x(1/2) cuboids on the level below the original one, and so on), make one step in the X direction, and 10 levels upwards.
Basically a three dimensional binary tiling, with different axes of compression depending if you're going up or down.
en.wikipedia.org/wiki/Binary_tiling
I have not seen a more interesting video on RUclips!
Thank you
I've never really understood non-euclidean geometries, but this video helped with all the different perspectives.
I absolutely love this. So playful and so informative. Very well done
Love to see this is blowing up on the algorithm! I got sucked into the mathematics of non-euclidean space traversal and rendering a couple years ago and came across your video "Ascending and Descending in Nil". When seeing this video just now I was reminded of that one, and I was delighted to see it is the same channel! Love your dedication to the subject of non euclidean space, it's an incredibly cool subject of mathematics.
tldr; Love the channel, earned a sub!
awesome, thanks for sharing :)
Watching this video made me realize I'm really not the target audience for anything Lovecraftian; this is more fascinating to me than scary.
Thank you for making this! That's what the Internet and 3d rendering really is good for.
Also, the colors and textures are nice.
The most astounishing thing is, that everything in these portals still makes sense
This music is on point. Well done, very entertaining.
It feels so pleasent when you get back to euclidian geometry, it feels like home!
That is a lovely voice!
And easy to hear what you're saying as well, very good English!
Absolutely incredible! 😁
Thank goodness
This is wild. Imagine something like Control 2, which already delved into non-euclidian spaces, adding this kind of geometry into The Oldest House. So cool.
Many games are referred to as non-Euclidean but they are usually based on portals or perspective tricks; I have not tried Control 2 but I believe it is one of these. Actual non-Euclidean geometry is rare (in 3D: HyperRogue, Hyperbolica, some games in development), even games pretending to take place of spheres often do it wrong. Hopefully existing games and videos will inspire game developers to create more :)
This is fantastic, I loved the commentary! The explanations were detailed, but terse enough to be easily digestible.
Maybe I'm just slow, but it took me awhile to realize what she meant by "hybrid" geometries.
"these edibles aint sh-"
3 seconds later
Amazingly incredible & incredibly amazing!
I like it when my sleep paralisis demon takes me to weird geometry land to show me some dubious mathematical blasphemies