Hey Zeno, have you tried Hyperbolica yet? Thought it might be neat to hear your experience of it given your pretty unique perspective 🙂

@ZenoRogue

2 жыл бұрын

No, but following it closely. HyperRogue started as a weird technical experiment and I did not expect it to be a good game, but it turned out that the combination of basic roguelike gameplay, hyperbolic geometry and open-world turned out to be unexpectedly deep and allowed the game to grow. Since the basic non-Euclidean engine was ready, we have experimented with using it for experiments with other genres/geometries -- but the basic roguelike gameplay is much more fun in top-down 2D, some of our experiments with other genres are fun but more on a "gimmick" level. According to a poll in the HyperRogue community, the players are most interested in extending its main gameplay (79%) rather than new experiments with genres (58%)/geometries (47%), and they don't want better 3D graphics (19%). AFAIK CodeParade created Hyperbolica to experiment with various genres in 3D hyperbolic space -- so similar to the experimental part of HyperRogue (I believe he did not know about such experiments in HyperRogue -- we advertise mostly the 2D roguelike because it is the best by far). It seems the reviews of Hyperbolica praise the polished graphics but they find the gameplay not satisfying. It is great that Hyperbolica got somewhat popular and people are discovering hyperbolic geometry, or want more and try HyperRogue :) Our experiments regarding the use of hyperbolic geometry in game design gives an impression that it shines the most with open-world designs (closed worlds tend to feel claustrophobic and closed manifolds tend to be confusing). Hyperbolica is closed world, although they did a great job to make it not feel that claustrophobic. Most roguelikes take place in dungeons, open areas are less interesting. HyperRogue manages to use hyperbolic geometry to make the roguelike gameplay fun in open world, but it seems to be restricted to 2D. There is a very successful game design based on a few of roguelike principles and open-world 3D -- it is called Minecraft. So I think a Minecraft-like in hyperbolic space could be great, and there are projects like that (Hypermine by Ralith et al, Hyperblock by Kayturs) although unfortunately they do not get as much interest as Hyperbolica and both seem to progress very slowly despite very promising start (to be fair, 3D hyperbolic open world is a huge technical challenge).

@zenzizenzic

Жыл бұрын

+

@PhillipAmthor

Жыл бұрын

@@ZenoRogue when do you make the first racing game in euclidian space?

@ZenoRogue

Жыл бұрын

@@PhillipAmthor HyperRogue has a non-Euclidean racing mode since January 2019... Euclidean racing works too :)

Explaining Nil visually like this is a lot easier to understand, & makes me want to see more. Thanks!

I think it was a very smart idea to have Ms Rogue narrate the videos, it makes these a lot more approchable than, say, text.

@tehorarogue786

2 жыл бұрын

Thank you so much for your kind words (:. I usually present our results in conferences -- so my voice was a quite natural choice -- but making videos is completely different. You do not see your audience (ok, the same was with online teaching but it was live!) and sometimes have to capture the same sentence a few times in a row due to random errors :D -- a perfectionist's hell!

@jakesanchez6621

9 ай бұрын

@@tehorarogue786 yeah your voice is perfect. I feel like a being from a higher dimension is explaining this to me.

@astrostar000

5 ай бұрын

@@tehorarogue786i love your accent

Thank you! Suddenly the surreal Nil world makes a kind of sense. Looking forward to Nil Rider!

Thank you, I've seen nil geometry on this channel but I've never quite understood exactly how it works, it's nice to finally get a simple explaination

@allenportilla

Жыл бұрын

11

@CallMeChili456

5 ай бұрын

simple???? god i shouldve paid attention in school@@allenportilla

The explanations of geodesics in non Euclidean geometries really goes a long way to helping understand the visualizations! Thanks for the great video!

I can't believe I just now discovered your channel, I've been looking for things like this for AGES! god bless you for posting this!!

@SimonClarkstone

2 жыл бұрын

Zeno also made a roguelike game that uses hyperbolic space: Hyperogue.

Fantastic video, thanks for finally shedding some light on Nil geometry!

Thanks for this video! I hadn't heard of Nil geometry before discovering your channel, and I feel like this was a good intro to the concept

Fantastic, great video! I wish I had been able to see this when I first found Thurston's conjecture during my undergrad. Love these videos, especially with a bit of voice over explanation!

It's gonna take a while for me to really get this. It seems like everything I really need to know, but I guess it'll take a bit more mental neural training before I really get used to the idea. We need more games and interactive simulations in nil space and other Thurston geometries

Oh MAN, my brain is not happy with this 😂 It's like "THAT shouldn't be possible!"

Thank you for this very well explained video! The demonstration was useful, and the topic is very interesting

Keep up the good work, really love your stuff!

Cool stuff, thank you!

This is lovely. I adore your explanation videos and encourage you to make more. But if this is all we get, i am still more than happy :)

Love your videos. Keep them up!

This is awesome, thank you! Congratulations on another video with voice explanations :). Question: it seems like in this geometry the Z direction is special, is it? Or it just a simplification for viewers and space has this symmetric property in all directions?

@spacelover4106

2 жыл бұрын

this is actually second zeno's video with voiceover and it isn't zeno's voice, nice to see you here too, btw!

@optozorax

2 жыл бұрын

@@spacelover4106 wow, which is the first?

@spacelover4106

2 жыл бұрын

@@optozorax it's the previous video called "Portals to Non-Euclidean geometries"

@optozorax

2 жыл бұрын

@@spacelover4106 I completely forgot, thank you.

@christopherking6129

2 жыл бұрын

Z is a special direction. However, there are no special locations, so in a Nil world you can have weird stuff everywhere, not just at the origin.

The visuals are incredible

This was such an interesting video!

great video!

THANK YOU SO MUCH THIS WAS EXTREMELY HELPFUL :D

Nil is related to S3. So IIUC we can construct the infinite staircase in mere S3 which is much more regular

@ZenoRogue

2 жыл бұрын

Indeed, the relation of Nil to the Euclidean plane is the same as the relation of S3 to S2. Here is a staircase model in S3 by Albert Chern: gallery.bridgesmathart.org/exhibitions/2019-icerm-illustrating-mathematics/achern (so AFAIK it is not based on a Euclidean square, but rather a spherical one)

In highschool I stumbled upon the concept of different kinds of geometries and asked my teacher about it. From that second on I became her favorite student. Lol she eventually became the administrator (our version of principal).

So if you're standing on such a encircled staircase and drop a ball down, it will move down all the cycle and strike your back with great speed?

6:00 POV - Your grandparents walking to school uphill both ways

Cool, thought the Penrose staircase still only works if you _define_ your gravity vector as g=. If you do it properly and define gravity as the gradient of the classic flat-gravity U=z potential, then the gravity vector is actually g = -∇U = i.e. there's a tangential component that pulls counterclockwise around the Z axis. And under that sort of gravity, each step "down" the penrose staircase would really be a short ramp sloping upward followed by a drop-off to the next ramp, something you can already build in normal euclidean geometry.

I've heard of Nil geometry, but I could never figure out what it was. Thanks for the video! Can you do Sol geometry?

This is great

This is so interesting! In these spaces you can actually extract an infinite amount of energy from gravity

My brain now have ptsd from this background music

I knrw nothing about noneuclydian geometry before this channel. now Just two videos in and I feel like I understand too much.

I wonder if this is the secret as to why the trucks in the game 'Big Rigs: Over The Road Racing' are able to infinitely accelerate in reverse gear? 😄😄

This reminds me of dreams I used to have as a small child.

Back in MY geometry, we literally had to walk uphill both ways to get to school!

I feel like a Non-Euclidian physics sandbox would be interesting to play around with

I’d like to see a similar video covering some of the other Thurston geometries, specifically Solv and the other one with a weird name; most of them (the X^3 and X^2xE) are pretty straightforward expansions of 2D spherical/hyperbolic geometries, but I don’t understand those two.

@ZenoRogue

Ай бұрын

Our video "Non-Euclidean third dimension" includes a similar explanation of Solv. The one with the weird name is also called twisted H2xR, while Nil is twisted E2xR -- just take Nil as explained here and replace the NESW plane with the hyperbolic plane. But there is more to say about it (like, why that weird name), so we will likely make a video about it at some point.

@KnakuanaRka

Ай бұрын

@@ZenoRogue Thanks; that helps some. I’ll check that one out and look for the other if it comes out.

How would gravity actually work in this space, I wonder? even in such a bizarrely chiral geometry, that unicycle's obvious violation of energy conservation seems suspicious. it probably still points to the center of the earth along the geodesic, but... i'm having trouble picturing it edit: there's probably a clue in the perpetuum mobile video

@ZenoRogue

2 жыл бұрын

Indeed, this kind of gravity probably does not make physical sense, since it is not based on a gravitational potential (and thus the conservation of energy no longer works). I do not think pointing to the center of the earth works either. There is something similar to conservation of energy though -- instead of the usual mv²/2 + mgh = const, we have mv²/2 + mgh + kA = const, where A is the area encircled (and k is a constant).

That geometry would be really good for Megarace or F-Zero like games.

Very good Video and very nice explanation! But is there any practical use to All of this? Just wondering

I wonder if a game of archery or paintball or something set in, uh, where the motion of things works like in Nil, but where maybe it is rendered in the Euclidean way you showed (or maybe rendered using the geodesics? I’m not sure) could be fun? I guess in the “paintball” one I’m imagining that there’s no gravity, but one can only thrust/swim/whatever in the “horizontal” directions, but where you can aim up/down as well, So that, in order to move vertically, you travel in a loop horizontally. Would it work to have the uh, “how much vertical motion corresponds to how much area?” change from location to location? So that it would be like, the integral of some function over the enclosed region? Of course, the space wouldn’t be the same everywhere in that case, but, still.

Most of the questions about Nil geometry seem to relate to how gravity and physics as we know them would operate, if at all. However, I believe there are a few other things to consider: - Assuming all the rules of Nil apply to subatomic particles equally as much as larger-scale objects, would light change frequencies as it travels? Would a red apple not appear red (or not even appear at all) depending upon which angle you look at it from? - On a similar note, would sound distort in different ways for different points of perception? Seeing as sound travels via vibrations of physical media (usually air), perhaps such vibrations could shift particles around on the X/Y plane, thus altering their Z coordinate and corrupting the original sound accordingly. - Sorry if this last one is a bit on the morbid side: if I stood parallel to the Z axis with my arms outstretched and spun around, would my body warp in accordance with Nil? It would seem that even mere travel in this geometry would disfigure an Euclidean being beyond what could be considered compatible with life.

@MichaelDarrow-tr1mn

Ай бұрын

if you draw it in the normal x,y,z coordinates it looks like distorted, but it's not. that's just a consequence of the way we're looking at the geometry from the outside. the light and sound does not change. every location behaves the same. the local distances are not the usual local distances. if you do the math with the correct local distances everything is fine.

This technology could create the greatest LSD themed video game ever

It’s interesting that as soon as someone changes where they placed the emphasis on a word, it becomes pretty hard to understand what is being said. The narrator of this video is a great example.

Can you make a video on solv geometry?

@ZenoRogue

9 ай бұрын

Our video "Non-Euclidean Third Dimension in Games" includes an explanation of Solv geometry similar to this one.

Thanks for this. Now I have a clearer understanding of both Nil and SL(2,R) now. One question I have is why do geodesic balls in Nil geometry flatten out the further away from the observer they are in one of the nontwisted direction? Secondly, a similar video for Solv would also be very cool. The 6 direction colored compass roses really helped me understand the twisty nature of Nil in this video.

@ZenoRogue

4 ай бұрын

I think explaining this flattening would require a thorough analysis... but for a somewhat similar explanation of Solv, see our "Non-Euclidean Third dimension in Games", although there are no compasses there.

This subject is deliciously creepy.

Magic power of ray tracing, now also in Nil

This rules I'm an award winning composer Can I score your next video? I love this stuff, fractal patterns are sweet too

The waiting room is a place people often go in a DMT trip, its staircaise like and nil like when you are in it

so weird. i love it

4 dimension and Hyperbolic space MIX!

Thank you and Tehora for this! Is the Z-axis unique in this property? Inversely, if I tilt myself so that X is vertical, does moving in a Y-Z loop then affect my X axis same as Z before?

@ZenoRogue

Жыл бұрын

Great that you like it! The Z-axis is indeed special (and unique). Nil is an anisotropic geometry (which means that directions do not work the same). It could be seen in the scene with compasses -- the "vertical" direction is easy to recognize even if the camera is tilted.

Would it be possible to create a similar video like this explaining the Solv geometry in similar manner?

@ZenoRogue

Жыл бұрын

Our new video "Non-Euclidean Third Dimension in Games" includes an explanation of Solv. (A short one, but it seems there is less to talk about Solv.)

6:50 kind of looks like a spinor, is it just the curl?

Joel g must hire you

what is the relationship between nil geometrics and non euclidean geometry with fractles?

a type of non-Euclidean that exists outside the curvature dimension (hyperbolic, euclidean, and spherical/Elliptic).

I like to draw abstracts with optical illusions. Some are very simple. but impossible. For instance the lines that create one form are also the absence of space for the rest of the object. In this way I can create images that suggests that I've folded the paper into a impossible shape. People always say I must do lots of math.

@microponics2695

10 ай бұрын

But the real secret is the pretzel.

Imagine a race game in nil geometry

@ZenoRogue

Жыл бұрын

Well, Nil Rider is a racing game...

It looks like the further away you move from the x and y axes, the more warped things get. Is this right?

@ZenoRogue

2 жыл бұрын

The geometry is actually the same at every point, so you can consider any point to be (0,0,0) and everything will work the same. But yes, a point which is far away in x and/or y coordinates will look "warped" in comparison to what it would look like if you were close.

What about the last of the 8 thurston geometries, the universal cover of SL2(R)?

@ZenoRogue

4 ай бұрын

Take Nil as described here, but replace the "NESW" plane with a hyperbolic plane, and you get this geometry.

I wonder what Euclidian axiom this geometry breaks. I'm guessing it's the rule that straight lines form the shortest distance between all points mainly.

@ZenoRogue

Жыл бұрын

This is not an axiom of Euclid, and is actually true in every geometry including NIl (depending on some definitions, at least). Which axioms are broken, it is a bit hard to say, because Euclid's postulates are not very formal, and also designed for 2D, not 3D... you could use e.g. Hilbert's axioms [Wikipedia] instead which do not have these drawbacks. Some of these are broken, e.g. incidence-2 and incidence-4.

1:45 looking like nuerons

and how about solv?

what are the practical applications of this geometry?

@ZenoRogue

Жыл бұрын

Not sure what you call "practical" but it appears in Thurston's geometrization conjecture (proven by Grigorij Perelman), which is used to classify 3D manifolds. Nil is also called the Heisenberg group, and is used in the description of quantum systems (but I do not know much about this). And of course creating art is an application too :)

@exsurgemechprints2671

Жыл бұрын

@@ZenoRogue like how can we use this in real life? does this make calculations easier (ie engineering, computer science)?

Does nil break conservative of energy?

@ZenoRogue

Жыл бұрын

Nil is just a geometry, so it does not break conservation of energy by itself, you would need to define physics in some way. The model of physics used in Nil Rider (a rather natural Newton-like interpretation, with gravity along the 'z' coordinate) does break conservation of energy. But there is still some form of conservation of energy -- if you return to the same location, your kinetic energy changes exactly by the area of the projection of the loop you took. But you could also have gravity along the 'x' or 'y' coordinate, and have conservation of energy as usual. Probably more reasonable physics, but it would be less interesting.

Something that strikes me as inconsistent or at least asymmetrical here. If moving in x/y entails a an extra corrective change in z, shouldn't the up/down transform contain an equivalent and opposite correction to the x and y coordinates. In other words shouldn't the change in coordinate position be independent of how an outside observer has chosen to label the axis? In even more words, if going around the staircase in a closed loop is equivalent to moving downwards, shouldn't any attempt to jump off the ledge and down staircase result in a forward translation causing you to land back on the staircase as if you had simply walked in the loop? Is it not the case that circulating around the stairs in x/y and falling directly along z are equivalent translations connecting the same pairs of points together on the stairs?

@ZenoRogue

11 ай бұрын

Geometries which work in the "consistent" way you have described here are called isotropic geometries. The only isotropic geometries are Euclidean, hyperbolic and spherical. Anisotropic geometries are less known but also interesting and important. Nil is an anisotropic geometry. It is still rotationally symmetric (one axis is special), while Solv geometry is not even rotationally symmetric.

@IIAOPSW

11 ай бұрын

@@ZenoRogue I suppose then I have something of a counterexample from my own game about movement through impossible spaces, albeit with the caveat that the position coordinate along the direction into the screen is only defined in so far as objects are ordered on screen as either in front or behind each other. In the case of two objects that do not touch from the cameras perspective, the third dimension is wholly ambiguous (and indeed it is possible to use vertical movements to jump between platforms that appear to have lateral separation and vice versa). You could try to model this geometry as if the 3rd dimension is integer valued, though inconsistent constructions like a penrose tribar are allowed so even violating assumptions of continuity doesn't quite shoehorn what I've made into the traditional formalism. This was many years ago and the implementation was in Flash.

Lil correction:Impossible Staircase was created by Oscar Reutersvard However it was popularized by the two Penroses. Credit should be given to the actual creator

Hey zeno, have you tried 4 dimensional geometries?

@ZenoRogue

Жыл бұрын

Higher-dimensional geometries are harder to visualize. "Higher-Dimensional Spaces using Hyperbolic Geometry" is about visualizing higher-dimensional (Euclidean discrete) spaces using hyperbolic geometry.

@TheAdhdGaming

Жыл бұрын

@@ZenoRogue ah oki, i was just asking because h4 didnt exist as anything physical yet, thank you for responding tho!

@Kaiveran

2 ай бұрын

​@@TheAdhdGamingApparently from 4 dimensions on up, infinite families of distinct geometries exist. So 3D is the highest dimensional space where we can have a finite, comprehensive list of all homogeneous geometries.

Shouldn't the light just follow a straight line? Or is a helix in Nil the same as a straight line in Euclidean?

@ZenoRogue

Жыл бұрын

Depends on what you call a "straight line". I would say straight line = geodesic (since geodesics are the shortest and also the straightest), and thus, yes, the light does follow straight lines in this animation.

@AstroEli133

Жыл бұрын

@@ZenoRogue What's a geodesic?

@AstroEli133

Жыл бұрын

@@ZenoRogue Also, when you try to walk forward, do you go in a helix? If so, then light would definitely not go in a helix, and you can't convince me it does.

@ZenoRogue

Жыл бұрын

@@AstroEli133 In differential geometry, "geodesic" is a curve that is locally shortest and straightest. In Nil, the helix curves constructed in this video are geodesics.

@AstroEli133

Жыл бұрын

@@ZenoRogue But why is it the straightest? If you jump off into a void with no gravity, will you go in a helix?

But why is it called "Nil"? How did it get that name, and for that matter, how did Sol geometry get its name and why is it also sometimes called "Solv"?

@ZenoRogue

Жыл бұрын

There are relations between geometry and group theory. (See geometric group theory and Lie groups.) So for example in Euclidean geometry when you perform move A then move B, the result is the same as when you perform B then A, which corresponds to commutative groups (aka abelian). Likewise hyperbolic geometry corresponds to a class of groups that has been called hyperbolic groups. We also have solvable groups (this name refers to Galois theory and the proof of unsolvability of quintic equations), and nilpotent groups. The geometries based on the properties of such groups are named Solv and Nil. Nil is also sometimes called the geometry of Heisenberg group.

This sounds like hell.

I notice it is not possible for the universe to have this geometry. I mean it's kinda obvious because we probably would have noticed by now we were in it given it's obvious consequences but this geometry could create/destoy energy since you could theoretically generate infinite energy by moving water "down" forever. Granted, this is assuming gravity makes sense in this kind of world, which I can imagine it being really unstable as going fast enough in a certain direction could theoretically thrust one upwards, but it's interesting to think about to say the least

@ZenoRogue

Жыл бұрын

I would say "not possible for the universe to have this geometry" is too strong, I would rather say this argument shows that "gravity as implied in this video is not consistent with the law of conservation of energy" -- but we can have Nil geometry without any gravity (and also while the conservation of energy seems quite solid, we can never be sure that the science we know is accurate). We could also have gravity acceleration in another direction (no free energy then), or gravity between massive objects. It does not really thrust one upwards, it is how straight lines work here, so we would just feel like going in a straight line. :) (assuming straight line = geodesics)

But wait, couldnt you abuse that to create a perpetuum mobile like eg with water flowing constantly down but in a circle?

@ZenoRogue

Жыл бұрын

You could, this is used in Escher's Waterfall and in Nil Rider. So gravity in the Nil down direction does not satisfy some physical assumptions. But it is still fun!

@gtctv7000

Жыл бұрын

@@ZenoRogue fascinating

but it could also spiral the other way, so it should seem twisted both ways, and blended between them. oh wait, that would be negative area

nil rubiks cube

Gives me a headache lol

so this is why psychedelic ingesters claim to reach higher dimensions

holy crap þe click i got when i realised what you meant by area- i had almost no idea until you mentioned about þe area 'being 25' and þen it hit my like a rock- þat is ridiculous edit; i mean it still breaks my brain, like i dont þink i will ever be able to comprehend 'what is happening' if yo uknow what i mean,but i get it now, ty!

Having light rays as 360° helices rather than straight lines just because (in Euclidean geometry) they always take the shortest path (which is a straight line) makes no sense. It's like they know exactly where and when they'll be deflected, and then pick a direction either randomly or deterministically and move in a helix to get there. You could put an obstacle in the middle of a helical ray path, and to account for the law, the ray would have to completely change it's path to one that does a 360° helix into the obstacle. You could keep adding layers of obstacles to force it to reflect from different points, but when you move the first obstacle, the original path will be unblocked and be able to catch the ray! Also, if you create a long tube too thin to fit a 360° helical ray, you will literally never see the end of it. That might just be a normal thing that happens (that was probably tested already) but I dunno, it just seems too wacko. Oh also, if you shine a ray onto a point and keep moving it forward toward the origin, the ray path would have to shrink, and if you put an obstacle blocking some of the helices there, it would literally block the point only at a given distance range. Is all this stuff that really happens in nil space?

0:25 I DIND'T SEE ANY PENTAGON.

non-euclidean geometry is literally math on drugs

This that roblox mining map

Huh?

Reality will always give you geometry or something better with a delay. The only thing that seperates real life from a dream is that in a dream when you go lucid and you intend to experience something, it just appears immediately. In Finding Nemo, Marlin loses his son. He is caught by a fisherman uh or a scientist and he's like taken away on a boat and the point of the movie is that we're trying to find Nemo. That's what Marlin wants to find. His son. Now the problem here is that Marlin doesn't know how to find his son but he just knows Beyond The Shred of a doubt that he's going to find his son because he's a dad and he lost his only son. So his whole life he knows for a fact, his heart and his mind have come into Union, they've both agreed that 100% we're going to find our son but I don't know how. So this is the same for you with your goal. You want something 100% for sure without a doubt but I don't know how I'm going to get it. That's okay normally you don't have to know how to get something. All you need to know is the first step. This is exactly how reality creation works. You hold your goal (P Sherman 42 Wallaby Way Sydney) in your mind and then you trust that the universe is going to take you there in its own mysterious way. Trust the universe that it's going to bring you to your goal but you won't know how. On the surface it's going to seem like everything is falling apart but once you look back you're going to say oh I actually needed to go through that failure because now I'm successful but I couldn't have been successful without that failure. You say oh this is going to bring me to my goal and you just kind of affirm to yourself that the Universe looks out for me and that the universe is on my team and is helping me manifest what I want. You want to be like Dory. You want to have radical trust in the universe and you want to hold in your mind your vision. That's why she keeps repeating P Sherman 42 Wallaby Way Sydney over and over again. Hold your vision in mind and trust that whatever happens you're gonna be brought there. That's how you create your reality using your imagination. Go after your goal in the same way that you go to get the post from the letterbox. Stop imagining the post to be a problem and simply put one foot in front of the other in the direction of the letterbox. Any goal you have if you can think and feel about it the same way as you do when you go to get your post from the mailbox then it's a done deal. (P Sherman 42 Wallaby Way Sydney) 📬📮

Well this didn't help me at all

Hey, if you died today, where would you go? Jesus Christ is the only Way to God the Father. All have sinned, and done evil in front of God who is a righteous judge, but Jesus died for your sins, taking your penalty for the evil you did, and it is written, if you will confess with your lips, Jesus is Lord and believe in your heart God raised Him from the dead, you shall be saved. Ask Jesus to hear His voice speaking to you, because God still speaks to His people, and Jesus's sheep hear His voice. 😊