Nil geometry explained! 

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

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.

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

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

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

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!!

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

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

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!

The visuals are incredible

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).

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

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.

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?

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?

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

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

Keep up the good work, really love your stuff!

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!

great video!

Cool stuff, thank you!

This was such an interesting video!

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

My brain now have ptsd from this background music

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.

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.

THANK YOU SO MUCH THIS WAS EXTREMELY HELPFUL :D

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

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

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

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

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.

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

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

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

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

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.

4 dimension and Hyperbolic space MIX!

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

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.

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.

This technology could create the greatest LSD themed video game ever

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

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

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?

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

Can you make a video on solv geometry?

This subject is deliciously creepy.

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

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

so weird. i love it

Joel g must hire you

Imagine a race game in nil geometry

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.

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

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

Magic power of ray tracing, now also in Nil

1:45 looking like nuerons

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"?

Does nil break conservative of energy?

and how about solv?

what are the practical applications of this geometry?

nil difficulty is truly confusing

Hey zeno, have you tried 4 dimensional geometries?

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.

This sounds like hell.

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

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

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

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?

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

nil rubiks cube

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.

Gives me a headache lol

so this is why psychedelic ingesters claim to reach higher dimensions

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) 📬📮

This that roblox mining map

non-euclidean geometry is literally math on drugs

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. 😊