Thanks for the shout-out! Here are some comments: * you say that "the shortest line on a sphere is not necessarily a straight line" but what is a straight line? It is a kind of meaningless concept until you define it. In my opinion a straight line is one that is (locally) shortest, making this "axiom" a definition. For a creature actually living in a non-Euclidean world, the shortest lines are indeed straight. If you are a creature living in a (two-dimensional) spherical geometry, the third dimension simply does not exist for you, and the great circles are perfectly straight lines, because they curve neither to the left nor to the right. Also, if you try a computer simulation of a spherical or hyperbolic three-dimensional space, the shortest lines will look straight (this is not the case in non-isotropic geometries though). * I definitely agree that all the games are just tricks. However, it does not matter! It is the effect which is important, not how it was achieved. The problem with games such as Antichamber or Superliminal is that they do not give a feeling of being in a non-Euclidean space at all. You do not see the visual or geometric effects typical for non-Euclidean geometry when playing these games. The effects you see have nothing to do with non-Euclidean geometry. * you sound as if non-Euclidean geometry was something accessible only to geniuses, and game development was easy. Most people are born with great math skills, which then deteriorate because of bad teaching. The math of non-Euclidean geometry is not really much more difficult than the Pythagorean theorem or trigonometry. The bigger problem is conceptual, not mathematical: people have their Euclidean intuitions so deeply ingrained that if you show them that they are wrong, they will not believe you and make the same Euclidean assumption again. * Also it is the best to just play a true non-Euclidean game and see for yourself. That is way better than watching videos or reading books. Everything can be experienced in HyperRogue.

The original non euclidean space is the infinite staircase in Mario 64

Before portal 2, valve experimented with a concept called "f-stop" it basically had the same rules as the game seen near the end. You had a "magic" camera that takes pictures, take a picture of an object and suddenly you can place a much larger or smaller version of that object by just using the portrait. It was an interesting concept that never saw the light of day but at least its idea exists in many games today.

Dude, if my geometry teacher explained it like this, I wouldn't have done summer school

Hyperbolica is an actual true Non-Euclidean game with curved space instead of a locally Euclidean game which occasionally breaks it's own space

Q: How can games be non-Euclidian? A: It’s software. It doesn’t have to model the real world.

TL,DR: Euclidean: "Makes sense to me" Non-Euclidean: "How tf does that work?"

Non-euclidian geometry: hi, whats up! My brain: *panic*

He’s back boys! So excited to watch

For a moment I thought CodeParade uploaded when I saw a non-euclidean themed video.

4:34 So my early 3d drawing program wasn't faulty, it was just simulating spherical space

It's funny how so many people imagine weird, eldritch stuff when hearing "non-euclidean"... Scared of a term they don't know, like with chemicals. Not realising they encounter non-euclidean geometry on a daily basis. Drew a face on a balloon? Had a tattoo? Congratulations, you made non-euclidian geometry. I guess we partly have to blame Lovecraft for that.

I love the way these "impossible" things are happening in a world that has taken decades to tune so that it didn't routinely do these kinds of reality-breaking things.

We were discussing the basic Euclidian Geometry in class, and I mentioned how some video games use their platform in creative ways to bend those Euclidian rules. I shared this video with the teacher, and made a 10 point extra credit assignment for the class if we could give a 150 word reaction of this video, discussing the stuff you went over.

Anti-Chamber was so fun. My favorite mechanic is finding out that you are expected to break the game. Set aside proper notions and see how often you have to do the exact opposite of what you think.

There's multiple ways of being non-Euclidean. Portal and Antichamber are mostly flat and Euclidean as long as you aren't close to a portal, but globally are not simply connected and so the axioms don't hold. But hyperbolic and spherical spaces are curved, and so the axioms don't hold. I wouldn't say one is more truly non-Euclidean. But the former are not even smooth manifolds, having sharp edges where space breaks down. If you were to stand in a Portal portal and move sideways, would you be sliced in half by the sharp edges of space?

Imagine being so legendary that even after 2500 years they use your theories to describe geometry.

And he came back when we least expected him

This is one trick where explaining the magic has only made it cooler. Simple, yet extremely effective.

This video coming into existence at this point in my life has made my week

Bro you're giving us hope like this, making amazing videos and whatnot

What a great video, just like the other of the bitwise series, I always love to see and understand how these games mechanics works with a great explanation and plenty of examples

YOOOOOO I suggested this a while back, glad to say it so beautifully explained!

"Hey vsaue, michale here. These games are non-euclidean. _or are they?_ "

"On spherical surfaces parallel lines converge." Latitude lines: Am I a joke to you?

Nobody: Christopher Nolan: lets write a film around this

ayyy, glad to know you're back dude!! hope you continue with your work in these videos, they are really great. Have fun making them, 'cause we're sure having fun watching 'em

First video I checked out from you. Loved it and I'm here to stay, will be looking out for what ever you put out ther

You mentioned Zeno Rogue! He's awesome, and his game Hyperrogue is a way to help wrap your head around what hyperbolic planes are like interactively.

AN UPLOAD! THIS MUST BE WHAT THE PROPHECY WAS SPEAKING OF

This was a very interesting topic to cover. You always deliver with your videos!

materiał jak zawsze przyjemny w odbiorze. czekam na kolejną część!

Their teleportation had to be on point, literally. They make sure that you teleport not only to the hallway but to the corresponding position in the destination hallway. I love these games!

“The fastest way to get from one point to another is a straight line with no curvature” Bhoppers: Well, yes but actually no

Great video! I definitely think the in engine examples you show put your vid a step above other explanations that I've seen of antichamber.

2:51 - Actually... a straight line is still the shortest in the curved space shown (sphere). The curve you see is extra-dimensional and is not an actual curve within the curved space.

This was a fun video, especially since I just finished writing a ray tracer yesterday, and I have a PhD in math and have worked a lot with non-Euclidean geometries. People should keep in mind that we're (basically) living on a spherical geometry: it's just a topological manifold as it's locally Euclidean (i.e. we get all the effects of Euclidean geometries on small scales: if you draw a triangle with chalk on the ground, it's going to look like the angles add up to 180 degrees even though in reality it's going to be 180 + epsilon for some very tiny value epsilon), and it's large enough that we can't usually detect that it's not Euclidean. When you fly, though, as you said, the shortest distance between two points is actually the great circle between them (take those two points and draw a circle passing through both of them that is just the equator transformed) and lines on a sphere are all great circles, where as you say, parallel lines intersect. We perceive ourselves to be living in 3D, but the surface of a sphere is a 2D geometry. It's also part of the reason why making accurate maps are so hard: you can't "unfold" the earth into a rectangle (basically a plane which would be a Euclidean geometry), since the surface of the Earth is non-Euclidean. As for the cube room, this could just be done with projective geometries. You're basically living in a projective geometry if you close one eye and it's what ray tracing is based on, as are, say, movies watched on a flat surface: you're watching a projection of a 3D world onto a 2D one... so those cubes you saw could just be like a "six sided LCD screen" with each side being a unique projective geometry onto a completely different scene. When I play a game like Boggle, I always feel cheated that it's played on a 4x4 grid, since the letters in the corners are only adjacent to three other letters, the letters on a side are adjacent to five, and the letters not on corners or sides are adjacent to nine... so I play Boggle on the surface of a torus instead (which is essentially a donut): you can "roll over" the top of the grid to the bottom (pretend they're glued together), and if you think of it that way, then you turn the surface of the board into a cylinder. If you then do the same thing to glue the left and right sides of the board, you take that cylinder of finite length, curve it around, and connect the ends, which gives you the "donut" shape of a torus. A real mindf*ck is to try to play Boggle on the surface of a finite projective geometry: take the top of the board, flip it around, and glue it to the bottom of the board: they you get a Möbius strip. Do the same thing with the left and right sides and you get a shape that you can't even really imagine. I worked on an app where you can choose the surface you want to play Boggle on, and when you click a letter, it shows what letters are considered adjacent to it... playing on a torus feels really natural, but playing on a finite projective geometry is very disorienting. Fun stuff and good video.

Yay, you're back! Thanks! :D

Such a great introduction of how physical geometry could work in game engine. Inspiring and fantastic, appreciate your work:)

Your content is truly amazing. I wish there was more of it.

I love non-euclidean puzzle games. Working out something that breaks everything you're meant to know is immensely satisfying.

This is so much fun to play with! I'm doing an infinitive rooms walk, the gimmick is that everything you did in previous rooms is saved. And the rooms loops back in an illogical manner, so you can't draw a map of it. With a 2x10^6 rooms I don't know how to make a game of this or how to make it interesting, but the technical aspect of it is fun!

Welcome back! High quality content as always

Omg! I just watched your portal video like 2-3 months ago and got so sad you haven’t posted in 3 years! I’m happy to see you back

so nice this video, has explaned so many of my questions.

I don't know why youtube has recomended this to me, but, man, thank you. This is amazing.

He's back! Always love to see content from this channel

Always a joy when you upload

actually, it was disappointing to know that non-euclidean games are actually euclidean lol

"you don't have to be incredibly smart or talented to create one of these things" lol what a backhanded compliment though I agree

great little demos you worked up there!

I love this content man, incredible work.

"a love to create and a passion to learn" I thought you were gonna transition into a skillshare sponsorship

I remember an old Duke Nukem 3D map, where there's two skyscrapers with pools on top. There's a vent in each pool that you can swim through to get from one pool to another. But there's no structure between the buildings connecting them. The buildings aren't even the same height, but the tunnel from one pool to the next is a straight line.

Love your content. Glad you are still posting.

funny you just mentioned my 3 favorite games at the start of the video xD, i love the immersion in these trippy worlds

Everytime i see antichamber i get reminded of when someone posted a “Portal 3 Gameplay” video and it was just antichamber

Great video! You teach people by using an unusual and most importantly an interesting example - games. As a 15 year old student, I am VERY interesting to watch this, thanks. I played Antichamber almost 5 year old, but still remember that masterpiece, i should play it again!

i love the videos on this channel and how you explain things!

I found this video very well explained and the examples were very helpful!

Really interesting games. Highly recommend Monument Valley as well. It uses isometric perspective with Escher-like tricks, where things that look connected are corrected.

0:46 a brazilian meme! here's my like

glad you're back Digi, great video!

this is actually very helpful, thanks for the great explanation!!

*watches whole vid* Digi: "Now youre a "bit wiser" me: "Ohhhhhhhhhhhhhh"

0:46 maybe the most famous Brazilian meme good taste

Really appreciate this vid. it's something I've wondered about for years.

I love these videos so much I'm so glad you're making more

I agree that we can't really fault '3D' videogames (or VR games) for trickery since it's literally all trickery in the first place ;p ''3D'' games aren't truly 3D, visually speaking, as we all know and VR is possibly even more trickery and illusion. Adding in more clever illusions to portray concepts such as non-euclidian geometry is brilliant. There's a few VR titles that mildly lend from these concepts as well. It's mostly portal/movement speed trickery in those, but still very clever and weird to experience. Whether it's ''Tea for God'' or ''Shattered Lights'' (both of which you will walk around your own playspace, despite traveling larger distances in game) or other roomspace/playspace trickery it's quite weird experience.

"You could say you are a bit wiser" 😅❤❤❤

Welcome back, dude Great vid, as always

Bro throughout all of highschool i struggled with finding angles in triangles cause i never understood and you literally just helped me figure it out in less than 3 seconds

Hey, around 1:35, perspective geometry claims that parallel lines intersect at infinity. On another note, there is another game similar to those showcased in which the player has a camera that takes photos, then can put the picture anywhere, and the picture becomes physical.

As a brazilian, it's fun to see a Nazaré Tadesco(the math woman) meme.

Trippyyyy!! I kinda did that effect in Unreal Tournament long time ago, when they had "portal" zoning mechanics. It did exactly that by a geometry plane displaying the other geometry plane in another part of the map. So you could literally walk from one part of the map to the other without noticing space bending.

Fantastic video! Thanks for the free knowledge!

Wouldn't be surprised if he just said "First, we need to talk about parallel universes." Then just started talking about SM64

I'm sure some are looking for this and were disappointed that it's not provided in the video. So: Use Möbius transformations of quaternions. (Look it up if it's new to you. I won't give full explanations here.) Quaternion rotations are represented as Möbius transformations of the form q 0 0 q A parallel transport (translation) of distance s in the direction of unit vector v is... in a flat Euclidean space: 1 sv 0 1 in a spherical space of curvature 1: cos(s) sin(s)v sin(s)v cos(s) in a hyperblic space of curvature -1: cosh(s) sinh(s)v -sinh(s)v cosh(s) This works with coordinates in a polar projection for spherical and a Poincaré disk for hyperbolic space. Geodesic surfaces are represented as surfaces of constant curvature (spheres and planes). If you want to use standard Z-buffer based 3d graphics, then for rendering you need to transform the hyperbolic coordinates to the Beltrami-Klein model and the spherical coordinates to a central projection. The central projection can only map half of the sphere, and that's ignoring limits of floating point precision. So you'd need think about how you can slice and dice your frustrum to render beyond that. In all of this we use the imaginary space of the quaternions as our three-dimensional world. But we're using four-dimensional quaternions internally. So this can easily be expanded to a four-dimensional world by allowing rotations into the real-valued axis of the form q 0 0 q* (where q* is the conjugate of q) and allowing v to have non-zero real values.

"Cameras & the Geometry of Vision" (2015) 📐 ukposts.info/have/v-deo/h6qlqX-Ilp-VsYU.html

in fact i tought this was a code parade video, until i saw the thumbnail, and i was happy to see your channel with a new video

"Light rays hate your eyes"

You forgot to mention Hyperrogue, one of the original noneuclidean geometry games.

hope you can continue making such amazing videos!

Very cool video! Love that you showed us how to actually do it! :)

You gotta play "manifold garden", this game is mesmerizing af

2:54 nitpicking, but that's not the shortest line!

dude you earned a sub from me. wow! what a compelling breakdown and stellar explanation

Thank you for making us a bit wiser!

I’m going to buy Superliminal on the Epic Games Store after having seen this video. Do you have an Epic Games Store creator tag?

I talk to some people about this kind of thing and they actually think games should simulate 100% accurate real world rules, but the game don't actually need to do that, cause you can totally fake it in the code and the player wont ever notice the difference if done right, and in the end you save some performance. I was discussing about orbital mechanics and FTL drives and the dude went nuts saying the FTL should totally calculate the velocity + orbit change + gravitational pull, etc, etc, i just said like: all it need to do is make the ship move forward in a straight line, lerping the movement, that's all. But he insisted it should be applied real formulas, i was like omg dude you over complicating this xD

This video is worth more than it sets itself out to be. Thank you!!!

Euclidean Space is considered to be any space in which Euclid's five postulates are true. Those five postulates are: 1, a straight line segment can be drawn between any two points (AKA The Space is Continuous) 2, any straight line segment can be expanded into an infinite straight line (AKA The Space is Infinite) 3, given any straight line segment, a circle can be drawn with the line segment as a radius, and one of the points as the center (AKA Rotation can happen) 4, all right angles are congruent (AKA Angles can be measured) 5th is the Parallel Postulate, which can be phrased a couple different ways. Basically, Parallel Lines have equal slopes. Generally speaking, "non-Euclidean" space is used to describe space that does not conform to the fifth postulate. This type of space can be separated into CONVEX and CONCAVE space. My favorite method of phrasing the Parallel Postulate illustrates them well, "For a given line AB, and a given point C that is not on AB, there exists a single line that passes through C, but does not intersect AB." Thus, a Euclidean space follows this rule, a Convex Non-Euclidean Space would have no lines that pass through C but don't intersect with AB, while a Concave Non-Euclidean Space would have more than one such line. You call the Convex space "Spherical" and the Concave space "Hyperbolic". Technically, Spherical is just one type of Convex space, while Hyperbolic is just one type of Concave space. There do exist other types of non-euclidean geometry, which violate one of the other four rules, but those are much harder to visualize for us. Things just get weird when the first four aren't true.

2:59 that point doesn't really work. We can't tell that thee shortest paths aren't straight lines just by looking at them, because for the space to be non Euclidian, this 2D space isn't allowed to be 3D, so any curvature into the third dimension doesn't matter. That's why using a sphere as an illustration is kinda bad. Edit. Ok I was too stupid to read the pinned comment. That already talked about that problem.

And then there's me who has never heard of antichamber

One of the first examples of non-euclidean geometry in a game that I recall seeing was a deathmatch map in Bungie's game Marathon in the mid-90's. I believe it was called "4D Space". Was a fun one to play. Possible or even likely there were examples prior, but that was my introduction to warping the 3D world in this fashion.

Im not a game developer but this is the most interesting video series on youtube :P I love the insight you provide!

Theres one game you didnt mention but it also does this, almost identically to antichamber. Its called "the Stanley parable" it doesnt focus on these instances, but rather just casually has them strewn about the world making it feel just that more surreal.

"Luckily the answer isn't every complex" He said as if it didn't take mathematicians hundreds of years to figure it out :D

Amazing video, I'm going to have to try these games out.

Throughout the 90's it was common for many RPGs to have an overworld, a sort of miniature map your character walked around that had towns and mountains and such at a size relative to the character that of a small building or shed, when entering them it teleported you to a larger cityscape map, and likewise when you entered a building it was often much bigger on the inside. Many overworlds would also have random encounters that would have you transition into a battle screen or battle map in some games. Overworlds like this became less and less common as more seamless open worlds became more common. but this came at the cost of travel times being longer and the worlds seemingly more empty, as you could travel faster with an overworld mechanic. Fast Travel mechanics were added to mitigate this, but most of them don't have random encounters and you will miss out on exploration in the process. Using forced perspective mechanics to literally make cities shrink as you leave them and mountains grow as you approach them, a battlefield stretch out as you encounter enemy armies or other random encounters and the like. It may be possible to merge the concept of an overworld with an open world for the best of both worlds. Make the distance between cities less than the distance from one end of a city to the other without making it look like that's the case. This could also be used for planets, space stations, and ships, either at sea or in space. If you are playing a game with fighters you don't want them to move too fast or dogfighting becomes more difficult than fun, but you also don't want to make the big ships too slow if you want them to be playable as well or else playing on them will become boring for travel. Travel Speeds that are super fast but prevent you from engaging in combat is the traditional solution to this. Whereas I like the idea of making the planets, space stations and big ships bigger or smaller as you approach them as, ironically, a more free form and sandbox solution. That way you can more easily intercept enemy fleets and the disconnect between combat and travel is mitigated. You might even allow fighters to "orbit" around capital ships so that they can treat the ship as being stationary once they get close enough. Allowing ships to still be pretty fast but fighters still have a massive maneuverability and speed advantage against capital ships.