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.
Thanks for checking out my video! I keep underestimating what a reach my videos can have, I think it's awesome that you found my video. Sorry for not including HyperRogue as an example, not sure how I forgot to include it in the video. I at least added a reference card and added a link to the description. Thank you for the insightful comments, I'll pin it in the hopes more people will read it :)
@@MatthijsvanDuin well, non Euclidean means some of the axioms don't work. Normally the axioms that don't work out lead to a non-zero curvature geometry. But any geometry that doesn't follow the axioms is non Euclidean. Even if all the axioms do work locally.
well considering it uses that teleport trick (this is part of the reason the Bowser paintings on the side repeat periodically), I see no issue with this
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.
Hyperrogue as well. Hyperbolica sorta dropped the ball. Hyperrogue shows off tons of concepts explorable in-depth, while hyperbolica only briefly touches on most concepts.
@@louisrobitaille5810 Not sure, but hyperrogue is open source and has been used in research for applied hyperbolic geometry. It's clunky to use, but it has so many more features than hyperbolica. It has support for various tilings of the plane, including even spherical tilings, or 4d hypercrystal tilings.
Hyperbolica was fun to play but limiting the game world's size kind of makes it just feel like it's all just set in a town where everything is set really far apart even though it's technically close (perhaps inspired by American urban planning :P). I found myself using the minimap much of the time as my primary mode of navigation, thus turning it into a top-down 2d game like HyperRogue.
pyropulse - If the title had been “How can a Euclidean game engine be tricked into providing a non-Euclidian game experience” you might have a point, no matter how triggered you come across as being. I invite you to watch the video again and see if the games shown model any real world experience you have ever had. However, the video title is “How do non-euclidian games work?”, and the true answer to that is better reflected by my comment than the content of the video. That’s a professional opinion, BTW. pyropulse, chill mate. Take a break if you’re stressed. 2020 will finally end and hopefully the world can become a friendly place again. Stay well until then.
The non Euclidean world demo that shows at 5:09 is pretty close to what a non Euclidean world would really be like, but I get your point, there are no laws of physics to follow. you can do allmost anything with software, but still by definition the games shown are non Euclidean, some of them at least
@pyropulse The comment is meant to communicate the following: Software needn't follow what us humans see as normal. I don't see a reason to be toxic about the comment.
@@markgearing Well at the rate 2020 is going some science experiment will go haywire and break into the 4th or 5th dimension and we will be able to go beyond our euclidean realm while inviting aliens from some place like control...
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.
@@jordanwardan7588 right. And 3d graphics, where we had to figure out projections that looked realistic, how to avoid drawing the backs of things, how to avoid drawing things that had other things in front of them, how to rotate things in such a way that they didn't lose all their integrity, the right way to move a camera so as not to spoil the illusion, etc.
@@japanpanda2179 yes, if we become able to make wormholes that have inertia we can make non euclidean spaces in earth, or a portal to the moon or something
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?
I personally think you would still collide with the wall, even though the wall should be supposed to be infinitely thin. If you push yourself hard enough you would slice yourself I guess.
There are multiple ways of being non-Euclidean, you are correct about that. However, this word has a specific meaning in mathematics, and Portal and Antichamber do not conform to this meaning. Portals and Antichamber tricks change topology, but the geometry remains Euclidean.
@pyropulse "Euclidean space" usually means this specific thing: en.wikipedia.org/wiki/Euclidean_space However, "non-Euclidean" means "Riemannian manifold which is not an Euclidean manifold" i.e. "Riemannian manifold whose geometry is not Euclidean". So Portal is an Euclidean manifold, but not the three-dimensional Euclidean space, and it is not non-Euclidean.
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.
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.
4 года назад+2199
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.
Glân von Brylân or maybe schools that don’t teach us this. You would expect a school to teach you more than RUclips can, but what can ya do. I still need to know the names of the wives a king killed thousands of years ago. Memorizing their names is far more important.
@@andersnaugle4105 History is far more important for regular life than understanding non-euclidean geometry. I bet you've managed to draw a smiley face on a balloon before without ever being taught how.
Masketta Man yeah I guess you’re right. Just yesterday I communicated with some ancient polytheistic gods. None of my friends knew their names, but luckily I had learned to tell, the difference between the Greek and Norse gods. We all would have been struck down if not for my extensive knowledge of what I had previously thought were two dead religions. How amazingly lucky I was to have learned that in school. I admit that learning Euclidean geometry is pretty useless, but I think learning about fake gods from a dead religion that is now only relevant in statues and literature trumps that. So does learning the legal system of a dead civilization just so I can understand the origins of the term “an eye for an eye”. I could be learning how to do taxes or a business runs but instead I’m learning about... ziggurats. Instead of learning something useful I now know every single one of Heracles’ trials, as well as why he did them and how he died. I repeat, I learned the life story of a fake person from literally millennia ago before I learned how vote. Like seriously WTF?!?! Our society prioritized the life cycle of a butterfly before it’s own legal system!
4 года назад+74
@@andersnaugle4105 History helps understanding today's world, but if you don't know History, I guess you can't realise that. Or maybe you simply don't care, in which case we don't need to further this conversation, since it obviously won't lead anywhere.
Glân von Brylân I understand WHY we learn history. I understand that we learn why Henry VIII killed his women because the ability to get a divorce was a significant event in women’s rights, and it also sparked the beginning of a newish religion. What I don’t understand is why we need to get quizzed on their names. I understand the impact the Greeks and Romans had on our governments, architecture, and more. I understand that a lot of that is because of their mythology and that we can understand more about them because of their mythology. What I don’t understand is why I need to know about Janus, god of doors. Wtf does a door have to do with anything. They literally already have a goddess of cross roads, why would they need another? He’s the most pointless, boring, and all around useless god ever. Why would I need to memorize his name for a test? I understand why we should learn about the history of native Americans. What I don’t understand is why I DIDN’T learn about all the amazing things that Geronimo did and all the hundreds and thousands of people who died because of the negligence of the US government. Learn from your mistakes and al, that right? Isn’t it all about that one quote “he who doesn’t know his history is doomed to repeat it” or something? But nooooo. It’s not like everyone knows his name without knowing any of his absolutely amazing bravery fighting for his people or anything. He’s not famous at all. I understand why I’m learning a lot of these things, but only generally. Most of the specifics are utterly useless and waste hundreds of hours teaching completely pointless garbage that will never be used by anyone in any context.
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.
If you imagine the 3D sphere - cutting through the sphere (the straight line) IS still the shortest, just the diagram shows the line along the surface as opposed to cutting through
Nitzan Bueno they’re 2d lines on a 3D object - they’re parallel in terms of their dimensions - but in a 3D object, they’re not actually parallel. They’d have to cut through the sphere
pyropulse dude no need to trash this person with your pseudo-intellectualism just because they don’t fully get a relatively complicated math concept. Everyone’s mind works differently.
Back in the 90s the game Descent 2 had a level editor where you just connected stretched cubes to each other. With a bit of fiddling I found out that you can stretch to cubes through each other and only one would render depending on the directions that you entered from. This let me design a whole level of non-euclidean geometry without any teleports or changes in distances. There was no limit to the number of things that you could overlap. Portals are just a literal doorway where the inside of the doorway connects to something that you can't see otherwise. As far as I know they didn't really use that in the core game but it was a really elegant solution.
Some maps from the Descent 1 add-on "levels of the world" used this technique. One particular DM map contained two overlapping copies, with a "side" to switch between them. With the tools available at the time, you couldn't connect faces, unless they overlapped, so you couldn't create non-euclidian shortcuts. Given the Descent engine is based on portals, it might be possible in a modern source port tho.
@@fonesrphunny7242 Now I'm a little tempted to get the source code and see if it's possible to salvage the game engine. Nothing stopping us from using hi-res textures, I bet we'd be capable of a lot more complexity just using a GPU.
elegant indeed. using non-euclidean functions to generate a series of euclidean spaces that flowed into each other. versus merely presenting a series of unconnected spaces. from the viewpoint of the player i imagine the two approaches would be indistinguishable ?
@@vorpal22 fun fact Pythagoras was a crazy spiritual leader guy who created a cult and had nothing to do with mathematics. “His” theorem was well known for almost 1300 years before he was even born
@@zokushatech It has been discovered by different cultures at different points throughout history, but the ancient Greeks had a system of logic in place that allowed for proving things rigorously, and made many important mathematical discoveries. While I already knew that Pythagoras was a "spiritual leader," that fact doesn't matter to me when discussing math. History is littered with charismatic people who started a religion / who a religion was started around. I mean, look at Christianity and Islam... both incredibly stupid religions, one started around a figurehead, and one started because of a figurehead. Same with Mormonism, Scientology, and many others. The fact that Catholics believe that the Pope is the spokesperson for their god is just as ridiculous, especially since each Pope has advocated for different things, and the mind of their god Yahweh is supposed to be unchanging and yet it clearly changed according to all branches of Christianity when Jesus died, and according to Catholicism, keeps on changing.
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!
It may sound difficult, but you really just always teleport the same distance on every axis and the distance is simply dependend on the size of the map
stop worrying about non-euclidean geometry. it is just different curvatures. ever written a face on a balloon before? congrats, you made non-euclidean space
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!
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
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.
@@fim-43redeye31 I never quite finished it... most of the logic is in place (and it has a border around the board that shows the across-board adjacencies so you can wrap your head around the different geometries), but I never got to the scoring or the configuration UI, and then I ended up moving on because it was in Java using JavaFX. I might go back to it at some point. Here's a video that shows how it's laid out, if you're interested: ruclips.net/video/XSBidnP8Jg0/видео.html
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.
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!
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
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.
@@Mate_Antal_Zoltan Whilst Build games do use a lot of teleporter tricks, the engine innately allows for overlapping sectors and non-Euclidan geometry and there are plenty of examples of it if you know where to look. The only restriction is that the renderer cannot draw two overlapping sectors on screen at the same time as the rendering will be corrupted. Examples where you can see such geometry are things like spiral staircases. Blood has plenty of levels where there are hallways and rooms that literally cross each other and occupy the same space and there is no teleporting. It is perfectly possible to build an endlessly looping figure 8 on a flat plane in Build without teleporters, even though the cross over point in the centre intersect, the engine will treat them as separate spaces oblivious to each other. Blood also allows the player to look from one sector down or up into another one and drop through, but that is indeed a teleport and rending hack to create a portal that uses two passes to render the player side and side behind the portal. You can always tell when this it being used because the enemies cannot see though portals. Blood E2M5 The Haunting is a great example of a map there loads of non-obvious non-Euclidean and non-obvious portal trickery is going on.
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
I'm a newbie game dev and just teleporting the player to an identical hallway is such a stupidly simple solution that I never would have thought to do. Honestly respect to the anti chamber dev
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.
5D Space-you were close. I got a t-shirt for winning a match playing it at MacWorld 1995 (keyboard only, no extra mice available). Everyone from Bungie was there-all 6 guys!! ruclips.net/video/9WxeeiqWrn4/видео.htmlsi=Qvc8Uz4grW6y--3l
A new non Euclidian game I found recently is space flux, its an indie fps with a focus on maps that can bend your sense of space, for example a certain map is built based on fractals, the closer you get to the center the map will repeat and expand itself like falling into a portal almost, and another is where you can see a smaller version of the map your playing on and jump INTO said map, it really blew my mind and I can't wait to see this game be fully realized as it still needs some kinks to iron out, but really hope it manages to find traction in the coming years
The good thing is that there exists some real non-Euclidean games and you can play it. See HyperRogue for example. It is even free when it's without some minor bonuses.
They aren't Euclidean. He even said that going by the axioms of Euclid, they violate them. But he suggested that they were less legitimate somehow. Which is a wrong way to put it. What he should have said is that they are non-Euclidean in different ways compared to the non-Euclidean games based on hyperbolic or spherical geometry.
These games make me so giddy, not only do I love the brain stimulation I get from the puzzles in these games, but I love feeling like a medieval peasant watching reality as I know it be shattered before my eyes. Regardless of whether it’s “clever trickery” or not.
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.
Really interesting games. Highly recommend Monument Valley as well. It uses isometric perspective with Escher-like tricks, where things that look connected are corrected.
Nice video. Few things remained unexplained. Such as "making" of the objects in Superliminal. You have a painting, clearly. Then you reach a way point, and there is no more paint or wall, there is only that object. Or how portals in Antichamber change where they lead when you're not looking at them. Keep creating.
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.
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.
A little off-topic: as a kid, I always thought how cool it would be to have that same power as the forced-perspective mechanic in Superliminal, but I create a copy of the object the size of my perspective in my hands. Like, I'd look at something, place my fingers around it from my perspective, and boop! I now have it in my hands the same size as I saw it from my perspective. I'm the only one I've ever known to have that as my answer to the question "If you could have one super power, what would it be?"
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.
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
Correct. The axiom should refer to straight lines. The oddest thing about latitude lines (from a Euclidean perspective) is that two latitude lines are everywhere parallel even though they are curved with different radii. Two latitude lines in opposite hemispheres are parallel even though they are curved "away" from each other. The oddest example is the equator and any other latitude line. There we have a straight line that is parallel everywhere along its length to a curved one.
Gotta love how the video goes into great detail explaining what a euclidean- and a non-euclidean-world is but fails to mention we already live in a non-euclidean world or how the geometry lesson relates to these games at all. These are just perspective tricks and teleports.
Iv been looking for the game anti chamber for like 8 years I never remembered the name because I was too young you literally solved one of my life mysteries thank you so much
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.
I learned about non-euclidian geometry my last year of high school and everyone I know insists im wrong but the intro to this video is validating cause the 3 right-angled triangle on a sphere is my go-to example
9:33 - Or they designed a virtual mirror image box. They sell these for real. Its half an image on one side of the v and mirror on the other, so it looks like a whole image, but you can have up to 4 images, one on each size.
Till 4:53 i was so engrossed in to it and unknowingly got in to kind of pressure to understand everything you were explaining as if i am learning it for an upcoming exam then suddenly i realized ohhh i am just watching these for fun.
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.
Parallel lines can be drawn around a sphere and stay parallel, it's all about how you divide the sphere. The method you show you are using longitude and latitude lines as with the global coordinate system, when you follow two longitude lines to the poles they converge. But that is based on the planetary model of a sphere where the latitude lines start at the equator and the longitude lines are 90 degrees from that equator line. If you follow two latitude lines they stay parallel so the 2nd rule can still apply to a sphere. You can make a coordinate system that does obey rule 2. If you took your triangles on a sphere example that followed the parallel lines one, the red line you used would be the equator line. All latitude lines run parallel from that line round the sphere creating slices. Now do the same for the blue and green lines, creating slices parallel from those start lines. You now have a 3 axis coordinate system, let's call them latitude, longitude x and longitude y. Or just L,X,Y for short. All L lines are parallel with each other, all X lines are parallel and so are all Y lines. It's just easier to use 2 coordinate values instead of 3.
Those directions do not form a tangent space and so do not parameterise the local behaviour of the manifold also you cannot parameterise a 2 dimensional manifold with 3 variables and preserve affine transformations (you are forced to have one of those variables be linearly dependent on the other two, thus making it irrelevant) plus all you need to show is that not *all* lines stay a fixed distance, even if you could show that there are *some* that do
Asteroids is, technically, also non-euclidean, as you can get out of one side of the screen and reappear on the other - you are playing on a torus (the surface of a doughnut). It just doesn't seem too weird, as you don't see the world from your character's perspective. Manifold Garden does something similar, but in three dimensions, so it does start to look quite bizarre with everything infinitely repeating in every direction.
Actually no, the torus is a Euclidean manifold when embedded in 2D space like in pacman and asteroids. You can verify for yourself that the axioms hold.
@@ObjectsInMotionYou can have a triangle with angles summing to more than 180°, though, by having one of its sides wrap around the edge, or are we using a stupid definition of triangle? Also, isn't the line between two points assumed to be unique in Euclidean geometry? On the torus, we have got infinitely many, instead.
@@theprofessionalfence-sitter I think you are confusing a regular 3D torus with the Flat toruses we are talking about here. A flat torus like the kind in asteroids and pacman have gaussian curvature of zero everywhere, meaning they truly are Euclidean. All triangles on these toruses do add up to 180 degrees and parallel lines remain parallel. The representation of 2D flat torus's in 3D is only an approximation. The "doughnut shape" you are thinking of, while having an average curvature of 0, has negative curvature on the inside and positive on the outside. A triangle on that surface will have interior angles according to the integral of the curvature inside its bounds.
@@ObjectsInMotion Do we require that triangles bound area? On the flat torus, you can have a triangle (if were using the simple definition of any three non-colinear vertices connected by line segments) which does not have an interior and an exterior, so this integral wouldn't even be well defined. Anyway, it seems that the axioms of Euclidean geometry (or, at least, Hilbert's formalisation thereof) indeed require there to be at most one line between any two points, whereas, on the flat torus, there are infinitely many such lines, so it is certainly at least non-euclidean in that regard.
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.
Moreover, HyperRogue is a great game in itself!
It annoys me when people call things "non-euclidean" when they're really just euclidean (zero curvature) with somewhat weird global geometry
Thanks for checking out my video! I keep underestimating what a reach my videos can have, I think it's awesome that you found my video.
Sorry for not including HyperRogue as an example, not sure how I forgot to include it in the video. I at least added a reference card and added a link to the description.
Thank you for the insightful comments, I'll pin it in the hopes more people will read it :)
I agree because longitude lines aren’t even straight like the example in 2:55 but everything else was great
@@MatthijsvanDuin well, non Euclidean means some of the axioms don't work. Normally the axioms that don't work out lead to a non-zero curvature geometry. But any geometry that doesn't follow the axioms is non Euclidean. Even if all the axioms do work locally.
The original non euclidean space is the infinite staircase in Mario 64
lol
well considering it uses that teleport trick (this is part of the reason the Bowser paintings on the side repeat periodically), I see no issue with this
Why portal? Why? WHY? W H Y
Duke Nukem 3D had a secret level before that called "Tier Drops".
I believe it is pacman
You got teleported back to the right if you go to left and vice versa
The same goes to up and down
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.
Actually, I think that was supposed to be the premise of Portal 2. But playtesters got confused when a game named "portal" has no portals in it.
A bit off topic but Portal 2 is an amazing game
Seems a lot like Superliminal, definitely a head trip of a game.
@Julia Li sadly, no
@@Vindextra Viewfinder.
Hyperbolica is an actual true Non-Euclidean game with curved space instead of a locally Euclidean game which occasionally breaks it's own space
Hyperrogue as well. Hyperbolica sorta dropped the ball. Hyperrogue shows off tons of concepts explorable in-depth, while hyperbolica only briefly touches on most concepts.
@@godlyvex5543 Code Parade documented how he made Hyperbolica. Idk if the Hyperrogue dev did too 🤔.
@@louisrobitaille5810 Not sure, but hyperrogue is open source and has been used in research for applied hyperbolic geometry. It's clunky to use, but it has so many more features than hyperbolica. It has support for various tilings of the plane, including even spherical tilings, or 4d hypercrystal tilings.
yeah the incorrect use of "non-euclidean" irks me. when someone says non-euclidean they almost always mean escherian.
Hyperbolica was fun to play but limiting the game world's size kind of makes it just feel like it's all just set in a town where everything is set really far apart even though it's technically close (perhaps inspired by American urban planning :P). I found myself using the minimap much of the time as my primary mode of navigation, thus turning it into a top-down 2d game like HyperRogue.
Dude, if my geometry teacher explained it like this, I wouldn't have done summer school
Allot of RUclipsrs are better teachers than real teachers
Jim??
Summer school? I feel bad for you...
@@Thor_the_Doge Twice man, and don't I was pretty lazy as hell
Lol Same.
Q: How can games be non-Euclidian?
A: It’s software. It doesn’t have to model the real world.
pyropulse - If the title had been “How can a Euclidean game engine be tricked into providing a non-Euclidian game experience” you might have a point, no matter how triggered you come across as being. I invite you to watch the video again and see if the games shown model any real world experience you have ever had.
However, the video title is “How do non-euclidian games work?”, and the true answer to that is better reflected by my comment than the content of the video. That’s a professional opinion, BTW.
pyropulse, chill mate. Take a break if you’re stressed. 2020 will finally end and hopefully the world can become a friendly place again. Stay well until then.
The non Euclidean world demo that shows at 5:09 is pretty close to what a non Euclidean world would really be like, but I get your point, there are no laws of physics to follow. you can do allmost anything with software, but still by definition the games shown are non Euclidean, some of them at least
@pyropulse The comment is meant to communicate the following: Software needn't follow what us humans see as normal.
I don't see a reason to be toxic about the comment.
@pyropulse oh my god you're so stupid
@@markgearing Well at the rate 2020 is going some science experiment will go haywire and break into the 4th or 5th dimension and we will be able to go beyond our euclidean realm while inviting aliens from some place like control...
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.
Yeah it would be quite fun if these things actually did happen IRL though.
I had to keep reading this be cause my brain just didn't understand it
the "world" they mean is the medium of video games
@@jordanwardan7588 right. And 3d graphics, where we had to figure out projections that looked realistic, how to avoid drawing the backs of things, how to avoid drawing things that had other things in front of them, how to rotate things in such a way that they didn't lose all their integrity, the right way to move a camera so as not to spoil the illusion, etc.
@@japanpanda2179 yes, if we become able to make wormholes that have inertia we can make non euclidean spaces in earth, or a portal to the moon or something
He’s back boys! So excited to watch
For a moment I thought CodeParade uploaded when I saw a non-euclidean themed video.
Same.
Im glad he mentioned Hyperbolica
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?
I personally think you would still collide with the wall, even though the wall should be supposed to be infinitely thin. If you push yourself hard enough you would slice yourself I guess.
There is an orange and blue portal frame around the portals, perhaps that provides a buffer between being cut in half.
i ask myself that every day
There are multiple ways of being non-Euclidean, you are correct about that. However, this word has a specific meaning in mathematics, and Portal and Antichamber do not conform to this meaning. Portals and Antichamber tricks change topology, but the geometry remains Euclidean.
@pyropulse "Euclidean space" usually means this specific thing: en.wikipedia.org/wiki/Euclidean_space
However, "non-Euclidean" means "Riemannian manifold which is not an Euclidean manifold" i.e. "Riemannian manifold whose geometry is not Euclidean". So Portal is an Euclidean manifold, but not the three-dimensional Euclidean space, and it is not non-Euclidean.
4:34 So my early 3d drawing program wasn't faulty, it was just simulating spherical space
That is so cool! If you think about it bugs are just unwanted features! Like you can accidentally make a non euclidean game by messing up your code.
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.
If you expected to break the game isn’t that just playing the game?
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.
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.
Glân von Brylân or maybe schools that don’t teach us this. You would expect a school to teach you more than RUclips can, but what can ya do. I still need to know the names of the wives a king killed thousands of years ago. Memorizing their names is far more important.
@@andersnaugle4105 History is far more important for regular life than understanding non-euclidean geometry. I bet you've managed to draw a smiley face on a balloon before without ever being taught how.
Masketta Man yeah I guess you’re right. Just yesterday I communicated with some ancient polytheistic gods. None of my friends knew their names, but luckily I had learned to tell, the difference between the Greek and Norse gods. We all would have been struck down if not for my extensive knowledge of what I had previously thought were two dead religions. How amazingly lucky I was to have learned that in school.
I admit that learning Euclidean geometry is pretty useless, but I think learning about fake gods from a dead religion that is now only relevant in statues and literature trumps that. So does learning the legal system of a dead civilization just so I can understand the origins of the term “an eye for an eye”. I could be learning how to do taxes or a business runs but instead I’m learning about... ziggurats.
Instead of learning something useful I now know every single one of Heracles’ trials, as well as why he did them and how he died. I repeat, I learned the life story of a fake person from literally millennia ago before I learned how vote. Like seriously WTF?!?! Our society prioritized the life cycle of a butterfly before it’s own legal system!
@@andersnaugle4105 History helps understanding today's world, but if you don't know History, I guess you can't realise that. Or maybe you simply don't care, in which case we don't need to further this conversation, since it obviously won't lead anywhere.
Glân von Brylân I understand WHY we learn history. I understand that we learn why Henry VIII killed his women because the ability to get a divorce was a significant event in women’s rights, and it also sparked the beginning of a newish religion. What I don’t understand is why we need to get quizzed on their names.
I understand the impact the Greeks and Romans had on our governments, architecture, and more. I understand that a lot of that is because of their mythology and that we can understand more about them because of their mythology. What I don’t understand is why I need to know about Janus, god of doors. Wtf does a door have to do with anything. They literally already have a goddess of cross roads, why would they need another? He’s the most pointless, boring, and all around useless god ever. Why would I need to memorize his name for a test?
I understand why we should learn about the history of native Americans. What I don’t understand is why I DIDN’T learn about all the amazing things that Geronimo did and all the hundreds and thousands of people who died because of the negligence of the US government. Learn from your mistakes and al, that right? Isn’t it all about that one quote “he who doesn’t know his history is doomed to repeat it” or something? But nooooo. It’s not like everyone knows his name without knowing any of his absolutely amazing bravery fighting for his people or anything. He’s not famous at all.
I understand why I’m learning a lot of these things, but only generally. Most of the specifics are utterly useless and waste hundreds of hours teaching completely pointless garbage that will never be used by anyone in any context.
This is one trick where explaining the magic has only made it cooler. Simple, yet extremely effective.
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.
And then he doesn't pick two parallel lines. 3 mins in and I'm already wondering what the game is.
@@happinesstan The lines are actually parallel
If you imagine the 3D sphere - cutting through the sphere (the straight line) IS still the shortest, just the diagram shows the line along the surface as opposed to cutting through
Nitzan Bueno they’re 2d lines on a 3D object - they’re parallel in terms of their dimensions - but in a 3D object, they’re not actually parallel. They’d have to cut through the sphere
I think we said the same thing - please regard my response as moot aha 😅
This video coming into existence at this point in my life has made my week
TL,DR:
Euclidean: "Makes sense to me"
Non-Euclidean: "How tf does that work?"
Thanks
pyropulse dude no need to trash this person with your pseudo-intellectualism just because they don’t fully get a relatively complicated math concept. Everyone’s mind works differently.
@pyropulse Wow, can't believe someone is taking comments too seriously. Maybe you should blow off a little steam if a dumb comment upsets you so much.
So, are women non-euclidean? Lel
@pyropulse I'm impressed that you wrote this much without explaining anything, and instead managed to only insult people.
And he came back when we least expected him
Back in the 90s the game Descent 2 had a level editor where you just connected stretched cubes to each other. With a bit of fiddling I found out that you can stretch to cubes through each other and only one would render depending on the directions that you entered from. This let me design a whole level of non-euclidean geometry without any teleports or changes in distances. There was no limit to the number of things that you could overlap. Portals are just a literal doorway where the inside of the doorway connects to something that you can't see otherwise. As far as I know they didn't really use that in the core game but it was a really elegant solution.
Some maps from the Descent 1 add-on "levels of the world" used this technique. One particular DM map contained two overlapping copies, with a "side" to switch between them.
With the tools available at the time, you couldn't connect faces, unless they overlapped, so you couldn't create non-euclidian shortcuts. Given the Descent engine is based on portals, it might be possible in a modern source port tho.
@@fonesrphunny7242 Now I'm a little tempted to get the source code and see if it's possible to salvage the game engine. Nothing stopping us from using hi-res textures, I bet we'd be capable of a lot more complexity just using a GPU.
elegant indeed. using non-euclidean functions to generate a series of euclidean spaces that flowed into each other. versus merely presenting a series of unconnected spaces. from the viewpoint of the player i imagine the two approaches would be indistinguishable ?
Bro you're giving us hope like this, making amazing videos and whatnot
Dont leave again? Deal.
Really? Your hope lies in computer games? Welcome to the simulation, sir.
Imagine being so legendary that even after 2500 years they use your theories to describe geometry.
Well, the Greeks formed a lot of the basis of modern mathematics. Look at how important the Pythagorean theorem is.
@@vorpal22 fun fact Pythagoras was a crazy spiritual leader guy who created a cult and had nothing to do with mathematics. “His” theorem was well known for almost 1300 years before he was even born
@@zokushatech It has been discovered by different cultures at different points throughout history, but the ancient Greeks had a system of logic in place that allowed for proving things rigorously, and made many important mathematical discoveries.
While I already knew that Pythagoras was a "spiritual leader," that fact doesn't matter to me when discussing math. History is littered with charismatic people who started a religion / who a religion was started around. I mean, look at Christianity and Islam... both incredibly stupid religions, one started around a figurehead, and one started because of a figurehead. Same with Mormonism, Scientology, and many others. The fact that Catholics believe that the Pope is the spokesperson for their god is just as ridiculous, especially since each Pope has advocated for different things, and the mind of their god Yahweh is supposed to be unchanging and yet it clearly changed according to all branches of Christianity when Jesus died, and according to Catholicism, keeps on changing.
@@zokushatech (I did not know about Plimpton 322 formerly, though, so thank you for indirectly teaching me something new.)
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!
It may sound difficult, but you really just always teleport the same distance on every axis and the distance is simply dependend on the size of the map
@@iamwhatitorture the maps could also be on top of each other so you only change the Y position
@@RafaelMunizYT that was my first thought too
some clever alignment and you are good to go for a prototype
I don't know why youtube has recomended this to me, but, man, thank you. This is amazing.
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.
Non-euclidian geometry: hi, whats up!
My brain: *panic*
@pyropulse everyone doesnt care what did you hate, sorry
@pyropulse I guess it beats hating yourself.
stop worrying about non-euclidean geometry. it is just different curvatures. ever written a face on a balloon before? congrats, you made non-euclidean space
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!
AN UPLOAD!
THIS MUST BE WHAT THE PROPHECY WAS SPEAKING OF
"Hey vsaue, michale here. These games are non-euclidean. _or are they?_ "
OOf xDDD
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
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.
That sounds absurd. Is that app public? I bet people would love to try it.
@@fim-43redeye31 I never quite finished it... most of the logic is in place (and it has a border around the board that shows the across-board adjacencies so you can wrap your head around the different geometries), but I never got to the scoring or the configuration UI, and then I ended up moving on because it was in Java using JavaFX. I might go back to it at some point.
Here's a video that shows how it's laid out, if you're interested:
ruclips.net/video/XSBidnP8Jg0/видео.html
YOOOOOO I suggested this a while back, glad to say it so beautifully explained!
lIteRaLlY?
Literally literal
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.
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!
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
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.
that game already uses teleports to sell the illusion of rooms stacked on top of one another
@@Mate_Antal_ZoltanThe Build Engine allows you to build stacked rooms and overlapping rooms without teleporters. It is not all teleport tricks.
@@soulsphere9242 I'm pretty sure it still uses teleports to achieve that, it just does it automatically
@@Mate_Antal_Zoltan Whilst Build games do use a lot of teleporter tricks, the engine innately allows for overlapping sectors and non-Euclidan geometry and there are plenty of examples of it if you know where to look. The only restriction is that the renderer cannot draw two overlapping sectors on screen at the same time as the rendering will be corrupted. Examples where you can see such geometry are things like spiral staircases. Blood has plenty of levels where there are hallways and rooms that literally cross each other and occupy the same space and there is no teleporting.
It is perfectly possible to build an endlessly looping figure 8 on a flat plane in Build without teleporters, even though the cross over point in the centre intersect, the engine will treat them as separate spaces oblivious to each other.
Blood also allows the player to look from one sector down or up into another one and drop through, but that is indeed a teleport and rending hack to create a portal that uses two passes to render the player side and side behind the portal. You can always tell when this it being used because the enemies cannot see though portals.
Blood E2M5 The Haunting is a great example of a map there loads of non-obvious non-Euclidean and non-obvious portal trickery is going on.
I was really impressed with superliminal when it came out. Literally didn't know how it was achieved. Now it looks so simple, but it's great.
“The fastest way to get from one point to another is a straight line with no curvature”
Bhoppers: Well, yes but actually no
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
Nobody:
Christopher Nolan: lets write a film around this
plot twist christopher nolan made these games
Is that Inception you mean?
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
"you don't have to be incredibly smart or talented to create one of these things" lol what a backhanded compliment though I agree
It's accurate, but it still takes a bit of creativity to come up with
I'm a newbie game dev and just teleporting the player to an identical hallway is such a stupidly simple solution that I never would have thought to do. Honestly respect to the anti chamber dev
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.
5D Space-you were close. I got a t-shirt for winning a match playing it at MacWorld 1995 (keyboard only, no extra mice available). Everyone from Bungie was there-all 6 guys!! ruclips.net/video/9WxeeiqWrn4/видео.htmlsi=Qvc8Uz4grW6y--3l
Such a great introduction of how physical geometry could work in game engine. Inspiring and fantastic, appreciate your work:)
Everytime i see antichamber i get reminded of when someone posted a “Portal 3 Gameplay” video and it was just antichamber
Oof
A new non Euclidian game I found recently is space flux, its an indie fps with a focus on maps that can bend your sense of space, for example a certain map is built based on fractals, the closer you get to the center the map will repeat and expand itself like falling into a portal almost, and another is where you can see a smaller version of the map your playing on and jump INTO said map, it really blew my mind and I can't wait to see this game be fully realized as it still needs some kinks to iron out, but really hope it manages to find traction in the coming years
actually, it was disappointing to know that non-euclidean games are actually euclidean lol
It was annoying hearing them referred to as such when they were not.
I always knew that so it isn't that bad
The good thing is that there exists some real non-Euclidean games and you can play it. See HyperRogue for example. It is even free when it's without some minor bonuses.
They aren't Euclidean. He even said that going by the axioms of Euclid, they violate them. But he suggested that they were less legitimate somehow. Which is a wrong way to put it. What he should have said is that they are non-Euclidean in different ways compared to the non-Euclidean games based on hyperbolic or spherical geometry.
@@zzasdfwas They only violate these rules in very specific circumstances.
Seeing a Brazilian meme + having the guy whose name inspired mine in the video truly made my day lol xD.
"a love to create and a passion to learn" I thought you were gonna transition into a skillshare sponsorship
These games make me so giddy, not only do I love the brain stimulation I get from the puzzles in these games, but I love feeling like a medieval peasant watching reality as I know it be shattered before my eyes. Regardless of whether it’s “clever trickery” or not.
Yay, you're back! Thanks! :D
I want more games that explore this concept because it's so neat
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.
funny you just mentioned my 3 favorite games at the start of the video xD, i love the immersion in these trippy worlds
Really interesting games. Highly recommend Monument Valley as well. It uses isometric perspective with Escher-like tricks, where things that look connected are corrected.
Nice video. Few things remained unexplained. Such as "making" of the objects in Superliminal. You have a painting, clearly. Then you reach a way point, and there is no more paint or wall, there is only that object. Or how portals in Antichamber change where they lead when you're not looking at them. Keep creating.
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.
materiał jak zawsze przyjemny w odbiorze. czekam na kolejną część!
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.
How could my favorite channel betray me like this? You uploaded this 1 hour after i went to sleep
0:46 maybe the most famous Brazilian meme good taste
Excuse-me, what the porra kkk
zoa, é de fucker msm
very fuck
cuma? rsrsrsr
Eu tava procurando este comentário kkkkk
Brasileiro, o povo mais carente do mundo
A little off-topic: as a kid, I always thought how cool it would be to have that same power as the forced-perspective mechanic in Superliminal, but I create a copy of the object the size of my perspective in my hands. Like, I'd look at something, place my fingers around it from my perspective, and boop! I now have it in my hands the same size as I saw it from my perspective. I'm the only one I've ever known to have that as my answer to the question "If you could have one super power, what would it be?"
so nice this video, has explaned so many of my questions.
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.
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
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
short answer: they program until it works
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
"On spherical surfaces parallel lines converge."
Latitude lines: Am I a joke to you?
are you sure its a line or a circle.
those aren't straight in spherical space tho
ikr
Correct.
The axiom should refer to straight lines.
The oddest thing about latitude lines (from a Euclidean perspective) is that two latitude lines are everywhere parallel even though they are curved with different radii. Two latitude lines in opposite hemispheres are parallel even though they are curved "away" from each other.
The oddest example is the equator and any other latitude line. There we have a straight line that is parallel everywhere along its length to a curved one.
@@trueriver1950 although the latitude lines do curve when they’re not at the equator, if they were straight lines they would converge
these are my favourite type of games and i never knew what they're called, so glad i found this video
Wouldn't be surprised if he just said "First, we need to talk about parallel universes." Then just started talking about SM64
Gotta love how the video goes into great detail explaining what a euclidean- and a non-euclidean-world is but fails to mention we already live in a non-euclidean world or how the geometry lesson relates to these games at all. These are just perspective tricks and teleports.
There is a lot that goes into making good portals, but yeah. I think this video goes off in the wrong direction
As a brazilian, it's fun to see a Nazaré Tadesco(the math woman) meme.
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SIM KSKAKW
Welcome back! High quality content as always
You gotta play "manifold garden", this game is mesmerizing af
I would love all the yt channels organize the video description just like you
*watches whole vid*
Digi: "Now youre a "bit wiser"
me: "Ohhhhhhhhhhhhhh"
Iv been looking for the game anti chamber for like 8 years I never remembered the name because I was too young you literally solved one of my life mysteries thank you so much
"You could say you are a bit wiser" 😅❤❤❤
0:01 plot twist: those cubes are made of screens that display images that look like they're the inside of the cube.
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.
I learned about non-euclidian geometry my last year of high school and everyone I know insists im wrong but the intro to this video is validating cause the 3 right-angled triangle on a sphere is my go-to example
You forgot to mention Hyperrogue, one of the original noneuclidean geometry games.
What is Hyperrouge?
@@miguelbaltazar7606 a Hyperbolic rogue like. There is even a free version.
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.
9:33 - Or they designed a virtual mirror image box. They sell these for real. Its half an image on one side of the v and mirror on the other, so it looks like a whole image, but you can have up to 4 images, one on each size.
although it has sense, that doesn't work pretty well in the final result.
HyperRogue was my first introduction to non-euclidean geometry in a game. Exploring a hyperbolic world is quite fun.
2:54 nitpicking, but that's not the shortest line!
Till 4:53 i was so engrossed in to it and unknowingly got in to kind of pressure to understand everything you were explaining as if i am learning it for an upcoming exam then suddenly i realized ohhh i am just watching these for fun.
"Light rays hate your eyes"
5:24
Stop that 🛑 😂 ✋
It’s 2 am and RUclips is the coolest thing ever now!
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.
Parallel lines can be drawn around a sphere and stay parallel, it's all about how you divide the sphere. The method you show you are using longitude and latitude lines as with the global coordinate system, when you follow two longitude lines to the poles they converge. But that is based on the planetary model of a sphere where the latitude lines start at the equator and the longitude lines are 90 degrees from that equator line. If you follow two latitude lines they stay parallel so the 2nd rule can still apply to a sphere.
You can make a coordinate system that does obey rule 2. If you took your triangles on a sphere example that followed the parallel lines one, the red line you used would be the equator line. All latitude lines run parallel from that line round the sphere creating slices. Now do the same for the blue and green lines, creating slices parallel from those start lines. You now have a 3 axis coordinate system, let's call them latitude, longitude x and longitude y. Or just L,X,Y for short. All L lines are parallel with each other, all X lines are parallel and so are all Y lines. It's just easier to use 2 coordinate values instead of 3.
Those directions do not form a tangent space and so do not parameterise the local behaviour of the manifold
also you cannot parameterise a 2 dimensional manifold with 3 variables and preserve affine transformations (you are forced to have one of those variables be linearly dependent on the other two, thus making it irrelevant)
plus all you need to show is that not *all* lines stay a fixed distance, even if you could show that there are *some* that do
And then there's me who has never heard of antichamber
This was a very interesting topic to cover. You always deliver with your videos!
Asteroids is, technically, also non-euclidean, as you can get out of one side of the screen and reappear on the other - you are playing on a torus (the surface of a doughnut). It just doesn't seem too weird, as you don't see the world from your character's perspective.
Manifold Garden does something similar, but in three dimensions, so it does start to look quite bizarre with everything infinitely repeating in every direction.
And pacman.
Actually no, the torus is a Euclidean manifold when embedded in 2D space like in pacman and asteroids. You can verify for yourself that the axioms hold.
@@ObjectsInMotionYou can have a triangle with angles summing to more than 180°, though, by having one of its sides wrap around the edge, or are we using a stupid definition of triangle? Also, isn't the line between two points assumed to be unique in Euclidean geometry? On the torus, we have got infinitely many, instead.
@@theprofessionalfence-sitter I think you are confusing a regular 3D torus with the Flat toruses we are talking about here. A flat torus like the kind in asteroids and pacman have gaussian curvature of zero everywhere, meaning they truly are Euclidean. All triangles on these toruses do add up to 180 degrees and parallel lines remain parallel. The representation of 2D flat torus's in 3D is only an approximation.
The "doughnut shape" you are thinking of, while having an average curvature of 0, has negative curvature on the inside and positive on the outside. A triangle on that surface will have interior angles according to the integral of the curvature inside its bounds.
@@ObjectsInMotion Do we require that triangles bound area? On the flat torus, you can have a triangle (if were using the simple definition of any three non-colinear vertices connected by line segments) which does not have an interior and an exterior, so this integral wouldn't even be well defined.
Anyway, it seems that the axioms of Euclidean geometry (or, at least, Hilbert's formalisation thereof) indeed require there to be at most one line between any two points, whereas, on the flat torus, there are infinitely many such lines, so it is certainly at least non-euclidean in that regard.
DigiDigger talking about CodeParade? what is this, a crossover episode?
but for real glad that you're back :D
0:46 a brazilian meme! here's my like
The math meme earned you an easy like haha, and the content is pretty well produced, thank you.