Flatland: A Romance of Many Dimensions is an 1884 satirical novella
by the English schoolmaster Edwin Abbott Abbott. Writing pseudonymously
as "A Square", the book used the fictional two-dimensional world of
Flatland to comment on the hierarchy of Victorian culture; but the
novella's more enduring contribution is its examination of dimensions.
"A Square" feels sorry for poor, limited "A Line", but can't wrap his two-dimensional head around what "A Sphere" can do.
For a visceral and entertaining exploration of what a literal fourth spatial dimension might be like from our perspective, try Rudy Rucker's hilarious 'Spaceland'.
'In a tribute to Edwin Abbott's Flatland, a classic mathematical fantasy about a 2-dimensional being (A. Square) who receives a surprise visit from a higher-dimensional sphere, Rudy Rucker's Spaceland describes the life of Joe Cube, an average, modern-day Silicon Valley hotshot who one day discovers the fourth dimension from an unexpected visitation. Spaceland contains scenes that are violent and sexual in nature as well as suggestive illustrations.'
I was really hoping for a continuous (rather than discrete) fourth dimension. The concept is cool and looks to be well executed, but it's not very different from other alternate reality implementations such as the Dark World in Legend of Zelda.
That would be really cool. The mathematics involved don't seem too difficult; you have to add another axis to your vectors, points, and transform matrices, and you can place and orient your models in 4D space, set up lights and a camera in 4D space, and project the camera's view into 3D space first then onto the 2D display. I think the hard part is creating the models and writing the rendering engine. Creating 4D models, rather than just putting 3D models into 4D space, would be hard to wrap your head around. And the tools, both for world creation and for rendering, are almost certainly limited to 3D so you'd have to write your own tools for everything. No more GPU acceleration, unless your 4D to 3D transformation creates models and textures on-the-fly to pass to the GPU.
This is something I've been working on as a side-project on and off for a while. It's complicated by the fact that every existing engine is built around the assumption that you only want to use 3 dimensions, not four, and not arbitrarily many, so I've been pretty much writing everything from scratch (and very limited GPU acceleration, as you said, because GPUs also assume 2 or 3D). So far, I'm punting on model building by just auto-generating 4D mazes with hypercubical cells, because maze-generation algorithms work in arbitrary dimensions.
For simplicity, my rendering engine is a dead-simple raytracer which doesn't even bother with reflections- just cast rays and record the first surface they run into. Handling rotations and projections is really easy, but there are two problems I have found:
1. It's just really frickin' slow. Raytracing just ain't fast enough, though I may be able to find some additional optimizations to make it better.
2. In terms of gameplay, navigating the space is really, really hard. It is so easy to get completely lost regarding which direction you're facing and what hyperplane you're on when you can make arbitrary rotations in all four dimensions. It might be something I could get used to, but so far I feel like it's practically necessary to have some mechanism that will snap you back into alignment with some set of gridlines.
One possible way of getting around that, stolen from machine learning, is to use colors to represent the fourth dimension, ie, make things darker (or change the hue/saturation) that are bigger in the fourth axis, and vice versa for smaller.
I would just avoid the rotation issue. Time works as a 4th dimension so if you have a charter that can flow backward or forward in time you can 'solve' 4d mazes without major issues. From a game-play perspective plenty of games have you vary limited control over time but you could make some vary interesting puzzles with full control.
EX: walls in the way. <Set bomb off> Walls gone but cave collapses. <walk though wall> <go back in time> <keep walking>.
Or simply let you flip times direction. Going vary slow or backward prevents changing things, but you get to move. So, you might be able to for example walk on water when moving backward in time etc.
That's pretty much the idea of Braid, an awesome indie game. It uses time manipulation in different ways to create mechanics in that vein. Recommended.
There are some simple games like that. 4d mazes, 4d tetris, 4d pong, etc. The problem is it's pretty impossible for people to wrap their mind around it, and it doesn't project very well on 2d screens. Research has shown that people don't get better at it with practice either.
I'm going to google around for this, but do you have any links to the research off hand? I'd be interested in seeing it.
My favorite 2048 variation was the 4D version. It was trippy but I felt like I got my head around that. I think it was because the game rules were so simple that instead of trying to visualize what was going on, my brain just accepted the patterns.
We already have 4-dimensional models; we just keep the 4th dimension separate and call it an animation path. In fact, Skyrim has 5-dimensional models: there are two versions of each walking animation, roughly "feminine" and "masculine," but each character actually uses a specific blend-point between the two. Similar analogies with facegen textures. All the pieces are there, they just need to be glued together right.
Can you clarify why having two animation paths makes 5 dimensions? Surely the dimension is time, you're just deforming the 3 spatial dimensions along two different paths as you progress uniformly along the 4th?
You could probably make something like Minecraft but out of hypercubes and besides turning in three dimensions you can also reorient your three dimensional subspace in the four dimensional world. As you mentioned, the interesting question is how you model objects - are our usual three dimensional things thin sheets in the fourth dimensions or do they have some interesting structure.
I believe there are some good libraries out there for handling quaternions. Can't remember the topic but there was a discussion here within the last 10 days about their application to a similar problem.
It's naturally confusing, because a 4th spatial dimension isn't something we naturally experience.
We can't really say what it represents because it's not real... it's not a model of reality.
We see in 2 dimensions and we infer the third from various cues. (The quote that we are "dimly aware of a fourth", which you may have heard, refers to time and is a different issue than a fourth spatial dimension.)
We have three dimensions, all orthogonal to each other. We can translate, rotate, and project things in between them.
So all we can do is apply those same principles and see what comes out - we shouldn't expect it to match anything recognizeable though.
Mathematically, they're identical to any other dimension. You can move a little bit into a fourth dimension just like moving a little bit left-to-right. Or you can rotate things around the fourth axis just like turning something clockwise. In the case of this game, he only shows a 2D projection of a 3D cross-section of a 4D world, so that might be adding to the confusion. What you see in the video is only 3D "slices" of a 4D object rotating.
One of the axes introduced by the fourth dimension. There are 3 in 3D, 6 in 4D. XYZ space has axes perpenicular to the planes xy, xz, yz. WXYZ space has axes perpendicular to the planes xy, xz, xw, yz, yw, zw.
No. There are only 4 axes in 4-dimensional space - every point is described as a 4d vector, i.e scalar positions along each axis: (x,y,z,w).
xy, xz, etc. are not planes in 4d space - they are 3d-hyperplanes, and perpendicular to them are planes, not axes. Imagine this: you have a 4d vector and you hold x and y constant. You still have 2 degrees of freedom: a plane, not a line.
It might help mentally, but it won't reduce the number of projections. You'd have to be able to see every point in the whole 3D volume, not just the surface.
The game is actually completely continuous. The crazy deforming shape at the beginning of the trailer is a 3D slice of an object called a 120-Cell, which is a true 4D shape.
Isn't that addressed in the additional notes below the video?
There are **infinitely** many 3D worlds stacked on top of
each other, even if in the Wall level the ground texture
makes it seem like there are only two.
Adanaxis is an open source space shooter game that lets you move in all 4 axis, or rotate continuously along any combination of axis. All models are 4 dimensional.
Needless to say, it is confusing to the point of unplayability past a certain point, but utterly fascinating.
You can grab a copy from your Ubuntu, Fedora, or Arch repos, and it is certainly worth trying to wrap your head around for an hour or two, if you are into that sort of thing.
But when he got to the 4th dimension it didn't make much sense as an explanation (rubble from 3rd dimension showing up in the 4th).
However, if the 4th dimension was time, then you would walk back (or forward) in time to when the wall wasn't there (isn't there anymore cause it is now rubble), move "past the wall", and then walk forward/backward to the original time.
I thought the same until I watched it a second time.
4th dimension is still time. 'The button' takes you to your character's view of moving across a 3rd dimensional plane. I'm assuming the two different textures on the ground represent the change in the 4th dimension (time).
So when you cross to that other side and hit the button again, you're back in the 3rd dimension across a different plane (time). So the desert area is either in the past or in the future (when the wall fell or was being built).
Then he goes through the wall and hits the button again and walks back to his original 3rd dimension.
That's what I took from it anyway. Seems really neat.
>4th dimension is still time. 'The button' takes you to your character's view of moving across a 3rd dimensional plane. I'm assuming the two different textures on the ground represent the change in the 4th dimension (time).
No, time is actually a 5th dimension in this 4D game projection.
The developer has actually done the proper 4 to 3D projections, like he has done for the 2/3D that he shows in the beginning.
In the 2D to 3D you can definitely see that he's not moving through "time", because you can see the 3rd dimension.
The problem is just that we can't intuitively visualize a 4th spatial dimension that makes this difficult.
But he's definitely not moving through "time" or in some "past" when the "wall wasn't there". And of course moving through time has different properties than moving through a 4th spatial dimension, so it's not really comparable.
E.g if we had a gradual rise in temperature from 0oF to 100oF in the time-dimension, then if you could move through the time-axis you'd experienced different temperature at will, whereas moving through any spatial axis would you'd only feel a partial rise in temperature.
That's how I saw it as well. When you rotate to the "side" (pivot) in the 4th dimension, you see the wall and its change over time as one entity. So then you can either pick going around the start (when construction of the wall began) or at the end (when the wall falls). Naturally the landscape would also experience change, hence the desert. When you pivot back, you are at the same location in time and space of the 3rd dimension, just in front of the wall now.
But the fourth dimension is timespace, is it not? At least that was what I gathered from Special and General Relativity and the relativity of simultaneity. X^t, Y^t, Z^t?
Whatever you decide "the" fourth dimension "is" (what's the first? second? third?), it doesn't stop someone else from imagining a world with an additional spatial dimension. Relativity does imply a close relationship between time and space but the game world doesn't appear relativistic.
I don't think dimensions are specifically numbered. You could call up one dimension, distance along a fixed track another dimension, and distance from the sun your third dimension. You could call temperature your fourth dimension and mass your fifth. This game specifically has 4 spatial dimensions.
In a lot of cases, the fourth dimension is describe as space. But if you were looking into string theory, often times there are description of many spatial dimensions, say 11 spatial dimensions and then a single dimension of time.
We could also have multi-dimension time dimensions as well. The idea of parallel universes where at least one decision is made differently from our own could instead be different dimensions of time. Move to the left or right in time and experience different choices as opposed to only going forward as I assume most of us do now. :D
No, it's actually not. For one, dimensions aren't numbered.
Time is only called "fourth" because we just add it as an extra to the 3 spatial dimensions we perceive intuitively. Not because there's some specific order.
There can be an arbitrary number of special dimensions in mathematics (hence hypercubes, etc).
And according to some (quite prevalent) physics theories, there are actually more than 3 spatial dimensions in the actual reality -- string theory calculations give 10 spatial dimensions.
That is one way of numbering dimensions, in a certain model of the universe. However, in a 2d game, time might be a "third dimension". Or time might be sufficiently distinct in some model that it does not behave like a dimension at all.
If I write an equation, and graph it, it has as many dimensions as variables. Those variables might have nothing to do with time OR space.
In the 2D/3D world, the rubble just fell off the wall and landed next to it. It's only strange because it fell in the direction that the player character doesn't normally perceive.
In the 3D/4D world, the rubble fell off the wall and landed next to it.
This is a very interesting way of explaining the 4th dimension by relating it to the difference between the 2nd and 3rd dimensions. However, I think what would really solidify my understanding is to see the 4D level editor.
I'm not sure how truly 4D this world is, but it appears to let you switch between a fixed set of different 4 dimensional positions, each of which has a different 3D representation. Just as a plane in a 3D environment is a 2D slice, a plane in a 4D environment is a 3D slice. You could effectively fake this by imagining a triangular prism (like a cylinder, but triangular instead of round). Each face of the prism contains a fully defined 3D world. Switching to the alternate face of the prism is effectively doing the swap to the other 4th dimension, where you can walk along the alternate 3D environment to avoid the obstacle, then switch back to the original 3D environment.
That's not quite what a 4th dimension is like. In your example, the faces are completely separate from one another. But really they would be connected, in a continuous line. You'd be able to push something a little bit along the dimension, or run a long distance. You could rotate an object in 4D so that parts of it ware poking a little bit into each dimension at the same time.
Of course not. You'll notice that I questioned how truly 4D the game was and said the game appears to have fixed sets of 4D positions (ie: not a continuous range). I also said that my example was fake 4D. If the developer were to show the level editor, I suspect that we'd see it's not really 4D at all, but my fake approximation of 4D via a fixed number of 3D environments that the player can transition between.
I've been getting progressively more excited about Miegakure over the past year, but I haven't heard anyone who's played the game talk about how easy it is to "grok". Does moving between dimensions become intuitive over time? Or do you always have to consciously think about what you're doing?
I've also played it semi-recently, and it's really, really intuitive – just like playing another game where you don't quite understand the rules. Things slowly start making sense. Marc (the creator) is an extremely talented designer for building a progression of levels that teaches you the concepts just through level design. You definitely don't get a formal understanding of the 4th dimension, though.
He said:
"Flipping through dimensions is remarkably intuitive- [...] I shifted dimensions, messing about with this strange toy until I quickly understood most of the problems that I faced."
I think it would become intuitive if there was some way to plug 4th dimension in our senses.
Have you read stories about people who implant magnets? After a while they develop a sense of direction. Brain adapts to new signals and starts to interpret them.
Different example, I remember reading about experiment where people had to wear glasses that flip the picture upside down. After not too long time they adapted and saw things normally. And everything would be upside down when taking the glasses off.
Yet another example is asking blind people what they see. They don't see black or grey, the sense of vision simply isn't there. Like our sense of 4th dimension isn't there.
So, one thing to try would be to give us sense of rotation and acceleration in 4th dimension and see how it goes. Next, maybe, a new "color channel" for our vision, and let the brain figure out how to represent it "internally".
One really interesting example of this is that of humans who perform echo location. Parts of the brain normally reserved for sight are activated, to some degree, by regions of the brain responsible for vision.
Basically, they're not just 'hearing' in the traditional sense, though that happens as well; they're also 'seeing' the sound that bounces back. Kind of crazy
People use 4D to go through walls all the time, when those walls take the form of sliding doors. They stand in one place in space, in front of the door, while naturally moving forward in the fourth dimension (time) to arrive to a slice of space-time in which the door isn't there. In this range of space-time they traverse space by a small distance (a meter or two), and also slip forward in time by a several seconds (which is not long enough to leave this special range of space-time). By so doing, they reach a point in space on the other side of where the door was. Then they slip forward in time again, to a time when the door is there again, at which time they smugly find themselves on the other side of the door.
I really want to believe this is possible and this is how we'll be able to travel far into space. Like how we have to spend time walking around a very long wall when a bird can just fly over it in a fraction of the time, we wouldn't have to think about traveling light-years because there would something else we can do that's faster than a straight line.
That's not very likely, because you don't have to walk around some kind of obstacle to get to a star. Even as a bird, you still have to travel the full distance to your destination.
You could make a shortcut in space, which is called a wormhole. You can either make one using extra dimensions, or in 3 dimensions with negative energy to stabilize it, but neither of those seem to exist. Edit: link https://en.wikipedia.org/wiki/Wormhole#Traversable_wormholes
Unless 3d space is 'wrinkled'; in 4D you could travel straight there. Imagine a crumpled sheet - the 2D traveler would have to trace a route through all the convolutions to reach the other end of the sheet. The 3D traveler may only have inches to go. 'A Wrinkle in Time' was a story about this.
Why not consider the idea that any "dimensions" are nothing but "[useful] concepts of the mind", like "universal time" and even "limited space", the way some ancient mystics (?) of various cultures believed?
The simplest example is about time - we (humans) have notion of time "entirely" due to the certain properties of our environment in which we have evolved - that there is day/night, Moon phases, seasons, periodical changes (due to rotation of the planet and its motion around the Sun, of course, but this is a very recent discovery).
Now imagine that you are somehow "suspended" in the outer space without any motion relatively to the Sun. What would be your notion of time?
The time of a certain process (a mass in motion) has nothing to do with time of another, completely different process (like radioactive decay) and the notion of some "common, universal time" is just a "creation of the mind" which is very handy and useful but "does not exist in reality".
Ancient Buddhist notions of "emptiness" or "void", and pre-Buddhist (Upanishadic) notions of "everything is mind" (which is wrong, but very close and accurate) are insights to the same "truths".
Your "dimensions" are "primitive concepts" of the same kind. Any coordinate system imposed on so-called "reality" is nothing but a "concept of the mind".)
> we (humans) have notion of time "entirely" due to the certain properties of our environment
That sounds very philosophical and all, but it seems to me that we have an internal perception of time based on the rate that electrical impulses propagate throughout our brains. Perhaps it would be distorted based on the lack of external stimulus, but I believe we would feel the passage of time all the same.
The game looks fascinating. I'm still trying to get my had around the analogies, but get the feeling that playing the game for real will bring a level of intuition to the experience.
In the meanwhile, can anyone help me out?
In the 2D->3D example, you have one shared axis (Z -- vertical), and an alternating pair for the second axis (X and Y -- horizontal). In effect, from the 2D character's point of view, 'jumping' into the third dimension is just swapping your horizontal axis. Your previous horizontal axis is a vertical section through your current one, and vice versa. As such, when swapping axes, you would expect one 1D line (ie. a 1-pixel-wide vertical band) to be the same before and after the swap.
When going 3D-> 4D, you have 2 shared axes. Your third axis alternates, as before. When switching between your 'third' axes, then, you would expect one entire 2D plane of your perceived world-view to remain the same[1]. This doesn't seem to be the case.
What am I missing?
[1] EDIT: This plane is likely to be a section through some concrete objects, so you won't necessarily have 'seen' it before, though it will have been there.
The plane exists - it is passing through the character body (perpendicular to the ground) and parallel to the desert, grass border in the example video.
You can see when the 4D rotation occurs there, that in that plane, everything stays constant (e.g. rocks stay there, trees, etc.) You just look at the objects from different angles as it rotates.
This is exactly right: when rotating in 4D, you can rotate "around" a plane, leaving it in place. You can also rotate around a line (permuting the other three axes).
So, in the video. The Z axis doesn't change (up and down) so that's one axis. The other one that doesn't change is at a 45 degree angle from lower left to upper right. Watch the tree: it is just as wide along this axis as it was before. Everything else sort of smoooshes around but that plane stays the same.
I suggest interested people check out adanaxis, a 4D space shooter. It's not particularly playable, but it is true four-dimensional space with four-dimensional spaceships zipping around.
In the first case, the 1D vertical line is the same, but it's "seen" from a different angle. It looks the same because it's 1D and it doesn't happen to change because of the rotation.
For the second case, maybe it's the ground plane that's the same; it certainly has some of the initial texture in it, but it's been rotated and it's now seen from a different angle. But now we can tell the different because it varies along the dimensions it rotated around.
Or maybe I just can't visualize the rotation through the fourth dimension... I'm not sure.
That video is incoherent nonsense, especially from the '5th dimension' and onwards, and in no way resembles the modern understanding of dimensions in physics or mathematics. I wouldn't bother watching it except for entertainment.
Care to expand? I'm not sure if it is possible to imagine these dimensions with the "modern understanding of dimensions in physics or mathematics", but it would be nice to at least know why this way is incorrect
One could waste one's entire life trying to correct rubbish attempts at physics on the internet, especially when it's full of non-sequiturs like the video. But as a start, the description of quantum mechanics is nonsense and I'm guessing the narrator got their understanding of it by reading popsci books (I'm trying to avoid a personal attack on the author, but it's hard when they've said a lot of wrong things so very confidently). The dimensionality of a typical QM Hilbert space is infinite, not five. People get wound up wondering what the N^th dimension is, when all dimensionality is, somewhat simplified, is how many numbers you need to describe the position of something.
Thanks for setting the record straight. The video looked interesting and coherent for layman like myself (gotta admit though I only watched at first the older version that didn't mention quantum mechanics), but I never cared to check what real physicist made of it. Gotta love physics stack exchange!
I remember being really excited about this game a few months ago. Part of me is a little concerned whether there is enough to gain some sort of intuition on where you are and what is exactly transforming with each input. I'm a little concerned it will just become a game where you try things and hope for the best and have a very incomplete mapping of whats really going on. But anyways, its neat to see such a well produced game attempt to take on these challenges.
I saw something similar on the demos of what was supposed to be the Sonic X-treme game. This was supposed to be the debut of the Sonic franchise in the 32-bit world.
Yep, except it is in 3d. I was hoping for some continuous movement, but it looks like it only switches between two worlds.
Maybe they will implement some more levels to make it look seamless. But there is simply too much work to design the world in 4d as a single piece of land holds a lot more information.
No, mathematically speaking the 4th dimension is no different than the 3 that we are used to dealing with, and is held to the same physical constraints.
There are no shortcuts. The straight line connecting 2 points in 3d space is shorter than any path connecting the points that goes through a higher dimension. The same applies to 2d compared to 3d. If you have 2 points on a sheet of paper, the shortest path is on the paper. Any path that went "off" the paper into the 3rd dimension would be longer, so no "shortcuts".
Imagine a strip of paper bent into a U. You can travel from one end of the strip to the other through 3D space with a much shorter distance than along the entire paper.
Sure, but that's not what we're talking about here. We're talking about Euclidean 4D space projected orthogonally onto Euclidean 3D slices, not some nonstandard projection onto some weird 3D surface embedded in 4D space.
We don't really know that. Let's suppose the universe was really 4D. We don't currently know if the universe is flat or not. A curved universe can be both unbounded and finite which would be very elegant.
Einstein proposed a test to find out. I'm hoping this test will eventually be Gravity Probe D (Probe C will be a repeat of the failed Probe B).
Looks like a neat game, but I'm skeptical how it would work in reality. For example, it looks like the 4th dimension (time) gets frozen (for the observer) when you want to traverse it. That means, the second you 'travel in time,' the Earth, at its current velocity, would already be 30km away.
You're one of the fools who thinks that the fourth dimension is time. Get over yourself. This is talking about a fourth spatial dimension, not a temporal one.
https://en.wikipedia.org/wiki/Flatland