I considered linking to the game's site at http://4dtoys.com/ , but this blog post introducing it has more design details that I think will interest HN.
These videos are such a treasure. All these years and nobody came even close (new Cosmos is ok, but not that profound).
This is how you make a nation great. You can do whatever you want with the economy, but without proper education you still have a nation of.. people.. who choose their president.. based on their beliefs and understanding of the world.
Shows the relative importance of a great presenter too. The props he uses are laughably cheap, yet he does a better job getting the point across than vastly more expensive productions.
This looks like fun, going to try this out over the weekend.
This is the same person who made this game (that also looks like good fun): http://miegakure.com/ that I remember reading about some years ago but never got a chance to play with
It's a shame it's only for iOS and Vive. I wonder how difficult it would be to make an open source desktop/browser version? Even if it's a lot simpler, it would be neat to feel what it'd be like to play around in 4 dimensions.
My favorite short story of that era. Still have not watched the movie out of sheer fear they didn't get it and I don't want it wrecked. (Still have no idea how that even made it out...)
The video explaining it is really well made, but for some reason I still can't grasp the 4th dimension. Perhaps it's one of those things that you only need to know they exist at the theoretical level.
Same here. I understand perfectly everything that was said in the video but I still don't understand how the fourth dimension quite works.
However, having only heard of hypercubes and not hyperspheres before I decided to see if there was anything useful about them online and I found this video that I just started watching and already 1 min 50 sec into the video something very interesting was said;
> Everybody knows what the sphere is. I'm thinking of a hollow sphere so like a basketball right. That's a two-dimensional surface living in a three-dimensional space. The hypersphere is generalized one dimension up, so in four dimensions you have this three-dimensional space called the 3-sphere or the hypersphere.
Already this is telling me something that I have not heard before, and which I find much more helpful than talking about what a shape looks like from the perspective of someone living one dimension further down. That being said it was still useful having that explained as well, which is what Flatland: The Movie (2007) was about as well. Just this fact I quoted above was even more useful IMO.
Another quote from the video I linked in parent comment.
6:19
> As complex numbers are to real numbers, quaternions are to complex numbers. It's like a way to build up even further. [...] Real numbers are one-dimensional. Complex numbers are two-dimensional. [...] For three dimensions there is no natural number system, but for four dimensions there is and it looks like this.
The rest of the video up until 16:26 has been about projecting down one dimension but it's doing so in different ways instead of just simple cross-section and it's using videos to show rotations and stuff.
That’s really misleading. The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise.
In a similar way, the quaternions are the space of transformations (scaling & rotation) of three-dimensional vectors. (It’s a little more complicated because 3-dimensional rotations are not commutative, and must be combined by sandwiching, so there are 2 choices of quaternion corresponding to every scale and orientation in 3-dimensional space. For an introduction see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)
>The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise.
You might be right, I don't know. Could you explain a bit more how you mean?
Complex numbers have multiplication defined on them. If you multiply two complex numbers, you get another complex number. (They compose via multiplication in exactly the same way as scaling & rotation operators on 2-dimensional vectors.)
If you just have 2-dimensional vectors, there’s no obviously well-defined way to multiply two vectors and get out another vector.
In other words, both 2-dimensional vectors and complex numbers are made up of 2 coordinates, but they don’t have the same mathematical structure.
As a person who studied knot theory and sitting through other peoples presentations about higher dimensional knots this looks like a neat treat! After hearing about the concept I bought the game and tried it out. I like how you replace actual physical actions to objects in 4 dimentions. Usually this is projected on the time axis but with this interface it makes it much more fun to play with it instead of having basically a generic slider.
I'm hitting the same problem -- the app only renders to a smaller rectangle of the whole screen rooted at the bottom-left (when held sideways), but touch events are scaled to the whole screen. I could probably live with the smaller view, but it's very hard to accurately manipulate the objects when it thinks I'm touching further left than I actually am.
This looks fantastic. If I had a VR device I would get this immediately. I've always had a fascination with trying to grok higher dimensions. I think it's just about impossible to have an intuitive understanding of it - 3D spacial reasoning is in our wiring through both nature and experience.
You know the theory of how language shapes your thinking? For example, in societies where there is no separate word for orange and red, they have extreme trouble telling the difference between them. In some native tribe where they use cardinal directions (North, South) - not relative (Left, Right) - they have an almost supernatural ability to know which direction they are without needing any other cues (sunlight, stars).
Point being - would being able to completely think in four dimensions have an impact on how you understand the world?
You have to look at how it was initially positioned. Linguistic determinism as a general purpose tool has not proved to be useful, but it doesn't mean that there isn't clear linguistic determinism in certain areas.
Colors may be the easiest way to see some limited sapir-whorf in action.
You're probably right - I actually suspected that may be the case - though I was only using that as a parallelism for what I'm saying.
Language, in my understanding, operates at a lower level of abstraction in the brain than say, Math. I would suspect that Spacial Reasoning is similarly low-level. It would be interesting to see if there is any impact to the higher-level reasoning when the foundational blocks are played with.
The difference probably wouldn't be very large, I'm now realizing. People who are blind since birth, as far as I know, don't have any observable differences in common patterns of thought. And that seems like a larger difference in low-level thought than being able to understand four dimensions versus three. (Not suggesting blind people lack spacial reasoning in three dimensions)
The way people think about the same things ("translation schemes") can be different (blind or not). As an experiment, try to measure a minute by counting and doing something else http://youtu.be/lr8sVailoLw
SW might have been disproven, but in this case, I think it would make a difference.
I think if you had to have 4D vision rather than 3D, it would require much more processing power and memory to account for the extra dimension. So I think at minimum, a being that is capable of thinking in 4D would have to have higher brain capacity than a being that thinks just in 3D.
And there is some strong evidence that our ability to navigate 3D world influences our general cognitive capabilities. For example, consider https://en.wikipedia.org/wiki/Method_of_loci
I think the problem is that the mild amount of support was the only claim it was ever making. It was everyone else who blew it out of proportion. It's like the whole thing with CERN making micro black holes, and everyone freaking out about a black hole eating the solar system.
You deal with 4 dimensions all the time -- up-down, left-right, forward-back, future-past.
In reality, you deal with high dimensional objets all the time: the position you're in is composed of dozens to hundreds of muscles rotating a few dozen joints.
We're fascinated by vision, but proprioception is where it's at if you want to think in high dimensions.
An easy way to imagine a 4-dimensional structure is to associate every point in 3-space with a color or a temperature. Your mental visualization processes should have no trouble with that.
But the system quickly breaks down; the imitation fourth dimension is not in the same class as the other three. You can easily visualize a three-dimensional object rotating around a line in 3-space, or a two-dimensional object rotating around a point in 2-space, or a two-dimensional object rotating around a line in 3-space. Try applying that process to a "three spatial dimensions plus one color dimension" model and you'll notice you have no idea what color a point should be in the rotated object. (You might think it should be the same color as it was before the rotation, but that's just as wrong as saying that if you rotate a point in 2-space 30 degrees around the origin, it should end up with the same x-coordinate that it started with.)
Proprioception is deeply tied to a three-dimensional conception of space -- that is how you perceive its output. And the evidence is not strong that it is a function of muscle activation, since it will work just as well if you relax while someone else moves your hand around.
That's why I used time not color: you have an inherent notion of what it means to move things along that axis. Your point about color also doesn't quite hold -- it's easy enough to map the rainbow colors to rotation. It obviously takes practice, but it's not hard. (Helps to start with two spatial and a color.)
And of course it's tied to 3 dimensional space -- the whole point is to couple the correspondence between a high dimensional phase space and a blob in 3 dimensions.
But you do have an awareness of the phase space, not just the 3D location -- and you can use it to think about higher dimensional objects.
I don't think my inherent sense of what it means for a point to move forward or backward in time is much different from my inherent sense of what it means for a point to get lighter or darker. Neither will help with a spatial rotation.
Often when I see something about 4d it uses this same analogy: a 2d being seeing a crosssection of a 3d world, in 2d. However, the video is in 2d, and it's able to show 3d in a much better/more clear way, clearly, cause when it's showing that example it's got a cutout of the "3d" view (which is still 2d!).
Why can't we do the same thing with 4d? Why does the object just disappear when it bounces into the 4th dimension, can't we maybe see a projection of it onto the 3rd dimension?
Maybe they could do something like a fog? That is, if the object was farther away in the 4th dimension, it would appear less distinct or more fuzzy/blurry, like being in the fog.
That's a neat idea! But I think it might be a bit overwhelming and confusing. It would help with getting the bigger picture.
Every 4d object would end up being a clear 3d cross-section where it intersects with our 3d world, plus a cloud of superimposed and increasingly hazier 3d cross-sections above and below us along the 4d ("w") axis, projected down onto w=0 3d-space.
Miegakure did something similar where it showed a shadow of the 3D space immediately adjacent to the currently visible slice. I think he since removed that, so I expect he must have experimented with something similar in 4D Toys and decided against it.
There’s a big problem with that, which is that we don’t actually see in 3 dimensions. Our eyes only get 2-dimensional projections of a 3-dimensional world.
We see in 2D, with a tiny bit of depth metadata. True 3D vision would allow you to look at someone, and see the entire volume of all their internal organs simultaneously.
It's somewhat like having the alpha on everything turned down to varying levels - you can see through the skin, bones and metal implants are still at 100% alpha - but fundamentally no different than what looking at someone gives you: a 2D plane.
The comparison here is, as usual, to go down a dimension. Imagine living in a world where everything is constant vertically, like those old 3D maze screen savers or Wolfenstein 3D. You're really seeing a 1D amount of information about a 2D world (in fact, this is how the calculation is for Wolf3D and other games of its era). You can infer depths to objects if you have two eyes.
Now contrast that to if you were plucked up vertically 'above' the game's level to look down upon it. Now you can see the entire 2D extent of the maze at once. Before, your vision was blocked by the walls, now you see the walls and what's on the other side of the walls simultaneously in a way that's entirely distinct from simply seeing through a transparent object.
Now, like seeing a 1D amount of information about a 2D maze while live inside it, we see a 2D amount of information about a 3D world around us (a picture demonstrate's this 2D amount of information - it's planar). Now imagine being lifted out the 3D plane of existence so that you could behold the entirety of the 3D world at once. That's the rough analogy.
Our eyes could get 2-dimensional projections of any more-dimensional world, actually, if it was available. The light-sensitive outline of that little guy in the video could accept photons from any 4-direction, and not only photons floating in his current 2-plane.
Ofc it is abstraction, I know that physicists aren't happy with heterodimensional settings at human scale.
Watch the part about the slicing of 2d world looking at the 3d objects. That slice is infinitely large in both dimensions, yet it can't see the 3D objects when they aren't in the same plane.
Now apply the same analogy to our world: our 3D world is just an infinitely large 3D plane in a 4D world. When the objects aren't in our 'plane', we can't see them.
A favorite novel of mine actually! Hard to believe that it was written in the 1800s, yet still Abbott understood the 4th dimension better than most people today.
I remember playing a flash or java game years ago that did exactly this. It was a 4d maze, rendered in stereoscopic 3d. It displayed the images side by side and you crossed your eyes to get the 3d effect.
It was really disorienting (the 4d, but also the eye-crossing), and like the post author says, just bundles of lines rather than solid shapes. Still very cool.
Idk where I found these. Not 3d though, just stereoscopic 2d. Tetris may be nice for stereo-training, as it doesn't involve any complex movements. But it was harder to me than seeing stills.
The 2D->3D analogy made sense to me -- only being able to see the cross-section that is visible in a current dimensionality -- but I got to wondering if you couldn't use some of techniques from information visualization techniques to show a representation of that extra dimension in the current dimensional landscape (ie fading the object as it gets farther away from the current dimension).
Purely theoretical, but the presence of a 3D object can be detected by a 2D creature, by it's shadow, depending on relative positioning of object, 2D surface, and light source. What would a 3D shadow look like, with no need for a wall or flat surface onto which the shadow would be cast? For that matter, what would 4D light look like, if it even differs. Maybe the light we see is only a shadow of something from the 4th Dimension, or maybe a portal or doorway we have not yet learned to use to it's potential.
My dissertation is about simulating conservative physical systems on a computer; but it is based in symplectic geometry, which exists only in even dimensions.
Put it this way: Riemannian (and as a special case, "ordinary") geometry is based around the inner product <a,b>, which gives you both angles and projection (and 2D shadows from 3D objects, for example) and lengths. You learn a lot about this (not in the "curve" differential-geometric sense, but most of the intuition carries over) in college linear algebra: you learn to think about entire vector subspaces having orthogonal complements as defined by the inner product. On the other hand: length (even if curved) in that direction is the same as length in this direction, i.e. the inner product is a symmetric bilinear map.
Symplectic geometry is built on a skew-symmetric bilinear map (the symplectic form) so that w(u,v)=-w(v,u). In 2D, the symplectic form is equivalent to the determinant: note that this is an oriented or signed area rather than an absolute-value one. And then in 3D, there is no symplectic form (it's rather fun to prove this) such that there is no vector (apart from the origin) that collapses all the other.
Anyway, the challenging thing about this, writing my dissertation, is that unlike with linear algebra where you can gradually extend intuition from the real line to the plane to 3D space and then think "ok, I think I can grok this in 500 dimensions" (and go do multivariate statistics, for example), in symplectic geometry there is either the trivial case (and again, this is the determinant and not particularly "new" to you) or the 4D case next.
And then you can't draw pictures. YOU CAN'T DRAW PICTURES. This is such an intuition-fucker. Books that need the symplectic form right away and can't waste time on the geometry merely note that the symplectic form at higher dimension is the sum of projections of 2D determinants. Ok, wait why?
But here's the fun thing (that maybe I've spoiled by talking about inner products first): while you're proving that there can't be a symplectic form in odd-dimensional spaces, you stumble upon a parallelism between symplectic complements, i.e. the sets
symp(W) = { u | w(u,v)=0 for all v in W}
and orthogonal complements, i.e. the sets
orth(W) = { u | <u,v>=0 for all v in W}
Namely that they're kernel sets of the bilinear maps <.,.> and w(.,.) (and because w(u,v)=0 implies w(v,u)=0, they're both left- and right-kernels). Moreover any vector z can be uniquely expresseed as
u+v where u is in W and v is in orth(W)
or alternately
p+q where p is in W and v is in symp(W)
So Riemannian geometry and symplectic geometry are like ways to split a vector space in complementary parts. Now, because there's Riemannian/euclidean geometry on a line and on the plane, you can build a geometry where there are planes orthogonal to lines, i.e. a 3D geometry. And maybe you have a timeline across which 3D spaces are strung together, ie. 4D space. But this 4D space of yours isn't symplectic. So it's realy hard to see.
So if you look for "symplectic geometry" on YouTube you're bound to find Dusa McDuff's lecture where she starts by writing in big bold letters in the blackboard:
4 = 3 + 1
4 = 2 + 2
... and that's a way to "see" four dimensions: to see entirely different geometries built on it.
Ok, think of two vectors in "three dimensions" (but not really). Draw X,Y,Z axes. So these pseudovectors will have a determinant which you can project on XY space and YZ space. Now, because our vectors have 4 dimensions and not 3, these projections will not be the same! Then your four-dimensional symplectic form (in some applications an "exterior product") is the sum of those two projections.
So you're pseudovisualizing a 4D object in 3D space as you might pseudovisualize a 3D object in 2D space but with a different concept of projection/perspective/etc. because we're not in the inner product geometry, we're in the symplectic geometry.
> Moreover any vector z can be uniquely expresseed as p+q where p is in W and v is in symp(W)
Of course, crucially, this is only true if `W` is non-degenerate; as you'll know, one of the important things about symplectic spaces is the existence of half-dimensional, totally isotropic subspaces. (In fact, there's nothing particularly symplectic per se about this issue; it is not the (conjugate-)symmetry of the usual inner product so much as its positive definiteness that means that no such extra non-degeneracy condition must be imposed. Indeed, the study of non-positive definite but symmetric inner products, as in the geometry of relativistic spacetime, carries its own challenges to intuition.)
That's a really cool point about language and how it influences our perception! Do you have a source for the orange vs. red thing? I'd love to read more about that.
I'm curious how this would look if the 4D space was projected onto the 3D space instead of taking a cross-section, much like we already project 3D space onto 2D space (your display), to create "3d" graphics.
Isn't that a 2d projection of the 3d projection of 4d space?
It gets weird because our eyes see almost in 2d, and even with VR, there is a 2d feed to each eye that gets combined to add slight 3d depth perception. So really we would be projecting a 4d world onto two 2d plains that our visual cortex would try to merge to add slight depth perception.
There are existing projects for this, for example [1].
I haven't tried OP demo yet (but I've spent a fair amount of time thinking about 4D and VR and implemented my own viewers), and I'm very convinced that the cross section approach is much much better if you want to deeply understand what is happening.
The perspective projection folds the fourth dimension into our hyperplane and you can't untangle it.
To paste something I wrote here [2], regarding the difficulty of groking the tesseract from the projected figure, I think it’s almost impossible to realize that there are points inside the tesseract that are not inside any of the eight cubes visible in the [projected] image, or that if you trace a diagonal between the corners of the tesseract you do not pass through any of the eight cubes.
There is much more space inside the hypercube than meets the eye in the perspective projection.
One application of these projections would be a point light source in the 4-th Dimension that would be obscured by solid 4-D toys and subsequently cast 4-D shadow "voxel maps" into the 3-D cube space. Having shadows and radiosity illumination would absolutely help with the "disappearing" cubes problem during game play. But it represents a substantial calculation for real-time physics and rendering engines.
If you make a Kickstarter to 4D print them, I would support it for my kids. I think for kids, it's more important to play with real-world physical objects rather than their virtual computer representation.
It's a joke - I don't actually have kids. And I mean the 2nd sentence seriously. Also, quick googling reveals that 4D printers haven't been invented yet, so presumably one would need a KS for those first.
As we represent on a 2d paper a projection of a 3d cube, we can print a 3d object that represents a projection of a 4d one. I'm down for the kickstarter.
George Hart made STL files for 3d projections of a number of 4d objects[0]. Some 3d printers include them as example prints One place I know even used Hart's 3d projection of 120 cell as a standard test print. It has a bunch of thin complicated features that machines have trouble with. One could tell immediately if something was wrong with the machine if the thin pieces were deformed or came out too brittle that normal handling broke them.
Does anyone have any resources on orthographically projecting 4D objects to 3D or 2D spaces? I'm curious if it looks better than taking a cross-section.
It seems like it should work similarly; deform a 4-frustum into a 4-cube and drop one or two of the axes. I guess the number of axes you can drop depends on the symmetry of the frustum...
What annoys me most: "thing disappear" I don't recall 3D -> 2D mapping making things disappear, just surfaces hiding other surfaces. But this might not work in 4D?
With the 2D->3D they are taking cross-section, I really don't like these. Just throw it all on there ! This would also mean you project your 4D world on a 3D camera, you project on a 2D surface to display.
I think the difference is projection versus cross-section. A video game is a 3D world projected on a 2D screen. The game is not a cross-section of the 3D world. In this game, you play within a cross-section of the fourth dimension, which you can move along a slider, sort of like a CT scan.
That having been said, I don't have any clue what a 4D projection into 3D space would look like. I suspect we would have much more difficulty understanding it.
Well look at the video for projection.
But you raise an interesting point, perhaps we could encode the 4the dimension in color or the stereoscopic channel (with VR gear) and remove this channel from the 3D part as we can do 3D without stereo.
That's if you do a "projection", like light rays from a 3D object focused onto a screen. This is more like taking a 3D slice of a 4D object. If you think of taking a 2D slice of a 3D object, you can see how objects disappear when they move out of the plane that you're drawing.
If you sliced a tesseract then presumably you'd encase it in the knife and a square, with no depth (the depth is in the 4th dimension that we're not experiencing), would appear?
Hypercubes have always been a difficult one for me to intuit, basically I have no 4th dimensional intuition. When I think of a 4D hypersphere all I can get is a simple sphere. I don't have any intuition as to whether that's wrong, it seems in a way it should be a sphere with infinite spheres on it's surface - from analogy with the tesseract - but from analogy of constructing a sphere from a circle it should be a case of rotating the sphere around itself perpendicular to the extra dimension?
A hypersphere has a similar description to a regular sphere or a circle, which is that every point a fixed distance from the center is part of the sphere. So I'd say it's a little boring. No matter what 3D section you take, it's just a larger or smaller sphere. It gets smaller if you take a slice off to the side, just like taking a slice of a sphere gives you a smaller circle near the side.
It might help if you consider spheres as a surface.
3D spheres are a 2D surface wrapped into a 3D space, likewise, hyperspheres would be a 3D surface wrapped into a 4D space. There's no "infinite spheres on its surface", I think the "rotation" is a better analogy.
Take a line rotated around an orthogonal axis and you have a circle, a circle rotated around an axis orthogonal to the other two is a sphere, a sphere rotated around another orthogonal axis is a hypersphere.
Now I'm happy to be completely wrong here, but I have a feeling that the disappearing of a 4D object in a 3D world cannot avoid using time to reveal the 4D nature of it.
Just like a in a 2D plane, to represent a 3D object and see all of it, you have to iterate though time to see it all, so parts will disappear from our perception at a given time interval.
I can't get this to run on iOS. Is it supposed to work? It just sits at the splash page playing a slightly interactive animation over and over. There's an arrow that I tried tapping and dragging and nothing happens.
One of the first things I thought of when I got a vive is building an app to let one intuitively navigate and understand four dimensional space. I never had the time or talent to hack something together though so I'm glad this exists.
Easily the hardest part of learning about string theory for me (via reading "The Elegant Universe" [1]) was grasping the idea of multiple other dimensions.
The book tried it's best to explain it by exploring a world starting with 1D and evolving to 3D, but it's still quite difficult to visualize, especially ones shaped like a "Calabi–Yau manifold" [2].
The one good thing I got out of learning about Calabi-Yau manifolds (and randomly reading another layman story involving Yau's clash with the guy who solved Poincaré conjecture) was a new interest in learning more about math and a getting a laymans grasp of topology. Although I later learned manifolds are quite an advanced subset of topology.
I enjoyed the linked video, I was looking for a way to better understand 4+D in a way I could wrap my head around and an interactive game makes a lot of sense.
it would be interesting to see multiple views at the same time, like how when we try to represent 3d shapes in 2d (think mechanical design drawing, archetictural drawings, etc)
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[ 2.7 ms ] story [ 136 ms ] threadThis is how you make a nation great. You can do whatever you want with the economy, but without proper education you still have a nation of.. people.. who choose their president.. based on their beliefs and understanding of the world.
This is the same person who made this game (that also looks like good fun): http://miegakure.com/ that I remember reading about some years ago but never got a chance to play with
My favorite short story of that era. Still have not watched the movie out of sheer fear they didn't get it and I don't want it wrecked. (Still have no idea how that even made it out...)
However, having only heard of hypercubes and not hyperspheres before I decided to see if there was anything useful about them online and I found this video that I just started watching and already 1 min 50 sec into the video something very interesting was said;
> Everybody knows what the sphere is. I'm thinking of a hollow sphere so like a basketball right. That's a two-dimensional surface living in a three-dimensional space. The hypersphere is generalized one dimension up, so in four dimensions you have this three-dimensional space called the 3-sphere or the hypersphere.
https://www.youtube.com/watch?v=krmV1hDybuU
Already this is telling me something that I have not heard before, and which I find much more helpful than talking about what a shape looks like from the perspective of someone living one dimension further down. That being said it was still useful having that explained as well, which is what Flatland: The Movie (2007) was about as well. Just this fact I quoted above was even more useful IMO.
6:19
> As complex numbers are to real numbers, quaternions are to complex numbers. It's like a way to build up even further. [...] Real numbers are one-dimensional. Complex numbers are two-dimensional. [...] For three dimensions there is no natural number system, but for four dimensions there is and it looks like this.
In a similar way, the quaternions are the space of transformations (scaling & rotation) of three-dimensional vectors. (It’s a little more complicated because 3-dimensional rotations are not commutative, and must be combined by sandwiching, so there are 2 choices of quaternion corresponding to every scale and orientation in 3-dimensional space. For an introduction see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)
You might be right, I don't know. Could you explain a bit more how you mean?
If you just have 2-dimensional vectors, there’s no obviously well-defined way to multiply two vectors and get out another vector.
In other words, both 2-dimensional vectors and complex numbers are made up of 2 coordinates, but they don’t have the same mathematical structure.
[0]http://www.gutenberg.org/ebooks/201
You know the theory of how language shapes your thinking? For example, in societies where there is no separate word for orange and red, they have extreme trouble telling the difference between them. In some native tribe where they use cardinal directions (North, South) - not relative (Left, Right) - they have an almost supernatural ability to know which direction they are without needing any other cues (sunlight, stars).
Point being - would being able to completely think in four dimensions have an impact on how you understand the world?
At least, that's what I remember learning in my linguistics courses in college. I'm willing to be wrong.
Colors may be the easiest way to see some limited sapir-whorf in action.
Language, in my understanding, operates at a lower level of abstraction in the brain than say, Math. I would suspect that Spacial Reasoning is similarly low-level. It would be interesting to see if there is any impact to the higher-level reasoning when the foundational blocks are played with.
The difference probably wouldn't be very large, I'm now realizing. People who are blind since birth, as far as I know, don't have any observable differences in common patterns of thought. And that seems like a larger difference in low-level thought than being able to understand four dimensions versus three. (Not suggesting blind people lack spacial reasoning in three dimensions)
I think if you had to have 4D vision rather than 3D, it would require much more processing power and memory to account for the extra dimension. So I think at minimum, a being that is capable of thinking in 4D would have to have higher brain capacity than a being that thinks just in 3D.
And there is some strong evidence that our ability to navigate 3D world influences our general cognitive capabilities. For example, consider https://en.wikipedia.org/wiki/Method_of_loci
In reality, you deal with high dimensional objets all the time: the position you're in is composed of dozens to hundreds of muscles rotating a few dozen joints.
We're fascinated by vision, but proprioception is where it's at if you want to think in high dimensions.
But the system quickly breaks down; the imitation fourth dimension is not in the same class as the other three. You can easily visualize a three-dimensional object rotating around a line in 3-space, or a two-dimensional object rotating around a point in 2-space, or a two-dimensional object rotating around a line in 3-space. Try applying that process to a "three spatial dimensions plus one color dimension" model and you'll notice you have no idea what color a point should be in the rotated object. (You might think it should be the same color as it was before the rotation, but that's just as wrong as saying that if you rotate a point in 2-space 30 degrees around the origin, it should end up with the same x-coordinate that it started with.)
Proprioception is deeply tied to a three-dimensional conception of space -- that is how you perceive its output. And the evidence is not strong that it is a function of muscle activation, since it will work just as well if you relax while someone else moves your hand around.
And of course it's tied to 3 dimensional space -- the whole point is to couple the correspondence between a high dimensional phase space and a blob in 3 dimensions.
But you do have an awareness of the phase space, not just the 3D location -- and you can use it to think about higher dimensional objects.
Why can't we do the same thing with 4d? Why does the object just disappear when it bounces into the 4th dimension, can't we maybe see a projection of it onto the 3rd dimension?
Every 4d object would end up being a clear 3d cross-section where it intersects with our 3d world, plus a cloud of superimposed and increasingly hazier 3d cross-sections above and below us along the 4d ("w") axis, projected down onto w=0 3d-space.
Yes, you can. This program just doesn't.
Now contrast that to if you were plucked up vertically 'above' the game's level to look down upon it. Now you can see the entire 2D extent of the maze at once. Before, your vision was blocked by the walls, now you see the walls and what's on the other side of the walls simultaneously in a way that's entirely distinct from simply seeing through a transparent object.
Now, like seeing a 1D amount of information about a 2D maze while live inside it, we see a 2D amount of information about a 3D world around us (a picture demonstrate's this 2D amount of information - it's planar). Now imagine being lifted out the 3D plane of existence so that you could behold the entirety of the 3D world at once. That's the rough analogy.
Ofc it is abstraction, I know that physicists aren't happy with heterodimensional settings at human scale.
Now apply the same analogy to our world: our 3D world is just an infinitely large 3D plane in a 4D world. When the objects aren't in our 'plane', we can't see them.
Edit: Didn't notice it was mentioned at the end of the comments, https://news.ycombinator.com/item?id=14472395, there's a free version linked from archive.org.
Aside from all the misogyny and the style of writing, you mean?
It was really disorienting (the 4d, but also the eye-crossing), and like the post author says, just bundles of lines rather than solid shapes. Still very cool.
Ooh, I found it!
http://www.urticator.net/maze/
http://www.hidden-3d.com/games_stereogram_tetris.php
http://www.hidden-3d.com/stereogram_games.php
But I'd still like to see the difference.
Put it this way: Riemannian (and as a special case, "ordinary") geometry is based around the inner product <a,b>, which gives you both angles and projection (and 2D shadows from 3D objects, for example) and lengths. You learn a lot about this (not in the "curve" differential-geometric sense, but most of the intuition carries over) in college linear algebra: you learn to think about entire vector subspaces having orthogonal complements as defined by the inner product. On the other hand: length (even if curved) in that direction is the same as length in this direction, i.e. the inner product is a symmetric bilinear map.
Symplectic geometry is built on a skew-symmetric bilinear map (the symplectic form) so that w(u,v)=-w(v,u). In 2D, the symplectic form is equivalent to the determinant: note that this is an oriented or signed area rather than an absolute-value one. And then in 3D, there is no symplectic form (it's rather fun to prove this) such that there is no vector (apart from the origin) that collapses all the other.
Anyway, the challenging thing about this, writing my dissertation, is that unlike with linear algebra where you can gradually extend intuition from the real line to the plane to 3D space and then think "ok, I think I can grok this in 500 dimensions" (and go do multivariate statistics, for example), in symplectic geometry there is either the trivial case (and again, this is the determinant and not particularly "new" to you) or the 4D case next.
And then you can't draw pictures. YOU CAN'T DRAW PICTURES. This is such an intuition-fucker. Books that need the symplectic form right away and can't waste time on the geometry merely note that the symplectic form at higher dimension is the sum of projections of 2D determinants. Ok, wait why?
But here's the fun thing (that maybe I've spoiled by talking about inner products first): while you're proving that there can't be a symplectic form in odd-dimensional spaces, you stumble upon a parallelism between symplectic complements, i.e. the sets
symp(W) = { u | w(u,v)=0 for all v in W}
and orthogonal complements, i.e. the sets
orth(W) = { u | <u,v>=0 for all v in W}
Namely that they're kernel sets of the bilinear maps <.,.> and w(.,.) (and because w(u,v)=0 implies w(v,u)=0, they're both left- and right-kernels). Moreover any vector z can be uniquely expresseed as
u+v where u is in W and v is in orth(W)
or alternately
p+q where p is in W and v is in symp(W)
So Riemannian geometry and symplectic geometry are like ways to split a vector space in complementary parts. Now, because there's Riemannian/euclidean geometry on a line and on the plane, you can build a geometry where there are planes orthogonal to lines, i.e. a 3D geometry. And maybe you have a timeline across which 3D spaces are strung together, ie. 4D space. But this 4D space of yours isn't symplectic. So it's realy hard to see.
So if you look for "symplectic geometry" on YouTube you're bound to find Dusa McDuff's lecture where she starts by writing in big bold letters in the blackboard:
4 = 3 + 1 4 = 2 + 2
... and that's a way to "see" four dimensions: to see entirely different geometries built on it.
Ok, think of two vectors in "three dimensions" (but not really). Draw X,Y,Z axes. So these pseudovectors will have a determinant which you can project on XY space and YZ space. Now, because our vectors have 4 dimensions and not 3, these projections will not be the same! Then your four-dimensional symplectic form (in some applications an "exterior product") is the sum of those two projections.
So you're pseudovisualizing a 4D object in 3D space as you might pseudovisualize a 3D object in 2D space but with a different concept of projection/perspective/etc. because we're not in the inner product geometry, we're in the symplectic geometry.
Of course, crucially, this is only true if `W` is non-degenerate; as you'll know, one of the important things about symplectic spaces is the existence of half-dimensional, totally isotropic subspaces. (In fact, there's nothing particularly symplectic per se about this issue; it is not the (conjugate-)symmetry of the usual inner product so much as its positive definiteness that means that no such extra non-degeneracy condition must be imposed. Indeed, the study of non-positive definite but symmetric inner products, as in the geometry of relativistic spacetime, carries its own challenges to intuition.)
It gets weird because our eyes see almost in 2d, and even with VR, there is a 2d feed to each eye that gets combined to add slight 3d depth perception. So really we would be projecting a 4d world onto two 2d plains that our visual cortex would try to merge to add slight depth perception.
I haven't tried OP demo yet (but I've spent a fair amount of time thinking about 4D and VR and implemented my own viewers), and I'm very convinced that the cross section approach is much much better if you want to deeply understand what is happening.
The perspective projection folds the fourth dimension into our hyperplane and you can't untangle it.
To paste something I wrote here [2], regarding the difficulty of groking the tesseract from the projected figure, I think it’s almost impossible to realize that there are points inside the tesseract that are not inside any of the eight cubes visible in the [projected] image, or that if you trace a diagonal between the corners of the tesseract you do not pass through any of the eight cubes.
There is much more space inside the hypercube than meets the eye in the perspective projection.
[1]: https://www.youtube.com/watch?v=BFmDLUUhZjE [2]: http://xn--1-2fa.fr/blog/down-the-4d-rabbit-hole-navigation/
[0]http://www.georgehart.com/rp/rp.html [1]http://www.georgehart.com/rp/120-cell-george-hart.stl
https://en.m.wikipedia.org/wiki/Flatland
https://en.m.wikipedia.org/wiki/Flatland
It seems like it should work similarly; deform a 4-frustum into a 4-cube and drop one or two of the axes. I guess the number of axes you can drop depends on the symmetry of the frustum...
Edit: In particular, this page shows a good summary of how the "slices" shown in the game relate to the hypercube projection you're familiar with. http://eusebeia.dyndns.org/4d/vis/09-interp-1#Interpreting_4...
With the 2D->3D they are taking cross-section, I really don't like these. Just throw it all on there ! This would also mean you project your 4D world on a 3D camera, you project on a 2D surface to display.
https://www.youtube.com/watch?v=BVo2igbFSPE <= this method is "saner" imo.
That having been said, I don't have any clue what a 4D projection into 3D space would look like. I suspect we would have much more difficulty understanding it.
Hypercubes have always been a difficult one for me to intuit, basically I have no 4th dimensional intuition. When I think of a 4D hypersphere all I can get is a simple sphere. I don't have any intuition as to whether that's wrong, it seems in a way it should be a sphere with infinite spheres on it's surface - from analogy with the tesseract - but from analogy of constructing a sphere from a circle it should be a case of rotating the sphere around itself perpendicular to the extra dimension?
A hypersphere has a similar description to a regular sphere or a circle, which is that every point a fixed distance from the center is part of the sphere. So I'd say it's a little boring. No matter what 3D section you take, it's just a larger or smaller sphere. It gets smaller if you take a slice off to the side, just like taking a slice of a sphere gives you a smaller circle near the side.
3D spheres are a 2D surface wrapped into a 3D space, likewise, hyperspheres would be a 3D surface wrapped into a 4D space. There's no "infinite spheres on its surface", I think the "rotation" is a better analogy.
Take a line rotated around an orthogonal axis and you have a circle, a circle rotated around an axis orthogonal to the other two is a sphere, a sphere rotated around another orthogonal axis is a hypersphere.
Just like a in a 2D plane, to represent a 3D object and see all of it, you have to iterate though time to see it all, so parts will disappear from our perception at a given time interval.
Looking forward to the be build.
Something similar but less polished can be found here: http://www.albert-hwang.com/blog/2016/6/what-does-vr-reveal-...
The book tried it's best to explain it by exploring a world starting with 1D and evolving to 3D, but it's still quite difficult to visualize, especially ones shaped like a "Calabi–Yau manifold" [2].
The one good thing I got out of learning about Calabi-Yau manifolds (and randomly reading another layman story involving Yau's clash with the guy who solved Poincaré conjecture) was a new interest in learning more about math and a getting a laymans grasp of topology. Although I later learned manifolds are quite an advanced subset of topology.
I enjoyed the linked video, I was looking for a way to better understand 4+D in a way I could wrap my head around and an interactive game makes a lot of sense.
[1] https://www.amazon.com/Elegant-Universe-Superstrings-Dimensi...
[2] https://www.wikiwand.com/en/Calabi%E2%80%93Yau_manifold