Physicists and mathematicians who think about higher-dimensional spaces are, if they allow their interest to somehow become public knowledge, inevitably asked: “How can you visualize more than three dimensions of space?” There are at least three correct answers: (1) You can’t. (2) You don’t have to; manipulating abstract symbols is enough to help you figure things out. (3) There are tricks to help you pseudo-visualize higher-dimensional objects by cleverly projecting them into three dimensions; see here and here.
But really, why can’t we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?
I can think of a couple of explanations why it’s so hard, with different ramifications. One would be simply that our imaginations aren’t good enough to project our consciousness into a constructed world so very different from our own. Could you, for example, really imagine what it’s like to live in two dimensions? Sure, you can visualize Flatland from the outside, but what about asking what it’s like to really be a Flatlander? The best I can do is to imagine a line, flickering with colors, surrounded by darkness on either side. But the darkness is still there, in my imagination.
The other possible explanation is that the process of visualization takes up a three-dimensional space in our actual brain, preventing us from “tuning a dimensionality knob” on our imaginations. The truth is certainly more complicated than that (and I’m not experts, so anyone who is should chime in); the visual cortex itself is effectively two-dimensional, but somehow our brain reconstructs a three-dimensional image of the space around us.
Maybe this could be a new tantric discipline: visualization in higher dimensions. Or maybe the Maharishi already offers a course?
Actually we do a great deal of visualizing in higher dimensions, we just don’t realize it as such because we get a little stuck on our word “visualizing.” What do I mean? For instance, we all are engaged in processes like planning, experiencing, feeling, creating, remembering, even –dare I say– loving. These life processes, and myriad others, take place in the four dimensions of space and time, and they are the very texture of higher dimensional “space” (probably only a metaphor after height, width, and depth), as well as the facts of life. Clearly, it is the infinite variety (and infinite reality!) of these processes that are wrapped up in the higher dimensions of space-time. From this point of view, the present moment is a black hole consuming the past (and spewing out the future?), compressing all time into an infinitely dense moment.
In fact we are all going “back” to 2D. IT people produce a picture in 2D and call it “3D” just because you can turn the object around and see previously hidden features. According to their concept all movies are in 3D!
As one of the older guys who have been trained in technical drawing before Autocad, I still can “see” a 3D object in a 2-view drawing.
>3 dimensions could be correctly represented by a correspondingly larger number of projections. Taking a subset of 2 views only could give you indefinition / aliasing effects. Side and front views would refuse to match. This is probably why you can’t carve a Donald Duck doll yielding the right cartoon-like views!
Sean, I can imagine living in 2 dimensions without much problem at all, since I grew up playing Nintendo. (I guess it wouldn’t work with this newfangled Wii…)
Four or more dimensions are of course harder. Penrose once gave a description of visualizing higher-dimensional space that matches my experience: you just picture a weird 3-dimensional space, with spheres, cubes, vectors, and whatever other objects you’re interested in, and then you set 3 equal to n. 🙂
The answer seem fairly obvious: we can only visualize 3 dimensions because we can only move in 3 dimensions, and the main (but not necessarily only) purpose of imagination is to simulate movement through space – our brains are optimized through evolution to do exactly that. Any creature that could only visualize two dimensions would be at a selective disadvantage to any creature that could also visualize a third dimension, such as height. On the other hand, being able to visualize more then 3 dimensions would also be disadvantagious to survival, since it would require much more complex wetware to simulate an extra degree of freedom that can’t even be moved around in. This logic implies that if we lived in a universe with 4 spatial dimensions, we most likely would have evolved the ability to directly visualize these.
Visualizing functions of a complex variable and 24 Views of the Complex Exponential Function are an exploration thinking about picturing functions C -> C in four dimensions in a variety of ways.
It’s easy. Just imagine n-dimensional space and set n to whatever you want. Even non-integers, and complex numbers. 🙂
I don’t think it is possible to visualize 4 spatial dimensions, but I have experienced something you could call “pseudo-visualization” of it.
Some years ago I saw a painting I was very fascinated by, because it had a special property: it was easy to view it as both being totally flat (2-dimensional) or having perspective (3-dimensional). Or to be more correct: you could freely choose to view it as being flat or not.
I found out, that if I systematically shifted back and forth between the two states for some time, the painting sometimes seemed to become “something more than 3-dimensional” in my imagination.
But this effect is probably only some kind of visual illusion. I guess that a true 4-dimensional visualization would be something you could keep constant for some seconds?
I think the best we can do with our poor 3D brains is imagine 3D slices of 4D objects, but let’s assume it’s not impossible to gain some kind of comprehension of a 4th dimension by watching these slices. I looked up an animation of a hypercube to help me:
http://www.youtube.com/watch?v=CtSNStVW81M
But I didn’t find it that helpful.
It got me wondering, though. When you think of it, this animation is a 2D projection of a 3D slice of a 4D object. It’s a projection of a projection, which means it’s got to really make those two dimensions work hard. Plus, it just rotates in a particular way. What if I could somehow grab the hypercube and push it around, turn it and roll it however I liked. What would it do? You know how in some graphics programs, if you’re manipulating the projection of a 3D object, it gives you little “handles” you can grab to rotate on all three of its axes? Would it be possible, say using virtual reality goggles, to give a person a stereoscopic view of a hypercube, such that they felt like they were actually looking at in in a 3D space, and give it those “handles” so that it could be manipulated in analogous way to a 2D projection of a cube in a 3D rendering program? Would that help us “see” the 4th dimension better?
Something else I’m pondering: I got thinking about an eye that could see in 4D last night, and realized I was cheating. I imagined this gelatinous sphere filled with rods and cones, and light can come in from all directions, somehow being focused by a 4D “lens” on different depths of the retina. But that doesn’t really capture the weirdness of it. I got thinking about a CCD in a camera. It’s essentially a 2D slice of silicon with an array of elements that absorb photons and turn it into an electrical signal. What kind of CCD would you need for a 4D camera? Could it be a solid block of silicon, filled in all three dimensions with photoelectric elements? I ask because it could be completely opaque beyond its outer surface, but presumably photons could still reach its interior elements via the 4th direction. They would have to for it to work, right? Well, this got me thinking, what if I wanted to take “stereoscopic” images, like the two cameras on the Mars rovers that let us get virtual 3D pictures of the planet? I’d need some number of 3D CCDs, I’m guessing more than one, but could 2 do the job? To perceive distance using parallax, I think two eyes work equally well for flatlanders and us (linelanders must be pretty challenged for depth perception, since their view of the entire world is a mathematical point). Would they do the trick for hyperspace denizens as well?
Fun questions, even if I never find out the answers. I think somehow addressing them could be part of a really fun museum exhibit.
It might be worthwhile to explore how one could construct a lens that could focus photons from a 4D space onto a 3D embedded disk (roughly speaking, of course).
You guys are forgetting that 3D visualization is not a conscious activity. I’m sure that if you brought up a baby with 4D blocks, toys, dolls, action figures etc. to play with, they’d be able to visualize them just fine.
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It could be that some autistic savants can visualize higher dimensional spaces. There are people who can tell you on what day of the week a certain date corresponds to within seconds. If you ask them how they do that, they say that they don’t do any computations, they can simply visualize the answer.
But they can’t explain to us what the picture they see looks like. This could be because the picture they are seeing, which indicates the weekday, is a 4 or higher dimensional picture.
It could be that such skills are due to privileged access to raw information that normal people don’t have access to. Some experiments have been done in which certain brain functions in normal people were inhibited which led these normal people to temporarily get savant like skills. See e.g. here:
http://www.centreforthemind.com/publications/SavantNumerosity.pdf
http://www.centreforthemind.com/images/savantskills.pdf
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We ‘see’ in 3D using eyes that form 2D images, so we should be able to visualise in 4D just using 3D images. We’d have to simultaneously be able to see all voxels of the 3D image though, not just the surface, but one can imagine training themselves to visualise all voxels in a 3D 10x10x10 grid (so only 1000 distinct points), and then using this mental visualisation surface to see the surface of 4D objects and manipulate them … further practice could then improve the resolution …
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Here you can see some animations I made of everyday objects rotating in 4-dimensions:
http://spacesymmetrystructure.wordpress.com/2008/12/11/4-dimensional-rotations/
(they are stereographically projected onto a 3-sphere, rotated and projected back down to flat 3-space)
I find the way the ‘poles’ of such a rotation appear particularly fascinating.
We cannot see in the fourth dimension is because time is an illusion.
Sean,
I played around with this a few years ago, and would say that it is quite possible to visualize four dimensions. After all, the images we receive from our eyes are two-dimensional, and we are able to visualize three dimensions based on that. Taking a step further is not so different. As a first step, I started out by imagining points in three dimensions, and pictured them darker or lighter depending on how close they were in the direction of the fourth dimension. This got me used to the concept. Eventually, I was able to drop this aid, and simply visualize the fourth dimension as any of the other three. One gets a feel for that not only is there space to move in as in up/down, right/left, front/back, but one more direction. It was definitely an aha-moment. I got to the point where I could rotate simple objects like squares around planes (rather than an axis as in 3d). But the main difficulty is the sheer amount of information contained in the most simple four dimensional objects. Try keeping track of all the corners of a 4d hypercube, and you’ll know what I mean!
Perhaps if the atomic structures of the material that forms some hypothetical newly evolved cortex of the brain – if this atomic material were to contain an axis lying in a fourth direction; let’s say a .0000006 seconds into the future and .0000005 seconds into the past, and with particles at these distances, then information four space might be able to be processed into perceptual interpretation.
Perhaps if the atomic structures of the material that forms some hypothetical newly evolved cortex of the brain – if this atomic material were to contain an axis lying in a fourth direction; let’s say a .0000006 seconds into the future and .0000005 seconds into the past, and with particles at these distances, then information of four space might be able to be processed into perceptual interpretation mediated through the 4-D neurons of this cortex.
Hi there,
Something that bothers me about the title of this article is that philosophers and psychologists have for centuries maintained that we can experience time as a dimension. Subjective time is known as the “specious present”.
See Time and conscious experience for an empirical description.
I used to advocate using time and color (and sometimes a second time dimension) to aid visualizing additional dimensions, but found that this approach has severe limitations when I needed to compute Voronoi cells of lattices in 6 dimensions (because the dimensions use different units I can’t compare distances meaningfully… also rigid body transformations are difficult (try rotating from the time dimension towards the color dimension while preserving distances)). Eventually I got a bit of a “knack” for visualizing lattices in 6 dimensions. Hard to describe, though; it “looked” 3D, but weirder. It hinges on the lattice being highly symmetrical.
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