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?
Having just worked on a project involving lots and lots 4dimensional rigid geometry I would say that while visualization is a hard coded feature of our brains, we can learn to imagine 4d. For example when talking to my coworkers I could say things like: “No there is a counterexample, imagine the rotation orthogonal to a face of the polytope, it will by neccesity intersect any other plain that has property xyz…”
I would say that anyone trained in mathematics is capable of imagining abstract structures for which we have no hardcoded visualization. This is the basis of our intuition, and the fact that we can make structurally correct conjectures that yet take significant technical work to corroborate.
If by “visualize,” you mean a true picture, you’re going to have a bit of a data problem to start with. By your analogy, a flat-lander in fact has 1-D imaging, constructed to allow for 2-D inferences. We have likewise a projected 2-D view, but with binocular vision, we can infer 3-D properties. But 1) consider the data difference between a colored flickering line and a full pictorial view. Going to 1 higher dimension would in effect require the ability to hold a tremendous number of instantaneous 3-D views. A 4-D being would have tremendous awareness of 3-D world. You could examine the interior of any solid object, read all the books on a self without opening them, etc. You quickly see how the visual information adds up just going to a 4-D world! It is interesting, that our 2-D binocular vision always presents projected views, but ones that allow 3-D inferences. For example, one can understand the information storage in a closed book, even if one cannot picture the ability to see it all at once except as laying out all the pages on a flat 2-D surface.
A interesting question, however, is the extent to which one can develop spatial intuitions in higher dimensioned space. I would claim that some level of this is possible, but like all such training it is likely to require prolonged immersion and practice.
Define “visualize.” How can we say that the mental “image” a mathematician constructs of a 4-dimensional polytope is qualitatively different that the image she constructs of a 3-dimensional polytope. These are mental constructions – one labeled 4-d, the other 3-d – and both consisting of sequences of firings of neurons in specific areas of the brain.
Perhaps that’s how our brain evolved just to survive in the environment. Having ability to visualize in > 3-d probably never helped to obtain food and have more sex to produce offspring. If understanding general relativity ever helped in attracting opposite sex, then I would think at least a fraction of human beings would have evolved to visualize things in more than 3-d by now. It just hasn’t happened.
It is certainly possible to visualize 4d by taking time as the fourth dimension. Back in high school, while working as a fry cook at Wendy’s, I used to amuse myself by visualizing 4d spheres and cubes. For example, a 4d sphere starts as a point, quickly “inflates”, passes the tipping point and starts to “deflate”, ending in a point.
Imagine a person blind from birth – suddenly able to see – they would have no idea of perspective or distances…………
Most probably it’s a combination of evolution and learning. All our ancestors lived in 3d world, so whatever wetware we have preinstalled in our brain it’s already configured for processing 3d environment. All our learning, especially in childhood also have as input mostly 3d. There are few multiparametric system human child or youth have to visualize a lot. Like human face expression reading, or hand-to-hand combat. BTW human face expression space is essentially multidimentional and was used successfully to label multidimensional arrays of data.
Everyone’s overlooking the obvious explanation here: our four dimensional overlords crippled our dimensional perception to keep us all in our place. About the same time they sabotaged our ability to self-manufacture vitamin C. If you translate the book of Genesis into hexadecimal, it’s all there.
Taking over from daisyrose:
There must have been work done on the visual acumen of children. Has noöne really evertried to make toddlers (and kindergarteners) understand 4D?
I can’t even doodle in 2D, myself, and my recent attempt to read my geometry book from five-odd years ago was pretty much a failure. I have a book on Riemannian geometry – I have no recollection of ever using it. I honestly do not know if I’ve taken a course and exam with it, or if I dropped out. I’ll have to check my transcript. (I’m scared …)
I think it is important to realize how amazingly GOOD we humans are at “visualizing” 3d objects – consider driving, catching a ball, or even walking, and then consider how nearly impossible it is to make a computer which does the same tasks. We process 2d images (together with stereo vision for some additional data) at incredible speed, making a 3d model of the world around us in milliseconds. So realize how much hardware and software our brains have at their disposal for dealing with 3d.
And it is obvious why it is so – for evolutionary purposes. We live in a 3d world, so it is obviously incredible *useful*, in an evolutionary sense, to be able to interpret that 3d world as good and as fast as possible. The reason 4d, and any higher dimensions, is hard to visualize, is because we simply have no hardware or software to deal with it. Asking the brain what 4d looks like is like asking it is what purple sound like, or asking a computer to make coffee. It simply makes no sense…
By the way, this can be generalized for just about all “intuition”… Intuition for non intuitive things, actually means, using the evolved parts of our brain which have grown for obvious *useful* reasons, for completely different purposes. Just like we “intuitively” feel that an electron is a ball traveling around an atom, does not mean it is anything like that at all. That is simply our “middle world” analogy for this non-intuitive scenario.
The easiest way to come close, I’ve found, is to think of it like a movie – a sequence of frame slices where the frames are 3-d cubes. Then you can imagine lining the cubes up or ‘playing’ them like a movie.
Well, I suspect the problem may well be a computational one. If we consider the simple fact that if we have seen an object from one angle, we can, with a very high degree of accuracy, recognize that same object from a different angle, it seems apparent that our brains have some mechanism for producing rotations in three dimensions. I would contend that the ability to mentally rotate objects is a necessary component for visualization in three dimensions.
Granted, I don’t expect our brains to do these rotations perfectly, but rather that they have some mechanism that produces a similar result, likely using some very interesting ad-hoc approximations.
Now, what would be required for us to do four-dimensional rotations? Well, things get a whole lot trickier. First, rotations in 3D can be fully represented by three numbers, while rotations in 4D require six numbers. So right away you need twice the amount of memory storage to even think about rotating something in 4D. Then there’s the problem that many of the shortcuts that work for 3D rotations just aren’t going to work for 4D rotations, such that much of our 3D machinery is likely to be completely useless.
So, I suppose it is [b]possible[/b] for us to visualize 4D objects, but only with dramatic changes to brain wiring that significantly alter the structure and size of whatever part of our brain manages visualization. I doubt it’d be possible to learn, but it shouldn’t be impossible with some fancy biological engineering (that we are very, very far away from).
I think Oded has it when he says we simply don’t have the hardware to visualize in more than three dimensions. The term visualizing simply means constructing an internal representation of objects in external three-dimensional space. We trust that our “visualization” is good because it imitates in many ways our experience of reality. Perhaps, since we never experience true four-spatial-dimension objects we could never in fact trust our visualization of such objects.
While it certainly seems that some people develop strong intuitions about objects in higher dimensions, intuitions are not identical with visualizations. Mathematicians have strong intuitions about all sorts of objects, and often fuel this intuition by mental pictures. But I think that it is misleading to say that a mathematician can therefore visualize, say, the monster group.
It’s interesting to speculate on whether or not a creature in 3 dimensional space could somehow evolve–or be created–to “visualize” four dimensional objects.
If an autostereogram allows you to see a simple 3D image by squinting suitably at a periodic 2D image, I wonder if one could make a suitable periodic 3D model (or even a 2D perspective image of one, possibly animated) and squint at that in the same way to see or clearly sense a rudimentary 4D image!
Yeah. It is hard to make two 2 dim planes intersect in a point as in >3dim, or two R3 intersect in a 2 dim plane (in>3dim) or in a a point (in>4 dim), without using a n-dim coordinate system (x,y,z,…). Also it is easy to count nfaces in m>n cube etc.
Or you can turn inside of a 3ball to outside via 4th dim and think that this is just the difference between night an day…
A parametrized view (3 dim movie):moving cube, rotating earth is not so good.
Actually, if we use the broader term of dimension, taking it from just spatial to any sort of observable quantity, then visualizing another dimension on top of what we can do isn’t so hard. Take a 3D object and add say… temperature to the object. If it’s hot on one side, cold on the other, then having the color vary from red to blue in your head gets you an extra dimension. You may say that’s not fair if you’re talking about a property with a variable dependence upon spatial position, but I think it still qualifies.
Michael
If you ask someone to visualize an object in their mind, while there is an absence of any visual stimulus, many of the same regions of the visual cortex light up as when they actually look at something. The higher faculties of the brain are built on the lower, and use the same basic structures, which ultimately boil down to the sensory apparatus. We have no senses for perceiving 4-d objects, so there is no way to ‘visualize’ them.
Being a highly visual creature, much of our cognitive ability is built on the logic of vision and sematosense.
It not so hard to imagine other dimensions, its not so hard to imagine that being born was just one probablity out of many that brought me into this world. Including an independent theory of the creating of my self; independent of all other events! If fact I find it hard to accept that, multi-dimensional thinking is not the norm, right up to using strings to create my own worlds and using nothing as the ultimate tool to stay alive. The hardest thing seems to be escaping from myself, but I see no reason why evolving into the imagination is not possible. Then only problem I have about imagining more dimensions is that; it’s not possible to imagine such things and live a “normal life”. I have to work, I have to look after my son and love my wife and I need to have friends. You can’t live looking at the all other times you died its too hard to accept, you can’t live knowing you created your own existence. Your better off just telling yourself your not god, even if you really are. Really you should think your selves lucky that you can’t imagine a world that has not even made up its mind whether you have been born or not. 3d living harsh enough, my advice is stay well away from the rest. Mankind is trillions and trillions of years away from being able carry on existing once you know and accept, what other dimensions mean to your place in the universe.
Qubit
On the other hand statisticians, probabilists and Hilbertinists “visualize” their own way?
Ideas shouldn’t project on each other. Kind of minimal information principle.
Maybe this is a silly question, but how would light propagate in a 4+1D space? If one imagines an idealized polarized beam of light, it’s got electric and magnetic waves oscillating at right angles, propagating orthogonally to both those vectors, and at a frequency, which, if it’s within a certain range, the photoreceptors in our eyes can detect, and which we perceive as having a color. I suppose with that color perception we “see” the time dimension indirectly, since the color is related to the amount of time between crests and troughs in the wave. But let’s just say there were an extra, macroscopic spatial dimension. What does that do to a classical path between a light source and the eye? In what direction could electromagnetic fields oscillate and still be recognizable to a human eye as such? Even if a fourth spatial dimension was there, could we see it with the wetware we’ve got? I guess I’m not sure what it even means to “see” a higher-dimensional space. My understanding is, with another direction to move in, I might be able to enter my house without opening a door. If fact, I could enter a box with no opening that a 3D person could perceive. I just walk “around” all the walls, floor, and ceiling. Could I see where I was walking? Would the inside of the box be illuminated via this other direction, like the interior of an open box being lit from above in 3 dimensions? We see with light. We use parallax from two 2D projections on our retinas to allow us to perceive depth. I think there may be something to the notion that we can’t “see” four dimensions because our brains lack the fourth direction. But the same is true of our eyes! Our “retinas” (whatever such an organ might be) would have to be able to capture some number of 3D projections, and use the equivalent of parallax (What would it be called? Hyperparallax?) to give this extra dimension “depth” (hyperdepth?). It find it rather mind-blowing to consider what it means, physically, and physiologically, to “see” four dimensions, when considering the question of why it is we can’t.
Anybody know much about the case of Alicia Boole Stott, daughter of George Boole? Her bio at Biographies of Women Mathematicians (online) says that she “possessed a great power of geometric visualization in hyperspace”. I don’t really know if her version of “visualization” coincides with what we’re talking about.
Low Math, that’s actually a very good question — or, more accurately, a large number of questions, which I can’t begin to answer. But for the issue of polarization vectors, first try to answer it yourself for only two dimensions of space. Clearly something has to go wrong, as you can’t have both the electric field and the magnetic field vectors perpendicular to the motion of the wave.
The answer to that one is that the magnetic field is not a vector in 2+1 dimensions, it’s just a scalar. Similarly, it’s not a vector in higher dimensions, it becomes a higher-rank tensor. There are a lot of details lurking there, as you might expect.
But light would still move in straight lines, even in higher dimensions. What the ramifications are for higher-dimensional life, I’m not sure.
I think Tod Lauer touched on something important. The visual system solves a problem of image reconstruction (loosely speaking, from a dynamic n-1 dimensional array of “pixels” to a dynamic virtual n dimensional array of “voxels”) that we know to be highly non-trivial. How do we know that this reconstruction can even work when n is not equal to 3?
At some level it seems that this question must be tied to a deeper question; can the laws of physics really “work” at anything other than 3+1 dimensions? In other words, can we say that the number of spatial dimensions is arbitrary and contingent? I believe there is considerable evidence to the contrary.
——————
Apparently off-topic, but I think you’ll see a connection:
The Truth About Autism (Wired)
If one attempts to train oneself to visualize a higher dimension – how could you ever know you got the visualization correct? None of us has an experiential reference point for this.
We can do all the math we want, but in the end, everything we present on paper gets flattened out to 2 dimensions, even if it attempts to co-erce out brain to get a 3rd dimension out the picture.
That being said, perchance when we truly use 3D representations to convery information, we might be able to coax a pseudo 4th dimensional feel or understanding to it.
Hmmm. Not quite sure how a magnetic tensor field would behave (I’m guessing it won’t be the same as gravity…but not entirely different), but thanks for the response! This stuff is definitely very interesting and challenging food for thought.