Why Can't We Visualize More Than Three Dimensions?

By Sean Carroll | March 30, 2009 10:29 am

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?

CATEGORIZED UNDER: Mathematics
  • fh

    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.

  • Tod R. Lauer

    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.

  • Mike

    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.

  • ts

    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.

  • Chris

    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.

  • http://www.dorianallworthy.com daisyrose

    Imagine a person blind from birth – suddenly able to see – they would have no idea of perspective or distances…………

  • http://mirror2image.wordpress.com Serge

    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.

  • Matt B

    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.

  • Sili

    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 …)

  • Oded

    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.

  • BlackGriffen

    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.

  • Jason Dick

    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).

  • Bruce S.

    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.

  • John R Ramsden

    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!

  • Hessu

    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.

  • http://www.reciprocalspace.net michael s pierce

    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

  • efp

    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.

  • http://quantumnonsense.blogspot.com/ Qubit

    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

  • Hessu

    On the other hand statisticians, probabilists and Hilbertinists “visualize” their own way?
    Ideas shouldn’t project on each other. Kind of minimal information principle.

  • Low Math, Meekly Interacting

    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.

  • SI

    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.

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • Chris W.

    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)

  • Tom

    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.

  • Low Math, Meekly Interacting

    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.

  • http://www.stefansaal.com Stefan

    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.

  • Snel

    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!

  • http://www.scottaaronson.com Scott Aaronson

    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. :-)

  • Chairbreaker

    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.

  • Ron

    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.

  • yesfm

    It’s easy. Just imagine n-dimensional space and set n to whatever you want. Even non-integers, and complex numbers. :-)

  • http://mogmich.blogspot.com/ mogmich

    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?

  • Low Math, Meekly Interacting

    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.

  • Richard

    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).

  • http://lablemminglounge.blogspot.com/ Lab Lemming

    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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  • Count Iblis

    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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  • Amir

    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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  • http://spacesymmetrystructure.wordpress.com/ Daniel

    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.

  • Thomas C.

    We cannot see in the fourth dimension is because time is an illusion.

  • invcit

    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!

  • schriAlphi

    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.

  • schriAlphi

    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.

  • http://newempiricism.blogspot.com/2009/02/time-and-conscious-experience.html John

    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.

  • S

    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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  • http://aclinks.wordpress.com/ Successful Researcher

    Great post and great discussion!

  • geraint

    hey im into this stuff: 2dimensions could be our eyes as we have 2 eyes, the third could be our imagination-brain as the brain does prossess what we se and therefore we can get a perception. and without our eyes we wouldnt see any dimention to imagin or process.

  • Nema Mansuri

    The nature of extra dimensions may be different from our standard three dimensions. This difference can explain how extra dimensions may not affect people’s experience. There are two ways of having the extra dimensions different from the 3D. First, one or more of the particles you see in the known universe may not be able to propagate into higher dimensions. This would explain why we’re not able to move in the fourth dimension. Secondly, the extra dimensions may be compact. If they are SO tiny (hopefully not) would explain why us (macroscopic being) do not notice them. Nema

  • Nema Mansuri

    I disagree with the one post above! Saying that “time is the fourth dimension” is an objection to searching for a fourth dimension. Time is a known dimensions, like the first 3 dimensions of space, time is NOT an extra dimension. A fourth spatial dimension or a second temporal dimension would be the “extra dimensions”, things beyond what has been observed in universe.

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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .

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