Time is the fourth dimension of space-time which bascially has the metric (when the space is Euclidan) ds[sup]2[/sub] = x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - t[sup]2[/sup], which you can see is NOT the same as the metric above for four dimensional Euclidan space.
Um, yeah. I see exactly what you’re saying there, MC and audi. I think I’ll go back to watching Benny Hill now.
Sort of my way too. It’s easier to see when you do a 2d and use time as a 3d streaching into the ‘z’ plane. With 4d the 3d object has absolute coordinated to define it in 3d space and the other dimention which I can’t describe that well but is somewhat superimposed on the other 3, but not really. It’s almost like it’s a very distorted image in that direction. I can break it down to an easier to understand and explain series of 3d images, each series reperesents the object in a particular 4d coordinate - this last one is particularly useful for using the 4d as time.
IIRC the 5th dimention appears as a further distortion of the 4th in a different direction. Direction is a bad word to use because I don’t assiciate it w/ the main 3.
Hopes this helps
Usually, I’d agree wholeheartedly about the nature of time as a non-spatial dimension, but something occurred to me the other day (which might just be utter bollocks, but…) - photons don’t ‘perceive’ time because, travelling at c, they experience complete dilation.
Another way to look at this would be as follows:
Consider a fishtank, filled with water; from various points to various other points in the tank are strung taut cords in all kind of random directions.
Each of the cords ‘perceives’ itself as being a complete instantaneous connection between the points at its two ends.
The water is slowly drained.
The surface of the water ‘perceives’ only the cross-section of the cords and, as the water level drops, the surface perceives these cross sections as being in motion; starting at one endpoint and making their way to the other endpoint.
From the point of view of the two-dimensional water surface, the third dimension (in which the surface is steadily moving) is the cause of movement - the third dimension is time.
Of course, this is probably nothing more than a fanciful daydream, but is it not possible that the fourth dimension is a spatial one, transposition in which merely manifests itself as the passage of time in our three-dimensional point of view?
Mangetout, that still doesn’t really explain how time is a different dimension than the three spatial ones. Or maybe it explains quite well…my head’s spinning too much to be sure.
We can see pretty well from that description that time is somehow different. I’m not sure how your analogy demonstrates otherwise…but like I said, I’m not sure what to make of that example. It’s been a long day, and I’m tired of trying to think four-dimensionally. 
I’d like some to see some stronger arguments, please, that just because the brain exists in three dimensions, it is not capable of generating an accurate internal representation of a 4D space.
Please also include a refutation of the proposition that, given sufficient computing power, a functionally two-dimensional imitation of the human brain could generate an internal 3D visualization.
Yes, time, while sharing properties with, is fundamentally different from the spatial dimensions in some way (also Mangetout photons don’t have rest frames in relativity). That said for an object inside the event horizon of a black hole the axis radially coming radially out from the centre of the black hole is a timelike dimension, also using the concept of imaginary time you can have 4 spacelike dimensions.
I’ve been trying to visualize four-dimensional objects since I was in grade school. One of my favourite techniques is to imagine a stack of glass plates, with a semi-transparent TV image on each plate. Then each plate represents a three-dimensional space, and the stack as a whole represents a four-dimensional one.
The way I figure it, we’re used to transforming a two-dimensional TV image into a three-dimensional mental picture; most of us do that all the time. So why not try to extend that little metaphor?
More generally, if you want to practice this sort of think I’d recommend starting with something simple: two planes which intersect in only a point. That’s only possible in four dimensions, but is simple enough to get a handle on.
My best effort ever was to enumerate all the cells in the 120-celled regular polytope (four-D analogue of the dodecahedron), and visualize which cells were adjacent to which, and how they all fitted together to form the polytope. That took a while.
Here’s what I was trying to describe.
The glass tank contains a length of green cord, fixed to the left hand pane, near the top/back, stretched tight and fixed to the opposite pane, but at the bottom/front.
There’s also a red cord, fixed to the right-hand pane near the top/back and stretching to a point on the opposite pane near the bottom/front.
The observer in the 3D world perceives each of these cords as being complete in length and joining two points in the tank all at once.
But shift the perspective to the two-dimensional surface of the receding water - only the cross-section of the cords can be seen and as we move downwards (‘downwards’ being a spatial dimension totally alien to our 2D surface), the cross sections are perceived to move from one corner of the tank to the other.
So, a movement in what we call the spatial third dimension is a movement in the non-spatial dimension of time to the 2D world, because it permits change and movement.
My own most serious sustained go at it was in the years when I was studying GR. As a result, I was less concerned with visualising higher dimensional polytopes per se as in finding ways of visualising curved spaces. Chronos’s technique strikes me as very clever, but alien, whereas Orbifold’s approach seems more familiar. In my case, there was a lot of reducing the problem by one dimension and then “trying to bump it back up.” That and a lot of switching around between different ways of visualising particular solutions of GR.
I’m not sure I can claim to have succeeded, but I did build up a IMO reasonable intuition of how things were working. And I think that intuition was the correct end goal.
Despite the fact that it usually involved working in 4 (or just slightly fewer than 4) dimensions, concentrating on QFT rather atrophied that intuition. All very flat and formal, except possibly if you’re into topological objects.
And I suppose someone has to mention Hinton’s The Fourth Dimension. I’ve owned a copy for some years, but have never got around to actually reading it.
I beg to differ. I find visualizing n-dimensional space to be a trivial exercise. Just because I am in a 3 spatial dimensions universe right now, doesn’t mean my mind cannot analyze higher dimesional objects, rotated them around in my head, ect.
What do we usually say here in the SDMB about proving negatives?
I don’t have to prove it’s not possible, you have to prove it is. The first step to that would be proving you know what 4D looks like, not what you imagine it looks like. All visualisations quoted so far are just models, interpretations, mind-tricks, they are only ‘accurate’ up to a point.
Not a valid comparison. Your 2D brain is an imitation of a 3D one, so already you are starting from a position of complete understanding of 3 dimensions. A more accurate comparison would be asking the 2D brain to come up with a visualisation of 3D on its own. To which it would say “Well I get the idea, but perhaps if we imagine this extra third dimension as a colour it would be easier to grasp…”
It is by-definition not possible for ANY human to accurately perceive how the fourth dimension would look. The human mind, even operating on all cylinders, which nobody can figure out how to do, cannot function on a higher-dimensional level.
It is, however, possible to view a three-dimensional analog of a fourth dimensional object, and although it takes quite a few glances to figure out, it’s quite simply - eight cubes fused together with edges extending through the fourth dimension. I can rotate it in my mind, and humans can perceive the concept, but it is entirely impossible for a human mind to visually grasp how anything in a hyper-space would function. I can be easily imagined mathematically, howeever.
I apologize, I must have missed the IMHO disclaimer at the top of your original post.
However, the purpose of my invitation was not to show that, when you failed to meet the invitation, it would mean the feasibility of a 4D visualization was proven. Its purpose was to show your claim about its impossibility was not as definitive as it sounded. (Which the IMHO accounts for.) And given the speculative and subjective nature of the question, there is no reason to apply some sort of “burden of proof” that would favor one side or the other.
As for the 2D brain, it is most certainly a valid comparison. I was just driving at a different conclusion that it must have appeared. I was addressing the physiological and computational side of the issue, because I felt that the actual 3D nature of the brain as a physical object were being called upon too much to support the impossibility of a 4D visualization. I agree that the mental models and techniques of 4D visualization (if an accurate 4D visualization exists at all) would probably be very hard to develop cold turkey, but if they could be implemented in a 3D brain at all, then 4D visualization would be possible.
But I think we agree: these days mental constructs and actions are still very subjective and inaccessible, and our dealings with 4D space very theoretical and speculative, such that there is no good broad answer to this question.
Simulpost, d’oh!
Your claims, however, SpACatta, have been made in such a way that you will probably want to back them up.
This sounds like a perfect matchup to me. owlofcreamcheese, do you have any such questions handy?
The direction perpendicular to a third dimension cannot be understood by someone who lives in a world of three dimensions. A four-dimensional object is a pair of its three-dimensional analogs connected by a number of edges equal to its total number of vertices, extending through the fourth dimension in a direction perpendicular to the three-dimensonal object’s alignment. No human mind can grasp in-which direction this fourth-axis can lie, as it doesn’t lie in a spatial dimension that fits our current perception of what perpendicular really is.
Although, being nothing but a high-school dilettante of hyper-dimensional study, I suppose that I can’t put forth an explanation as well as a psychology/theoretical geometry major could, dealing with the reason as to why we can’t perceive such a phenomenon in the way that would be required to understand exactly how it is constructed in reality, as-opposed-to simply in “Mathland”, where everything can be explained!
But that’s the explanation I can put forth, having studied this concept for only three years.
Just as a being living in a theoretical two-dimensional universe can not possibly understand a direction existing on either side of his flatter-than-paper universe, yet can perceive several of his universes existing next to each-other, as he sees his square friends existing next to, or rather floating above and below, another object, we can perceive the theoretical concept of how a fourth dimension would look, but there is no non-mathematical way to truly visualize in which direction our three-dimensional shape would extend to form our hyper-dimensional analog.
Preamble: Most of what I am about to say has been said already, but sometimes putting things slightly differently helps communication.
First, 3D. It seems generally granted in this thread that if one closes his eyes and pictures a 3D object (say, a cube), we would say that he has “visualized 3D space” (to use the language of the thread title.) Maybe he rotates the cube around in his head or maybe he slices it up, but in some sense he is mentally interacting with an object that has 3-dimensional properties, so we say he is “visualizing” that 3D object.
We all gain this ability as children, but we do so using only 2D receptors (eyes). However, because we live in a 3D universe, we have ceaselessly bombarded our brain, since birth, with 2D images of objects which behave in 3D ways. We see a zillion 2D images of things moving through our vision, and we learn – keyword: learn – to intuit 3D behavior. Before long, we can perform a mental trick that we call “visualizing 3D space.” All of this happens without us realizing it because we can’t help but get the requisite training.
The key point here: there is nothing special about 3D visualization except that the training comes for free and the training comes in boatloads since we live and move in a 3D world.
Now, 4D. Though we learn to visualize 3D space by watching plenty of 3D-behaving 2D images, that is by no means the only way our brains learn to do tricks. Indeed, 4D intuition comes not from our 2D eyes (which can only access 3D objects) but rather from working with 4D concepts for enough time in enough ways. The training “samples” of 4D-behaving things are much less plentiful, and one will likely need mathematical intuition of various sorts to have access to any 4D training at all, but that makes the eventual visualization of 4D-behaving objects no less real than the visualization of 3D objects.
–> Our 3D visualization training happens to come through our (2D) eyes. 4D visualization training comes from an understanding/intuition of 4D behaviour that we gain via the study of relevant mathematics and (presumably) through our familiarity with 3D behaviour.
And just to throw in my tally, I’ve accumulated no more than a minute or two of 4D visualization time total. I’m not saying it’s easy; I am saying it’s possible, and you definitely know when it happens. (That is, you definitely know that it is distinct from the various “false 4D” examples given by others.) My experiences sound similar to bonzer’s, as I also had my best success while studying GR.
Something to think about: Conway’s “Game of Life” has long since been proved to be Turing-complete. Specifically, given any computer program and a large enough board it is possible to construct an initial configuration of cells such that the system as a whole will “execute” the program.
So someday in the future, when we’ve figured out how to program a computer to process visual input and visualize three-dimensional situations, we can take that program and implement it in the Game of Life, which is an inherently two-dimensional system.
In other words, it is theoretically possible to program a two-dimensional computer to visualize three-dimensional space. So why on earth is it somehow impossible to program a three-dimensional computer, specifically our brains, to visualize four-dimensional space?
The fact that our brains our three-dimensional doesn’t matter. The nature of the hardware doesn’t constrain the nature of the software, and this is fundamentally a software question. And the fact that we never receive four-dimensional input from the real world also doesn’t matter. We can deduce mathematically what that input should be.
There is really no obstacle to prevent me or anyone else from visualizing four-dimensional space except patience, time, and imagination.