Sure they can. Take a four-dimensional Cartesian space defined by four mutually-perpendicular axes w, x, y, and z. Consider two planes, one containing the w and x axes, the other containing the y and z axes. What happens at the zero point where all four axes intersect?
In the introduction to his book The Fourth Dimension, Rudy Rucker said something that totally blew my mind when I read it as a young teenager. He said:
“What is really needed here is the concept of a fourth space dimension. It is very hard to visualize such a dimension directly. Off and on for some fifteen years, I have tried to do so. In all this time I’ve enjoyed a grand total of perhaps fifteen minutes’ worth of direct vision into four-dimensional space. Nevertheless, I feel that I understand the fourth dimension very well.”
(The Fourth Dimension, pp. 7-8)
I was greatly thrilled by the prospect that it might actually possible to get a vision of four spatial dimensions. And now it seems like Chronos can do it as well. And a few others here.
I suspect this is the book Fenris was talking about. There’s also The Planiverse by A.K. Dewdney (who has unfortunately gone off into 9/11 conspiracy theories these days) which posits a much more “realistic” 2D universe than Flatland. It goes into detail about how the 2D creatures eat, excrete, live, travel, communicate, build technology, etc. Quite interesting.
I made two statements which were not consistent. First I said, “Two planes do not intersect in a point regardless of how many dimensions you are in.” That was wrong, and I have to roll over in light of these two illustrative mathematical explanations. And although I understand the math I am boggled by the concept. This idea defies trying to work from an analogy of 2-space moving to 3-space, unlike talking about squares, cubes, and tesseracts.
However, I believe my second statement is correct, that “In four dimensions, two non-parallel 2D planes lying in the same realm (three-dimensional hyperplane) intersect in a single line, just like they do in our 3D world.” The two planes described in each of these two explanations are in separate realms. To be in the same realm (a realm perpendicular to three axes), one dimension in each of the planes would have to be the same constant. On reflection, that is equivalent to two planes in 3-space, and seems like it doesn’t really say much.
However, nonetheless I didn’t actually do the math, didn’t expect it was possible under any conditions that two planes would intersect in a point, and consider my ignorance fought. 
I think I started when I was around 9 years old, and was able to get 4 dimensions consistently by about the time I was 15 or 16. Mostly, it consisted of spending a lot of free time trying to imagine 4D objects, rotating them in my head, etc. Which, apparently, is indistinguishable from petit mal seizures, to an outside observer: All my time just sitting staring off into space had my mom worried about me for a while.
Well, sure, but that’s not a statement about 4D, just a statement about 3D. If you’re restricting everything to a 3D hyperplane, then of course it follows the 3D rules.
Incidentally, when working in varying numbers of dimensions, one has to be very careful about what one is taking for granted, since an awful lot of science and mathematics is predicated on the assumption of three spatial dimensions. For instance, Euclid’s Elements contains a “proof” that the intersection of two planes is a straight line. I was very curious when I got to that one, since I already knew by that point that Euclidean geometry was generalizable to arbitrary numbers of dimensions, and that two planes in 4D space could intersect in a point. It turns out that what Euclid actually did was that he tacitly assumed that two planes intersected in a line, and then proved that that line was straight.
In principle, it is really easy. You just imagine time. What does a hypercube look like? A cube. Then eight dots for a few minutes. Then another cube. Brilliant! What is much harder, because we are not familiar with it, is to know what it’ll look like if you rotate it, or to know what it’ll look like if you fold the time dimension into itself (ie, make a 3D or 2D projection).
Is there stereoscopic software that lets you manipulate the hypercube, rotate it, etc, to train your brain? (I’m already good at cross-eye 3D btw.)
To illustrate it physically: Imagine a 2D plane that exists in one moment of time. Imagine a line that crosses that plane. Except that line exists for a long time. The two hyperplanes intersect at a point. One point in one moment of time. The line doesn’t even have to be at a right angle.
Instead of a plane in one moment of time, you can imagine a moving line. The point at which your moving line intersects the eternal stationary line: also a point. Or hell, two moving lines. If they ever intersect, then that’s your two-planes-at-a-point moment.
Told you 4D is easy.
My brother-in-law claimed he could visualize a tesseract. When he told me, our conversation went something like this:
B-I-L: I know what a tesseract looks like – I can visualize one in my head!
Me: How do you know?
B-I-L: What do you mean?
Me: We can’t experience a tesseract in the real world. How do you know the thing you’re imagining is what a tesseract really looks like?
He got really quiet after that. But he’s a blowhard, so when I called his bluff he folded.
The people in this thread are not blowhards, but I must put to them the same question. So **Chronos **and **Stranger **and Turble, how do you know the thing you are visualizing is an accurate representation of that four- (or higher) dimensional object? I’m not saying you can’t; I’m asking, sincerely, how do you know you’re right?
I know, mathematically, what properties a tesseract (or other higher-dimensional object) should have, and the thing I’m visualizing has all of those properties. A thing with all of the properties of a tesseract is a tesseract.
Yes, thence my “on reflection” statement immediately thereafter. I am still trying to grok this idea of planes intersecting in a point even though I can do the pencil-and-paper math of it.
I am really, really trying to get this but I get stuck in the mud right away. You start out talking about a 2D plane and a line, but then you refer to “the two hyperplanes.” *What *two hyperplanes? The original issue is two 2D planes intersecting within 4-space.
One suspects that this is probably the only way to do it. Whilst the brain is still plastic enough to wire itself to manage a true 4D capability. For the rest of us it is almost certainly way to late.
One question. Do you visulalise a wireframe/connected discrete points topology, or can you manage manifolds? Even 4-branes?
In all seriousness, one suspects that Chronos’ brain is very rare. It would be really interesting to do some functional MRI studies whilst doing this visualisation. Just determining which parts of the brain were actually processing the imagery would quite possibly be worth the experiment alone. But more beyond.
Again, in all seriousness, this is could be a brain worth leaving to science. Don’t know how you would feel about spending a few decades floating about in a jar, but eventually neurophysiology will get to the point where it is possible to reverse engineer the synapses, and then a capability to visualise 4D might hold some quite worthwhile knowlege.
Is fMRI really sophisticated enough to get anything interesting out of that? I knew they could tell things like that a person was visualizing something, but I didn’t know that they could get any information about the thing being visualized. But sure, if it’s possible, and someone’s actually interested enough in my brain, I’d be game. And I already intend to donate my body to science when the time comes, but let’s not rush things.
EDIT:
For something like a hypercube, I usually go with the wireframe, but for things like intersecting planes it seems like it’d be way too difficult: Instead of just having two objects to keep track of, I’d have hundreds. Planes I visualize as continua.
What struck me was that you have managed to press some part of the visualisation system into understanding the extended topology. The question is, which part? And also, have you co-opted some part of an additional processing mechanism to help? Are the processing areas larger in size than typical (something that does seem to happen.) So although there is scant chance of working out anything deep, watching how your brain lights up when visualising different things, of different dimensions, and comparing that to ordinary folk, may turn out to be revealing. Not just for you, but it may help understanding of conventional visual processing as well. Just as imaging people with deficits in capability helps enormously, an extended capability should also be quite worthwhile.
The reason it’s impossible to visualize is because we see in polar coordinates. Our eyes see two angles (up/down and left/right) and (indirectly) distance. Pictures just remove the distance. Four-dimensional beings would see three angles and distance. Again, their pictures would just take away distance. They would look at this picture with the three angles in their eyes. We can’t possibly visualize that. By talking about making holograms and such, you’re swapping an angle for distance which won’t work.
For one thing, we can’t see the internal parts of a 3D object. A 4D being could see all points of it, including the interior. And they wouldn’t simply see some parts as farther away than others, they would see them at a different angle, like we see all the points on a painting at different angles.
That is, unless they were looking at it from the same 3D realm, which would be like looking at a painting edge on. That’s basically what you’re trying to do by creating a hologram. Like I said, we only see distance indirectly, due to our stereoscopic vision and visual cues. You might as well say we see in two dimensions. That’s why pictures look close to real life.
And I don’t care what anyone says; I contest that no one can visualize a four-dimensional object. Imagine it and what properties it would have, possibly. But truly visualizing it would require a sense that we don’t have. It would be like someone who was born blind imagining color.
Well, OK—I take back the accusation that you were whooshing us. And for the record, that’s exactly what I look like when I’m trying to do it. I’ve sat on the edge of a couch with one sock half off staring into space for 15 minutes. It’s no small wonder people think I’m nuts. But like I’ve said, I’ve been at it since junior high and I still can’t do it. I just can’t visualize four-dimensional space.
This depends upon the semantics of visualise, and also a philosophical ideal of what perception is.
[quoteIt would be like someone who was born blind imagining color.[/QUOTE]
No, there is a significant difference. You can’t describe colour to anyone who can’t perceive it. We don’t have any language or mathematics to do so. Additional dimensions we do. We have a number of advantages here. One, ordinary 3D people are already able to perceive in 1, 2 and 3 dimensions. So the jump isn’t new. Secondly, we have a well developed mathematics of what additional dimensions work like.
You have to ask, “how is it that we perceive 3D?”. We don’t come pre-wired out of the womb to perceive 3D. It is a learned skill. The best answer seems that as our brains develop we create internal wiring that provides the needed firmware to process spatial information. What is also clear, some people are much better at it than others. Some people have a lot of trouble with maps, and a lot of people simply can’t manage engineering drawings. Yet being able to visualise a 3D construct from a series of 2D drawings is a fundamental skill for many engineers and architects. It may be that people that can’t believe that it is possible to visualise 4D can’t manage 3D all that well, and simply can’t understand the leap.
What Chronos describes is very interesting, and believable. The internal wiring of the brain is very plastic at a young age, and it is teachable. Languages and musical skills are good examples. Music especially. There is no reason why the approriate spatial processing centres could not be trained to manage multi-dimensional toploogies. And I think topology is likely the key thing here. It is the connectness of the dimensions that matters, and the manner in which the brain can grok the additional dimensions of connectedness that makes it work. It is also how the brain can be taught. As Chronos wrote, when he visualises a tesseract, it has the correct topological and geometrical properties. It is thus a true tesseract.
I can manage a pretty good 3D, and have no problem with quite complex 3D spatial problems, many of which totally stump friends. What I find really interesting is that sometimes I can’t explain how I solve them. Clearly there is a deep processing capability that isn’t part fo the direct conscious visualisation mechanism that comes into play here, in addition to the conscious visualisation mechanisms.
This is partly why I would love to see Chronos studied with fMRI. I suspect, that we could find one or more interesting additional things happening. It may be we see just the deep visualisation centers light up, but maybe they are bigger, maybe we see some other processing centres light up, one we don’t see for 3D, and possibly conclude that they are processing some of the higher dimensional topologies. Also, there may well be some total surprises.
Humans have a wide range of interesting cognitive capabilities. Music is another one. Some composers can hear the notes as they compose, some to the point of hearing the orchestra play almost for real. So harmonic as well as melodic, plus timbre. Beethoven is perhaps the best known example. Some mathematicians have an inate grasp of deep mathematics in a manner that is just plain scary. In both cases it seems reasonble to conclude that some people can process specialised high level complex information structures in complex ways. A 4D space doesn’t seem all that outlandish.
On the contrary – the jump is new and 4D space is, by definition, outlandish.
I find this stuff fascinating. And I’m not a mathematician, but I wonder if the 4D visualizers aren’t confusing the models in the their heads with the reality of what a 4D object really is.
With all due respect, Chronos, what I hear you saying is that you know what a tesseract looks like because you know what a tesseract is supposed to look like. I keep going back to the Flatlander. Can he ever truly visualize a cube? He may think he can, but can he really, seeing as he is not equipped to experience a cube? The best he can hope to do, based on the constraints of his situation, is to visualize a 2D analog of what a 3D shape may look like – and that is not the same thing as visualizing a 3D shape.
I guess to me it’s like saying, “Count by ones until you get to infinity.” And then someone says, “Ok, I got it.” You sure about that? I see that you get what it means mathematically. You understand counting. You understand the concept of infinity. But do you understand that you, due to the limitations inherent in being human, cannot actually count to infinity?
To be perfectly clear – I have no doubt the visualizers are each visualizing something that has characteristics that lead them to believe it is a tesseract. And they may in fact be tesseracts. But until we can compare those mental images to a real tesseract, how can we be sure?
Perhaps the fault lies with me, or with the limitations of natural (not mathematical) language to convey concepts outside of ordinary experience. I’m afraid until we perfect telepathy or some technology to let us see inside each others’ minds we won’t be able to resolve this one.
And about here you have the nub of the problem.
But, how can we be sure that any of us visualise 3D any better? Our brains are brilliant at constructing a reality out of very limited and interrupted input. Our brains construct the reality that it itself perceives. There is a bit of a problem here.
Choosing infiniy as the concept that can’t be properly grapsed is interesting, as indeed infinite provides the interface where a very large part of the interplay between philosophy and mathematics breaks down, or mostly agrees to disagree. Infinite is special, and for the sake of argument about 4D visualisation we really should choose a different example. 4D isn’t the same jump as infinite. Nothing else comes close to the problems infinite brings with it. Not all mathematicians actually accept infinite. It is that big an issue.
The point about the terseract is really not much different to being able to visualise anything. Visualise a purple sphere. Imagine a green plane intersecting the sphere. Visualise the shape of the intersection. What shape is the intersection. Now do a cone. What is the shape? How do I know you got the visualisation right? How do you control the geometry? How do you know that you got it right?
My point is that all visualisation is learnt. And the rules of the visualisation are known. Some other examples. Chess players. What is it that a chess master carries around in their head that allows them to play so well? Somehow there is a deep multidimensional information representation capability. And it is learnt. They are not born chess masters. They don’t consciously work their way through the rules and moves each play. When one considers the extraordinary level of computational effort that has been needed to best a grand master this is clear. If a grand master can maintain this level of complexity why can’t another human’s brain learn the rules for 4D geometry as a deep capability?
People seem to learn different things, and have different aptitudes. I can do complex 3D, I can do very complex data structure and algorithm design that stumps many of my colleages, and can do parallel algorithms that also stump many programmers, all pretty much in my head. I can’t play chess to save my life. Kids of 6 who have played properly for a year can whip me. I cannot play cards. Any game more complex than Snap I can’t manage. I can’t remember a phone number long enough to dial it. If I don’t have it written down I am in trouble. On the other hand, I can roughly follow a tune on a musical score, and can hear the melodies in my head, and can also hear basic harmonies from a score. Very basic ones. I’m a hachi kyu to Beethoven’s juu dan. All that this tells me is that my brain got wired and learnt some complex things well, and lost out big in other ways. It also tells me that other people can have similalry wide variations.
If I see a simple major chord, and imagine the sound, how do you know I got it right? When I imagine the sound of flattening the third, how do you know what I hear is a minor tonality? How do I even know I got it right? In a century’s time we can microtome Chrono’s brain and map the synapses. Then reverse engineer the wiring and work out what he visualised. (Maybe.) Perhaps then we might be able to see whether what he sees is capable of being correct. Even then, we are very unlikey to actually be able to understand what he sees.
Visualizing 4d is a couple or orders magnitude harder than imagining how a 100khz-beep sounds