I thought it might be, and if I’d read it elsewhere, I’d have been sure of it—but this is LessWrong, which is chock-full of hyperintelligent people whose abilities to do math, reason and visualize are close to superpowers from where I am. You people seriously intimidate me, you know. (Just because I feel you’re so much out of my league, not for any other reason.)
It’s a standard joke about mathematicians vs everybody else, and I intended it as such. I can do limited visualization in the 4th dimension (hypercubes and 5-cells (hypertetrahedra), not something as complicated as the 120-cell or even the 24-cell), but it’s by extending from a 3-d visualization with math knowledge, rather than specializing n to 4.
For what it’s worth, my ability to reason is fairly good in a very specific way—sometimes I see the relevant thing quickly (and after LWers have been chewing on a problem and haven’t seen it (sorry, no examples handy, I just remember the process)), but I’m not good at long chains of reasoning. Math and visualizing aren’t my strong points.
Been there, done that. Advice to budding spatial-dimension visualizers: the fourth is the hardest, once you manage the fourth the next few are quite easy.
Well, I can elaborate, but I’m not sure how helpful it will be. “No one can be told what the matrix is” and that sort of thing. The basic idea is that it’s the equivalent of the line rising out of the paper in two-dimensions, but in three dimensions instead. But that’s not telling someone who has tried and failed anything they don’t know, I’m sure.
If you really want to be able to visualize higher-order spaces, my advice would be to work with them, do math and computer programming in higher-order spaces, and use that to build up physical intuitions of how things work in higher-order spaces. Once you have the physical intuitions it’s easier for your brain to map them to something meaningful. Of course if your reason for wanting to be able to visualize 4D-space is because you want to use the visualization to give you physical intuitions about it that will be useful in math or computer programming, this is an ass-backward way of approaching the problem.
Is it like having a complete n-dimensional construct in your head that you can view in its entirety?
I can visualise 4-dimensional polyhedra, in much the same way I can draw non-planar graphs on a sheet of paper, but it’s not what I imagine being able to visualise higher-dimensional objects to be like.
I used to be into Rubik’s Cube, and it’s quite easy for me to visualise all six faces of a 3D cube at once, but when visualising, say, a 4-octahedron, the graph is easy to visualise, (or draw on a piece of paper, for that matter), but I can only “see” one perspective of the convex hull at a time, with the rest of it abstracted away.
When I was 13 or so, my brains worked significantly better than they currently do, and I figured out an easy trick for that in a math class one day. Just assign a greyscale color value (from black to white) to each point! This is exactly like taking an usual map and coloring the hills a lighter shade and the low places a darker one.
The only problem with that is it’s still “3.5D”, like the “2.5D” graphics engine of Doom, where there’s only one Z-value to any point in the world so things can’t be exactly above or below each other. To overcome this, you could theoretically imagine the 3D structure alternating between “levels” in the 4th dimension every second, so e.g. one second a 3D cube’s left half is grey and its right half is white, indicating a surface “rising” in the 4th dimension, but every other second the right half changes to black while the left is still grey, showing a second surface which begins at the same place and “descends” in the 4th dimension. Voila, you have two 3D “surfaces” meeting at a 4D angle!
With RGB color instead of greyscale, one could theoretically visualize 6 dimensions in such a way.
Doing specific rotations by breaking it into steps is possible. Rotations by 90 degrees through the higher dimensions is doable with some effort—it’s just coordinate swapping after all. You can make checks that you got it right. Once you have this mastered, you can compose it with rotations that don’t touch the higher dimensions. Then compose again with one of these 90 degree rotations, and you have an effective rotation through the higher dimensions.
(Understanding the commutation relations for rotation helps in this breakdown, of course. If you can then go on to understanding how the infinitesimal rotations work, you’ve got the whole thing down.)
The way I would do it for dimensions between d=4 and d=6 is to visualize a (d-3)-dimensional array of cubes. Then you remember that similarly positioned points, in the interior of cubes that are neighbors in the array, are near-neighbors in the extra dimensions (which correspond to the directions of the array). It’s not a genuinely six-dimensional visualization, but it’s a three-dimensional visualization onto which you can map six-dimensional properties. Then if you make an effort, you could learn how rotations, etc, map onto transformations of objects in the visualization. I would think that all claimed visualizations of four or more dimensions really amount to some comparable combinatorial scheme, backed up with some nonvisual rules of transformation and interpretation.
Adding multiple temporal dimensions effectively how I do it, so one more shouldn’t be a problem*. I visualize a 3 dimensional object in an space with a reference point that can move in n perpendicular directions. As the point of reference moves through the space, the object’s shape and size change.
Example: to visualize a 5-dimensional sphere, I first visualize a 3 dimensional sphere that can move along a 1 dimensional line. As the point of reference reaches the three-dimensional sphere, a point appears, and this point grows into a full sized sphere at the middle, then shrinks back down to a point. I then add another degree of freedom perpendicular to the first line, and repeat the procedure.
Rotations are still very hard for me to do, and become increasingly difficult with 5 or more dimensions. I think this is due to a very limited amount of short-term memory. As for my technique, I think it piggybacks on the ability to imagine multiple timelines simultaneously. So, alas, it’s a matter of repurposing existing abilities, not constructing entirely new ones.
*up to 7: 3 of space, 3 of observer-space, and 1 of time
There’s a pretty serious gap between the idea of a person evolved to visualize four dimensions and it being capable of thoughts I cannot think. This might be defensible, but if so only in the context of certain thoughts, something like qualitative ones. But the original quote was inferring from the fact that not everyone can see all the colors to the idea that there are thoughts we cannot think. If ‘colors I can’t see’ are the only kinds of things we can defend as thoughts that I cannot think, then the original quote is trivial.
So even if you can defend 4d visualizations as thoughts I cannot think, you’d have to extend your argument to something else.
But I have a question in return: how would the belief that there are thoughts you cannot think modify your anticipations? What would that look like?
By itself? Not much at all. The fun part is encountering another creature which can think those thoughts, then deducing the ability (and, being human, shortly thereafter finding some way to exploit it for personal gain) without being able to replicate the thoughts themselves.
Try visualizing four spacial dimensions.
Just visualize n dimensions, and then set n = 4.
You might as well tell me to ‘just’ grow wings and fly away...
I believe wnoise was making a joke—one that I thought was moderately funny.
I thought it might be, and if I’d read it elsewhere, I’d have been sure of it—but this is LessWrong, which is chock-full of hyperintelligent people whose abilities to do math, reason and visualize are close to superpowers from where I am. You people seriously intimidate me, you know. (Just because I feel you’re so much out of my league, not for any other reason.)
It’s a standard joke about mathematicians vs everybody else, and I intended it as such. I can do limited visualization in the 4th dimension (hypercubes and 5-cells (hypertetrahedra), not something as complicated as the 120-cell or even the 24-cell), but it’s by extending from a 3-d visualization with math knowledge, rather than specializing n to 4.
For what it’s worth, my ability to reason is fairly good in a very specific way—sometimes I see the relevant thing quickly (and after LWers have been chewing on a problem and haven’t seen it (sorry, no examples handy, I just remember the process)), but I’m not good at long chains of reasoning. Math and visualizing aren’t my strong points.
Been there, done that. Advice to budding spatial-dimension visualizers: the fourth is the hardest, once you manage the fourth the next few are quite easy.
Is this legit and if so can you elaborate? I bet I’m not the only one here who has tried and failed.
Well, I can elaborate, but I’m not sure how helpful it will be. “No one can be told what the matrix is” and that sort of thing. The basic idea is that it’s the equivalent of the line rising out of the paper in two-dimensions, but in three dimensions instead. But that’s not telling someone who has tried and failed anything they don’t know, I’m sure.
If you really want to be able to visualize higher-order spaces, my advice would be to work with them, do math and computer programming in higher-order spaces, and use that to build up physical intuitions of how things work in higher-order spaces. Once you have the physical intuitions it’s easier for your brain to map them to something meaningful. Of course if your reason for wanting to be able to visualize 4D-space is because you want to use the visualization to give you physical intuitions about it that will be useful in math or computer programming, this is an ass-backward way of approaching the problem.
Is it like having a complete n-dimensional construct in your head that you can view in its entirety?
I can visualise 4-dimensional polyhedra, in much the same way I can draw non-planar graphs on a sheet of paper, but it’s not what I imagine being able to visualise higher-dimensional objects to be like.
I used to be into Rubik’s Cube, and it’s quite easy for me to visualise all six faces of a 3D cube at once, but when visualising, say, a 4-octahedron, the graph is easy to visualise, (or draw on a piece of paper, for that matter), but I can only “see” one perspective of the convex hull at a time, with the rest of it abstracted away.
Even better—play Snake in four spatial dimensions!
When I was 13 or so, my brains worked significantly better than they currently do, and I figured out an easy trick for that in a math class one day. Just assign a greyscale color value (from black to white) to each point! This is exactly like taking an usual map and coloring the hills a lighter shade and the low places a darker one.
The only problem with that is it’s still “3.5D”, like the “2.5D” graphics engine of Doom, where there’s only one Z-value to any point in the world so things can’t be exactly above or below each other.
To overcome this, you could theoretically imagine the 3D structure alternating between “levels” in the 4th dimension every second, so e.g. one second a 3D cube’s left half is grey and its right half is white, indicating a surface “rising” in the 4th dimension, but every other second the right half changes to black while the left is still grey, showing a second surface which begins at the same place and “descends” in the 4th dimension. Voila, you have two 3D “surfaces” meeting at a 4D angle!
With RGB color instead of greyscale, one could theoretically visualize 6 dimensions in such a way.
Now, if only this let you rotate things through the 4th dimension.
Doing specific rotations by breaking it into steps is possible. Rotations by 90 degrees through the higher dimensions is doable with some effort—it’s just coordinate swapping after all. You can make checks that you got it right. Once you have this mastered, you can compose it with rotations that don’t touch the higher dimensions. Then compose again with one of these 90 degree rotations, and you have an effective rotation through the higher dimensions.
(Understanding the commutation relations for rotation helps in this breakdown, of course. If you can then go on to understanding how the infinitesimal rotations work, you’ve got the whole thing down.)
I knew a guy who credibly claimed to be able to visualize 5 spacial dimensions. He is a genius math professor with ‘autistic savant’ tendencies.
I certainly couldn’t pull it off and I suspect that at my age it is too late for me to be trained without artificial hardware changes.
The way I would do it for dimensions between d=4 and d=6 is to visualize a (d-3)-dimensional array of cubes. Then you remember that similarly positioned points, in the interior of cubes that are neighbors in the array, are near-neighbors in the extra dimensions (which correspond to the directions of the array). It’s not a genuinely six-dimensional visualization, but it’s a three-dimensional visualization onto which you can map six-dimensional properties. Then if you make an effort, you could learn how rotations, etc, map onto transformations of objects in the visualization. I would think that all claimed visualizations of four or more dimensions really amount to some comparable combinatorial scheme, backed up with some nonvisual rules of transformation and interpretation.
ETA: I see similar ideas in this subthread.
Am I allowed to use time/change dimensions? Because if so, the task is trivial (if computationally expensive).
Ok, now add a temporal dimension.
Adding multiple temporal dimensions effectively how I do it, so one more shouldn’t be a problem*. I visualize a 3 dimensional object in an space with a reference point that can move in n perpendicular directions. As the point of reference moves through the space, the object’s shape and size change.
Example: to visualize a 5-dimensional sphere, I first visualize a 3 dimensional sphere that can move along a 1 dimensional line. As the point of reference reaches the three-dimensional sphere, a point appears, and this point grows into a full sized sphere at the middle, then shrinks back down to a point. I then add another degree of freedom perpendicular to the first line, and repeat the procedure.
Rotations are still very hard for me to do, and become increasingly difficult with 5 or more dimensions. I think this is due to a very limited amount of short-term memory. As for my technique, I think it piggybacks on the ability to imagine multiple timelines simultaneously. So, alas, it’s a matter of repurposing existing abilities, not constructing entirely new ones.
*up to 7: 3 of space, 3 of observer-space, and 1 of time
Either I can visualize them, and then they’re thoughts I can think, or I can’t visualize them, in which case the exercise doesn’t help me.
If you can, replace 4 with N for sufficiently large N.
If you can’t, imagine a creature that evolved in a 4-dimensional universe. I find it unlikely that it would not be able to visualize 4 dimensions.
There’s a pretty serious gap between the idea of a person evolved to visualize four dimensions and it being capable of thoughts I cannot think. This might be defensible, but if so only in the context of certain thoughts, something like qualitative ones. But the original quote was inferring from the fact that not everyone can see all the colors to the idea that there are thoughts we cannot think. If ‘colors I can’t see’ are the only kinds of things we can defend as thoughts that I cannot think, then the original quote is trivial.
So even if you can defend 4d visualizations as thoughts I cannot think, you’d have to extend your argument to something else.
But I have a question in return: how would the belief that there are thoughts you cannot think modify your anticipations? What would that look like?
By itself? Not much at all. The fun part is encountering another creature which can think those thoughts, then deducing the ability (and, being human, shortly thereafter finding some way to exploit it for personal gain) without being able to replicate the thoughts themselves.
Hinton cubes. I haven’t tried them though.
ETA: Original source, online.