thinking about more than 3 spacial dimensions
November 22, 2010 6:06 PM Subscribe
Is there anyone, anywhere, anywhen, who has been able to visualise 4 orthogonal spacial dimensions?
so i was wondering what an equilateral hyper tetrahedron would be like.
I figure if you start with one point, then place a new point at the same position, then move the new point until it is 1 unit distance away from the original point, then connect these two points you now have a line segment.
then repeat:
place a point in the centre of the line segment, move it on an axis which is orthogonal to the line segment, until the new point is 1 unit distance away from both the two original points, connect the new point to the previous points, and you now have an equilateral triangle.
repeat once more, starting by placing a point at the centre of your equilateral triangle, and you end up with a tetrahedron.
so far i am equipped with appropriate brain circuitry to visualise all of these.
but if i repeat the process one more iteration, i end up with what i think i can get away with calling a 4d equilateral hyper tetrahedron.
I can't visualise that, but i don't know that it's impossible to visualise, i suspect that if i could give my brain the right interactive datafeeds, it could figure out a rough sense of 4d.
Has anyone every put a really serious effort in trying to do this?
how did it go?
I'd love to see some 3d 'slices' as a hyper tetrahedron moved and rotated through our 3d 'plane',
If i could watch a 3d visualisation of a slice change as i adjusted position and rotation i would be very interested to see what my brain could do with that information.
Id also like to able to do the same with 3d 'drawings' which project a picture of a 4d object onto 3d space, just as we do with 3d objects onto 2d paper
failing that, i eagerly await the announcement, perhaps at a TED talk, of n-dimensional visualisation brain upgrade chips.
so i was wondering what an equilateral hyper tetrahedron would be like.
I figure if you start with one point, then place a new point at the same position, then move the new point until it is 1 unit distance away from the original point, then connect these two points you now have a line segment.
then repeat:
place a point in the centre of the line segment, move it on an axis which is orthogonal to the line segment, until the new point is 1 unit distance away from both the two original points, connect the new point to the previous points, and you now have an equilateral triangle.
repeat once more, starting by placing a point at the centre of your equilateral triangle, and you end up with a tetrahedron.
so far i am equipped with appropriate brain circuitry to visualise all of these.
but if i repeat the process one more iteration, i end up with what i think i can get away with calling a 4d equilateral hyper tetrahedron.
I can't visualise that, but i don't know that it's impossible to visualise, i suspect that if i could give my brain the right interactive datafeeds, it could figure out a rough sense of 4d.
Has anyone every put a really serious effort in trying to do this?
how did it go?
I'd love to see some 3d 'slices' as a hyper tetrahedron moved and rotated through our 3d 'plane',
If i could watch a 3d visualisation of a slice change as i adjusted position and rotation i would be very interested to see what my brain could do with that information.
Id also like to able to do the same with 3d 'drawings' which project a picture of a 4d object onto 3d space, just as we do with 3d objects onto 2d paper
failing that, i eagerly await the announcement, perhaps at a TED talk, of n-dimensional visualisation brain upgrade chips.
Charles Hinton was one of the epicenters of the brief fourth dimension craze in the late nineteenth century, the time period that gave us Flatland. He was rumored to have been able to visualize the fourth dimension, and even developed a set of blocks to aid others in that endeavor (modern example).
posted by Nomyte at 6:24 PM on November 22, 2010
posted by Nomyte at 6:24 PM on November 22, 2010
We discussed and tried drawing this kind of 4d stuff in my "Math for non-math majors" class in college.
The text we used was very good, Heart of Mathematics.
posted by HMSSM at 6:27 PM on November 22, 2010
The text we used was very good, Heart of Mathematics.
posted by HMSSM at 6:27 PM on November 22, 2010
I think a set of your 3d 'slices' would simply be a tetrahedron shrinking to a point, much like slicing the tetrahedron along its 'newest' axis would be a set of shrinking triangles.
One way to visualize it is to let time be the fourth dimension. Let it start of as a tetrahedron now, and imagine it shrinking to a point over the course of 10 seconds. Now imagine that these ten seconds of time exists at the same time, are compressed to now so to speak, so you get the idea of the tetrahedron extending in a fourth dimension much like it extended in time, only now independent of time.
Meh, it's hard to explain :P
posted by spr at 6:31 PM on November 22, 2010 [1 favorite]
One way to visualize it is to let time be the fourth dimension. Let it start of as a tetrahedron now, and imagine it shrinking to a point over the course of 10 seconds. Now imagine that these ten seconds of time exists at the same time, are compressed to now so to speak, so you get the idea of the tetrahedron extending in a fourth dimension much like it extended in time, only now independent of time.
Meh, it's hard to explain :P
posted by spr at 6:31 PM on November 22, 2010 [1 favorite]
this video is misleadingly confident, but it "visualizes" up to 10d
posted by acidic at 6:50 PM on November 22, 2010
posted by acidic at 6:50 PM on November 22, 2010
Response by poster: spr, that is useful, and it sort of was my starting point, but the way you have conveyed is clearer than my original set of thoughts. So its a start, but just a i can take a cross section of a tetrahedron which is not an equilateral triangle, if the plane is not aligned with one of the surfaces of the tetrahedron, i suspect that the slices of the hyper tetrahedron could have other appearances if it was aligned differently in 3d+time space
posted by compound eye at 6:51 PM on November 22, 2010
posted by compound eye at 6:51 PM on November 22, 2010
If you're asking if there's a way to use animated rotations of a four-dimensional object to construct an intuitive model of that entire object in your brain, I submit that the answer is "no." It might seem possible due to analogies like Flatland, which compare cross-sections like these to a 3D object passing through a two-dimensional plane, but it's not that simple.
Sure, given what a Flatlander would see (a series of shrinking triangles for a tetrahedron), you would be able to imagine the entire 3D object. It would be a little tough if you didn't know what the object was in advance, but it would be doable if you had a good memory and a strong sense of spatial awareness. Just stack the various cross-sections until you can visualize the whole. But to do the same thing, intuitively, for a four-dimensional object? "Stacking" 3D objects until you can intuit the whole thing? That would require being able to grasp the nature of four-dimensional space. And that's simply beyond human understanding.
We can always memorize what the various cross sections of a 4D object look like, how one seamlessly blends into another, and learn how to derive these cross-sections through mathematical reasoning -- just like a Flatlander could memorize that a sphere looks like a series of expanding and contracting circles. But it's impossible to conceive of the entirety of a 4D object at once. It would be like trying to imagine -- intuitively -- what it would be like to see in four dimensions -- able to see the inside and outside of everything around you simultaneously. Or to go in the other direction, what it would be like to have your vision restricted to a single one-dimensional line, with nothing -- absolutely nothing -- above or below it. There's an entirely new direction (or lack of a direction) involved, something completely outside your experience, and that gets in the way of any attempt to grasp the whole picture.
As Douglas Adams once wrote: "[A]nyone who has been to any of the higher dimensions will know that they're a pretty nasty heathen lot up there who should just be smashed and done in, and would be, too, if anyone could work out a way of firing missiles at right-angles to reality."
posted by Rhaomi at 7:29 PM on November 22, 2010 [3 favorites]
Sure, given what a Flatlander would see (a series of shrinking triangles for a tetrahedron), you would be able to imagine the entire 3D object. It would be a little tough if you didn't know what the object was in advance, but it would be doable if you had a good memory and a strong sense of spatial awareness. Just stack the various cross-sections until you can visualize the whole. But to do the same thing, intuitively, for a four-dimensional object? "Stacking" 3D objects until you can intuit the whole thing? That would require being able to grasp the nature of four-dimensional space. And that's simply beyond human understanding.
We can always memorize what the various cross sections of a 4D object look like, how one seamlessly blends into another, and learn how to derive these cross-sections through mathematical reasoning -- just like a Flatlander could memorize that a sphere looks like a series of expanding and contracting circles. But it's impossible to conceive of the entirety of a 4D object at once. It would be like trying to imagine -- intuitively -- what it would be like to see in four dimensions -- able to see the inside and outside of everything around you simultaneously. Or to go in the other direction, what it would be like to have your vision restricted to a single one-dimensional line, with nothing -- absolutely nothing -- above or below it. There's an entirely new direction (or lack of a direction) involved, something completely outside your experience, and that gets in the way of any attempt to grasp the whole picture.
As Douglas Adams once wrote: "[A]nyone who has been to any of the higher dimensions will know that they're a pretty nasty heathen lot up there who should just be smashed and done in, and would be, too, if anyone could work out a way of firing missiles at right-angles to reality."
posted by Rhaomi at 7:29 PM on November 22, 2010 [3 favorites]
Here is a video attempting to visualize a 4-dimensional tetrahedron, otherwise known as a simplex.
The clip comes from an amazing series of videos on visualizing dimensions, appropriately titled "Dimensions". I highly recommend watching the entire series. Some segments are more engaging than others, but you will definitely find many of your questions addressed therein. We even have a Metafilter thread about the videos.
To appreciate how difficult this problem is, take a moment to realize that even the visualization of ordinary 3D objects requires some kind of inherent parametrization (conceivably due to the 2D nature of our retina). Consider the apple, which is by all accounts a 3D object. You can't actually visualize the skin and the seeds of the same apple at the same time. You would never see both areas at once. Our vision is limited by surfaces, and our ability to abstract to 4D is probably a consequence of the necessity to abstract to 3D in the first place.
posted by abc123xyzinfinity at 9:16 PM on November 22, 2010 [1 favorite]
The clip comes from an amazing series of videos on visualizing dimensions, appropriately titled "Dimensions". I highly recommend watching the entire series. Some segments are more engaging than others, but you will definitely find many of your questions addressed therein. We even have a Metafilter thread about the videos.
To appreciate how difficult this problem is, take a moment to realize that even the visualization of ordinary 3D objects requires some kind of inherent parametrization (conceivably due to the 2D nature of our retina). Consider the apple, which is by all accounts a 3D object. You can't actually visualize the skin and the seeds of the same apple at the same time. You would never see both areas at once. Our vision is limited by surfaces, and our ability to abstract to 4D is probably a consequence of the necessity to abstract to 3D in the first place.
posted by abc123xyzinfinity at 9:16 PM on November 22, 2010 [1 favorite]
There's the game Miegakure: "Miegakure is a platform game where you explore the fourth dimension to solve puzzles." I haven't played, but it looks interesting. Mentioned in this XKCD.
posted by ShooBoo at 9:27 PM on November 22, 2010
posted by ShooBoo at 9:27 PM on November 22, 2010
So its a start, but just a i can take a cross section of a tetrahedron which is not an equilateral triangle, if the plane is not aligned with one of the surfaces of the tetrahedron, i suspect that the slices of the hyper tetrahedron could have other appearances if it was aligned differently in 3d+time space
WARNING: I have no idea what I'm talking about.
Yeah, you would. Slicing it up in time is like slicing it up in alignment with time, and you get tetrahedrons of decrementing size. So let's slice it up in a different direction. Imagine a 2d plane extended in 3d space at 'now' in time. Let this plane intersect the tetrahedron so you get two vertices on each side. The slice will now be some four-sided shape as opposed to a triangle (the plane intersects all four triangles of the tetrahedron).
Now visualize the slice that is "drawn" out on that plane as the ten seconds pass and the tetrahedron is shrinking. As time passes, two sides of the four-sided shape will be shrinking. At one point in time one of those sides will become infinitely small and then one vertex will cross the plane. From now on the slice will be a triangle instead of that four-sided shape. Let time keep passing, and this triangle will shrink until eventually the second point crosses the plane too.
If that made any sense, then try to read that last paragraph again, only this time also imagine you are constructing a 3d object from the slices drawn in the 2d plane as time passes by. The final 3d object then is (I believe) a 3d slice not aligned with any surface, and it's not a tetrahedron.
A more direct description of the 3d slice would be to start with a four-sided base (as you get from slicing the tetrahedron at the now point in time). Then extrude the four vertices upwards while slowly bringing two neighboring vertices together. Once those two vertices have been joined in a single point, then the three vertices (the other two plus this 'new' one that came from the first two being joined into one) will form the base of a tetrahedron. The final slice i composed of a four-sided base connected with two triangles and two four-sided surface.
Actually, now that I think about it, this description was probably unnecessarily convoluted. Slice the tetrahedron by the 2d plane as explained, and then the structure (a tent) on the side of the 2d plane opposite of the fifth point will be the slice that is drawn out in time (possibly with some linear transformation such as skew). So the slice is a tent.
REPEAT: I have no idea what I'm talking about.
posted by spr at 10:15 PM on November 22, 2010
WARNING: I have no idea what I'm talking about.
Yeah, you would. Slicing it up in time is like slicing it up in alignment with time, and you get tetrahedrons of decrementing size. So let's slice it up in a different direction. Imagine a 2d plane extended in 3d space at 'now' in time. Let this plane intersect the tetrahedron so you get two vertices on each side. The slice will now be some four-sided shape as opposed to a triangle (the plane intersects all four triangles of the tetrahedron).
Now visualize the slice that is "drawn" out on that plane as the ten seconds pass and the tetrahedron is shrinking. As time passes, two sides of the four-sided shape will be shrinking. At one point in time one of those sides will become infinitely small and then one vertex will cross the plane. From now on the slice will be a triangle instead of that four-sided shape. Let time keep passing, and this triangle will shrink until eventually the second point crosses the plane too.
If that made any sense, then try to read that last paragraph again, only this time also imagine you are constructing a 3d object from the slices drawn in the 2d plane as time passes by. The final 3d object then is (I believe) a 3d slice not aligned with any surface, and it's not a tetrahedron.
A more direct description of the 3d slice would be to start with a four-sided base (as you get from slicing the tetrahedron at the now point in time). Then extrude the four vertices upwards while slowly bringing two neighboring vertices together. Once those two vertices have been joined in a single point, then the three vertices (the other two plus this 'new' one that came from the first two being joined into one) will form the base of a tetrahedron. The final slice i composed of a four-sided base connected with two triangles and two four-sided surface.
Actually, now that I think about it, this description was probably unnecessarily convoluted. Slice the tetrahedron by the 2d plane as explained, and then the structure (a tent) on the side of the 2d plane opposite of the fifth point will be the slice that is drawn out in time (possibly with some linear transformation such as skew). So the slice is a tent.
REPEAT: I have no idea what I'm talking about.
posted by spr at 10:15 PM on November 22, 2010
"misleadingly confident"
I agree. The question was originally about orthogonal spacial dimensions, but the possible-future-selves concept of a dimension that the video portrays does not seem orthogonal to the others. Your future self is not independent of past locations.
[I don't know what I'm talking about, either.]
posted by klausman at 10:34 PM on November 22, 2010
I agree. The question was originally about orthogonal spacial dimensions, but the possible-future-selves concept of a dimension that the video portrays does not seem orthogonal to the others. Your future self is not independent of past locations.
[I don't know what I'm talking about, either.]
posted by klausman at 10:34 PM on November 22, 2010
Note the late and much lamented Martin Gardner wrote quite extensively about Hinton and his blocks. MG strongly advised against experimenting with the blocks--he believed they were hazardous to your mental health.
posted by Logophiliac at 11:43 PM on November 22, 2010
posted by Logophiliac at 11:43 PM on November 22, 2010
And remember that we denizens of the 3D universe cannot actually see the entire surface of an apple anyway, only 1/2 of it at one time (the other half is behind the part we can see). Wouldn't 4D viewers also be limited to only seeing 1 hyper-half of any object?
Hmmm . . . that seemed a lot more cogent before I stretched it out linearly.
posted by General Tonic at 6:43 AM on November 23, 2010
Hmmm . . . that seemed a lot more cogent before I stretched it out linearly.
posted by General Tonic at 6:43 AM on November 23, 2010
I think by definition it would be impossible to "visualize" a four dimensional object, since "vision" is a three dimensional experience.
My closest approximation would be to imagine an object as if you were looking at it through a camera with a shutter speed as long as the existence of the object itself. So say there's an object that was in existence for 20 years. The shutter would open at the creation of the object, and close 20 years later at the destruction of the object. You'd be left with a blur that represented 20 years of constant change and fluctuation.
Of course this assumes the object exists independently of all other objects, and has a definable "birth" and "death," and the observer (the camera) is somehow fixed in space/time...
So I vote no.
posted by tipthepizzaguy at 1:40 PM on November 23, 2010 [1 favorite]
My closest approximation would be to imagine an object as if you were looking at it through a camera with a shutter speed as long as the existence of the object itself. So say there's an object that was in existence for 20 years. The shutter would open at the creation of the object, and close 20 years later at the destruction of the object. You'd be left with a blur that represented 20 years of constant change and fluctuation.
Of course this assumes the object exists independently of all other objects, and has a definable "birth" and "death," and the observer (the camera) is somehow fixed in space/time...
So I vote no.
posted by tipthepizzaguy at 1:40 PM on November 23, 2010 [1 favorite]
You can try this great video of Carl Sagan explaining the strange picture on wikipedia and elsewhere of the hypercube.
posted by milestogo at 8:02 PM on November 23, 2010
posted by milestogo at 8:02 PM on November 23, 2010
Response by poster: but i finally understand the hypercube.
the fourth dimension is orthogonal to the previous three axes,
so its orthogonal to all the faces on all six sides of the cube.
so when you extrude a cube into a forth dimension each of the six faces is extruded into a cube with one axis in the new dimesion.
it's impossible to do in three dimensions, but with a forth dimension the new cubes share four of their faces with their neighbors.
i still think if i could feed my brain interactive 4d data you could figure it out,
the question is how could i feed that data in.
posted by compound eye at 10:44 PM on November 23, 2010
the fourth dimension is orthogonal to the previous three axes,
so its orthogonal to all the faces on all six sides of the cube.
so when you extrude a cube into a forth dimension each of the six faces is extruded into a cube with one axis in the new dimesion.
it's impossible to do in three dimensions, but with a forth dimension the new cubes share four of their faces with their neighbors.
i still think if i could feed my brain interactive 4d data you could figure it out,
the question is how could i feed that data in.
posted by compound eye at 10:44 PM on November 23, 2010
This thread is closed to new comments.
posted by Paragon at 6:12 PM on November 22, 2010