4D visualization

[quickfur]

As promised, I've written up the first 11 chapters of my thoughts about how one might go about visualizing 4D. (Don't worry, the chapters are short.) You can find it here:

Visualization of the 4D world

I haven't quite finished it yet; there are still a number of subjects that I'd like to cover before I consider the document "done". (Such as how a duocylinder might look to a 4D person, and the different types of 4D cones that are possible.) But I thought I'd post it here and see what others think about it. :-) The main idea is to approach 4D visualization not so much from a mathematical perspective, but from the perspective of how a 4D being might see its surroundings. Well, read it to see if it helps you see 4D. ;-)

The one big missing thing is good diagrams to accompany the text. Unfortunately I'm not a very good artist, and I'm not familiar with how to render 3D views of 4D scenes. For the type of visualization approach I'm trying to describe, I need more than just 4D wireframe projections, I'll need at least 4D hidden-surface clipping (e.g., a hypercube projected onto a rhombic dodecahedron should show only 4 cells, not 8 ), and preferably with 3D transparency so that you can see the entire scene all at once. :-P Any help would be appreciated.

Let me know what you folks think of my 4D visualization approach.

[jinydu]

I have tried, unsuccessfully, to access that website on 3 computers. Could you please check to make sure the link is right?

[pat]

dyndns.org is a site where you can get an name for a machine whose address changes frequently. It seems to me like the name is correct, but the poster has not updated his address with dyndns in awhile or maybe only has his machine up at certain times of the day....

[quickfur]

I have tried, unsuccessfully, to access that website on 3 computers. Could you please check to make sure the link is right?

Sorrrry... I accidentally deleted the wrong directory on that server, and so destroyed the files for that page as well. I've restored it from a backup. It should be working now. Sorry about that

[pat]

What I've read so far seems cool.... I'll have to read it more closely soon. In the meantime, here's a picture of a hollow hypercube with the front cell removed. To distinguish the sides, the +x side is red, the +/-y sides are green, the +/-z sides are blue, and the +/-w sides are yellow. The positive sides are striped, the negative sides are left unadorned.



If I had rendered enough frames, then this would basically be all of the pixels hitting Tetra's retina at one time. The resolution here is 75x75x11. I'd imagine Tetra's eyes would be more like 75x75x75. And, I didn't go to the trouble of mitering the edges, so the box is put together a little bit roughly.

[quickfur]

What I've read so far seems cool.... I'll have to read it more closely soon. In the meantime, here's a picture of a hollow hypercube with the front cell removed. To distinguish the sides, the +x side is red, the +/-y sides are green, the +/-z sides are blue, and the +/-w sides are yellow. The positive sides are striped, the negative sides are left unadorned.

Cool! Is this done using your multi-dimensional ray-tracer? :-) I downloaded it and compiled it, but didn't know where to start. Are there any online docs about how to use it?

If I had rendered enough frames, then this would basically be all of the pixels hitting Tetra's retina at one time. The resolution here is 75x75x11. I'd imagine Tetra's eyes would be more like 75x75x75. And, I didn't go to the trouble of mitering the edges, so the box is put together a little bit roughly.

Well, in my mind, when I imagine the image in Tetra's retina, it is slightly transparent, so that I can see all the objects in it from my 3D viewpoint. (Of course, in reality it is quite opaque, but I'm just trying to "see" the image from my 3D perspective.) Is it possible to render these images with partial transparency? :-)

Also, if Tetra's retina is anything like ours (and I'm working on that assumption), it would have higher resolution around the center, and lower resolution on the periphery. But we'll let that slide, since the computer deals better with an evenly distributed resolution.

[pat]

Cool! Is this done using your multi-dimensional ray-tracer? :-) I downloaded it and compiled it, but didn't know where to start. Are there any online docs about how to use it?

Not any good ones... You can glean some stuff about how to use it by reading this document:
http://www.nklein.com/products/rt/rt2.5.2004.03.29/rt2.5.2004.03.29.pdf
But that includes lots of stuff about the source code that you just won't want to know. There are lots of examples around though. On this page http://www.nklein.com/products/rt look for "input file". Here are the files I used to create the above image and the one in this post.
http://www.csh.rit.edu/~pat/lj/box.rt
http://www.csh.rit.edu/~pat/lj/box2.rt

Well, in my mind, when I imagine the image in Tetra's retina, it is slightly transparent, so that I can see all the objects in it from my 3D viewpoint. (Of course, in reality it is quite opaque, but I'm just trying to "see" the image from my 3D perspective.) Is it possible to render these images with partial transparency? :-)

Yes, but none of the pictures that I've ever rendered that way look good at all. I think it's mostly because the whole idea of front-and-back become muddled. An object that starts at a point in the kata direction and fans out to sphere in the ana direction looks exactly the same as a whole slew of other objects that have a point slice and a sphere slice.

Here's the same cube at a different angle:


Here's the same things with the transparency effect... (and a black sky)



And, for completeness, the input files:
http://www.csh.rit.edu/~pat/lj/box-trans.rt
http://www.csh.rit.edu/~pat/lj/box2-trans.rt

Also, if Tetra's retina is anything like ours (and I'm working on that assumption), it would have higher resolution around the center, and lower resolution on the periphery. But we'll let that slide, since the computer deals better with an evenly distributed resolution.

Agreed on both points.

[PWrong]

I'm not sure about this bit:

"Remember that Tetra does not merely see 3 faces of the cube the way we would see it. She sees all 6 faces simultaneously"

This is true, but she doesn't see all of the cube. You can only see one side of a piece of paper. Maybe you could see the whole thing if it had zero thickness, but it doesn't. Likewise, a wooden cube existing in 4D would have a very small tridth, but not zero. So a 3D cube in 4D would have two sides, and Tetra can only see one side at a time.

Other than that, it's very good so far.

[quickfur]

I'm not sure about this bit:

"Remember that Tetra does not merely see 3 faces of the cube the way we would see it. She sees all 6 faces simultaneously"

This is true, but she doesn't see all of the cube. You can only see one side of a piece of paper. Maybe you could see the whole thing if it had zero thickness, but it doesn't. Likewise, a wooden cube existing in 4D would have a very small tridth, but not zero. So a 3D cube in 4D would have two sides, and Tetra can only see one side at a time.

You're right, if I assumed that the front cell of the hypercube is an isolated object that can be removed, then it must have some "tridth". I was working under the assumption that it was merely a 3D cube.

Nevertheless, it is true that she still sees all 6 faces (and its inside) at once, because a 3D cube with non-zero tridth is essentially a very flat hypercube. As such, it would have 2 distinct 3D volumes, each of which is cubical in shape. (Well, technically it has 8 volumes, but only 2 significant ones...) She would only be able to see one volume at a time, but will see that volume in its entirety. I actually addressed this issue in the later chapter on surfaces, where Tetra shows us a piece of 4D paper which has two sides (i.e. volumes). I guess I'll have to revise the earlier chapters to not contradict this point. :-)

[quickfur]

(...) Here are the files I used to create the above image and the one in this post.
http://www.csh.rit.edu/~pat/lj/box.rt
http://www.csh.rit.edu/~pat/lj/box2.rt

OK, I'll take a look at those and try it out. Thanks!

(...) Is it possible to render these images with partial transparency? :-)

Yes, but none of the pictures that I've ever rendered that way look good at all. I think it's mostly because the whole idea of front-and-back become muddled. An object that starts at a point in the kata direction and fans out to sphere in the ana direction looks exactly the same as a whole slew of other objects that have a point slice and a sphere slice.

True. I was thinking, though, of color-coding displacement in the 4th direction, so that farther objects appear, say, more blue, and closer objects appear, say, more red. (Or maybe just different intensities of red would be good enough.) "Flat" things like the cells of a hypercube would have homogenous color when viewed "cell-on". I think this should turn out OK, although it might depend on the 3D viewpoint of the projected scene.

Here's the same cube at a different angle: (...)

Interesting. I wouldn't know how to use this to illustrate what I'm trying to describe, though, 'cos the reader would have to somehow piece these slices together in his mind. I was hoping for some way of illustrating the actual 3D objects that comprise the image in Tetra's retina.

Here's the same things with the transparency effect... (and a black sky) (...)

Hmm you're right, it's kinda hard to see. :? In the 2nd image, though, it looks like it's just super-imposing the slices with alpha-blending. Is there a way to do actual transparency of the projected 3D volumes?

And, for completeness, the input files:
http://www.csh.rit.edu/~pat/lj/box-trans.rt
http://www.csh.rit.edu/~pat/lj/box2-trans.rt

I can't seem to access these two files...?

[jinydu]

Ok, now I can view the web site.

Yes, some pictures are needed please. I'm struggling to visualize your description of how Tetra sees a rhombic dodecahedron. The unfamiliar terminology might be part of the difficulty.

[quickfur]

Ok, now I can view the web site.

Yes, some pictures are needed please. I'm struggling to visualize your description of how Tetra sees a rhombic dodecahedron. The unfamiliar terminology might be part of the difficulty.

If it's of any help, here are some images of the rhombic dodecahedron: (1) (2)

It doesn't quite show where the hypercube slices go, unfortunately; I'll have to hand-draw a diagram when I get back from the weekend.

[pat]

Interesting. I wouldn't know how to use this to illustrate what I'm trying to describe, though, 'cos the reader would have to somehow piece these slices together in his mind. I was hoping for some way of illustrating the actual 3D objects that comprise the image in Tetra's retina.

Yes, but showing a 3-D volume of pixels on a flat page is not so very easy.... 8^)

Hmm you're right, it's kinda hard to see. :? In the 2nd image, though, it looks like it's just super-imposing the slices with alpha-blending. Is there a way to do actual transparency of the projected 3D volumes?

Well, those are just blendings of the different frames. I can do semi-transparent objects in my raytracer, but none of the colors showing through other transparent colors ever come out looking right. There's no way in my raytracer at the moment to do anything where the output image has transparency (except that the parts outside of the viewing area can be transparent, but the sky cannot). I should add that in.

And, for completeness, the input files:
http://www.csh.rit.edu/~pat/lj/box-trans.rt
http://www.csh.rit.edu/~pat/lj/box2-trans.rt

I can't seem to access these two files...?

Yep, I realized this morning that I hadn't put those up there. They are there now.

[quickfur]

Interesting. I wouldn't know how to use this to illustrate what I'm trying to describe, though, 'cos the reader would have to somehow piece these slices together in his mind. I was hoping for some way of illustrating the actual 3D objects that comprise the image in Tetra's retina.

Yes, but showing a 3-D volume of pixels on a flat page is not so very easy.... 8^)

True... perhaps there's really no way to fully depict everything in a single image, since it's only 2D. I'll probably have to resort to multiple 2D images to fully describe the 3D image. For example, I could perhaps render a transparent rhombic dodecahedron, and set the color of one cell to red while the others are transparent white. Then I can show another image with a different cell colored red, etc.. (Sorta like using glass pieces to assemble the rhombic dodecahedron, if you will.) This should at least give some idea of how the 3D cells are distributed around the vertex. Maybe I can even show parts of the wireframe in the images so that the reader can see how the cells connect with each other.

The wood-grain structure would have to be in a completely separate image, of course, otherwise the result would be completely incomprehensible.

Well, those are just blendings of the different frames. I can do semi-transparent objects in my raytracer, but none of the colors showing through other transparent colors ever come out looking right. There's no way in my raytracer at the moment to do anything where the output image has transparency (except that the parts outside of the viewing area can be transparent, but the sky cannot). I should add that in.

Just wondering, how exactly does your raytracer handle 4D viewpoints? Does it treat the 3D viewpoint separately from the 4D viewpoint? There's a thesis by one Steve Hollasch that describes an approach where you can render multiple 3D viewpoints of a 4D scene projected into a 3D volume. I thought it was a nice idea, 'cos that gives you the freedom of examining the projection from different angles while keeping your 4D viewpoint static.

I can't seem to access these two files...?

Yep, I realized this morning that I hadn't put those up there. They are there now.

OK thanks. I won't be able to look at them until next week, though, 'cos I'll be away for the weekend.

[pat]

My raytracer does not distinguish at all between any spaces, realms, directions, etc. It is fully n-dimensional (for positive integer n). It just happens to output pixels so that the first axis is forward, the next axis is left, the next axis is up, the next axis is frames to the left, the next axis is frames down, the next axis is frames of frames to the left, etc.

I'll have to look at Hollasch's thesis. I hadn't seen anything from him beyond his 4-D raytracer.

[quickfur]

My raytracer does not distinguish at all between any spaces, realms, directions, etc. It is fully n-dimensional (for positive integer n). It just happens to output pixels so that the first axis is forward, the next axis is left, the next axis is up, the next axis is frames to the left, the next axis is frames down, the next axis is frames of frames to the left, etc.

I see. I wonder if it makes sense to ray-trace N dimensions based on N-2 viewpoints (the reason for 2 being that the output image is 2 dimensional). Each viewpoint reduces N dimensions to N-1 dimensions, so N-2 viewpoints will reduce N dimensions to a 2D image for any N. The flexibility in having many viewpoints is so that you can examine the projection of N from many different angles in (N-1) space while keeping the N viewpoint fixed. As we can tell from examining 4D objects, it's useful to be able to look at the same projection of a 4D object from different 3D angles, so that we get a clear idea of what comprises the image. I imagine this applies to higher N dimensions as well.

Just my thought.