Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Hello there. I've been studying the geometry of the fourth dimension for maybe a year or so now. It's really interesting. I'm not good at the mathematical aspect, but I can make sense of most 4D shapes at this point. However, when I became able to understand what 4D objects really looked like, I discovered a few things that surprised me. Most of you will probably have already known this, but whatever.

First off, from what I can tell, most perspective projections on almost all websites with information on tesseracts is incorrect, all because of a simple error in the pictures. See, almost all pictures of the tesseract depict the tesseract as a large cube with a smaller, internal cube inside it, such as this:

Now, #1. The "center" cube is too small to make this an actual tesseract. What this projection is telling me is that this figure is actually a hyper-rectangular prism. If it were an actual tesseract, the smaller cube would likely be bigger. But now on to the bigger problem with this picture:

#2. The lines connecting to smaller cube to the bigger cube are projected completely wrong. This aggravates me because it makes tesseracts much harder to understand, and also because this mistake is made in nearly every picture of this kind of projection of the tesseract. I will now explain why this is wrong. The lines that connect the two cubes are projected as leading INSIDE the bigger cube itself. The correct way to show these lines would be to put them BEHIND the larger cube instead of leading to the inside of it, like in this picture:

If you look at how the far cube is shown now, you might be able to make more sense of the tesseract's actual structure. To aid with this correct projection, I think of the two front and back cubes as completely flat, like paper. These cubes are then connected by lines to and from all of each cube's vertices. Of course, I know that 3-cubes aren't completely flat in 4D, retaining some kind of 3D depth. But that's about it for this kind of projection. Of course, I could be wrong about this entire thing. I could be looking at it the wrong way. But simply making inferences from lower dimensions seems to yield more logical results, so tell me what you guys think of this.

Also, I found a way to show an opaque tesseract. This has been found out before, but it's helped me with my studies a whole lot. Basically it shows that the tesseract itself is little more than a cube with cubes for sides.

http://img28.imageshack.us/img28/8785/imsuchagenius.png

Ignore the rather gloaty filename.

Now, I figured this out by doodling an opaque cube and then spontaneously getting the idea to "lift" each square side of the cube to the third dimension, successfully making an opaque tesseract. This outcome can also be reached by simply connecting two opaque cubes by their vertices.

Questions or comments anyone? I'd like to know what you fellows think of this.
Sir Puppum Hat
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### Re: Discovery About Tesseract Projections

Pictures of four dimensions, printed onto 2d paper, have to loose two dimensions in the process. Usually, this takes the sense of a projection onto 3-paper, which we then make 'transparent' to see all of the picture.

The tesseract for example, can be first a picture taken quite close to a face. The outer cube would appear very much larger than the inner one. It's looking at the edge-frame of the tesseract, that your mind might fill in the missing faces etc. It's pretty good for "walking the surface"

The opaque example that ye show is also valid, since it shows one side of the thing. It's how a 4d being would see a solid tesseract. It's not really good for grasping the geometry of the thing, though. You see that the 4d figure is shown in a isometric projection, (ie makes a 3-paper thing), and then that picture is "exploded" to show the interior of it.
The dream you dream alone is only a dream
the dream we dream together is reality.

wendy
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### Re: Discovery About Tesseract Projections

Hi, and welcome to the forum!

Unforunately, to be blunt, you are indeed

Think of the projection of a cube. If you look at a cube from any of its faces, so that you can only see one face of the cube rather than two or three, and now imagine removing that face so you can see through to the back of the cube, you see the back corners of the cube inside it. In other words, the face-first projection of a cube is a square inside a square. In the same way, the cell-first projection of a tesseract is a cube inside a cube.

The second point you made can't be refuted with a 2D analog, but wendy's post should be enough for that.

Keiji

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### Re: Discovery About Tesseract Projections

I get what you mean, but it still doesn't make much sense why the farther cube looks as if it was "inside" the other, rather than being actually BEHIND it. I know it would be foolish to try and question you guys, since you have waaayy more experience with this than me, but if you would take a look at this, I'd appreciate it if someone could tell me what's wrong with my reasoning here.

http://img708.imageshack.us/img708/4136 ... ngblah.png
Sir Puppum Hat
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### Re: Discovery About Tesseract Projections

It all depends on what ye mean by "behind". Recall that we are loosing two dimensions in the projection.

In the first projection (4D to 3D), the view is taken on the line through cube-faces. This makes the second cube appear inside the first, since they are concentric and the further is smaller. The second view is taken at some off angle, to reveal all of the detail.

In any case, the presentation is for people to "walk the surface", and see the cubes three at an edge. The eight cubes are clearly visible here. It would not be so if the projection is something like a single square, or a hexagon, with its centre connected to the outer edges.

It should be noted that there are a lot of people wrangling 4d and higher without the vaguest notion of what goes on up there. Many of the vague notions that we get from 3d go out the door in 4d. Even the new ones that we make between 4d and 8d go out the door in nine dimensions.

For example, in nine dimensions, the densest packing of spheres consists of two separate arrays of spheres, that if one gives it a decent shake, one could fall out of the other!
The dream you dream alone is only a dream
the dream we dream together is reality.

wendy
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### Re: Discovery About Tesseract Projections

Based on your last picture I think you might be right. Remember you're actually projecting 4D onto 2D, not onto 3D, which makes it complicated. If you think about building a projection of a tesseract in 3D (you can do this with toothpicks and blu-tack), then it's ok to have one cube inside the other. If you then take a photo or drawing of that construction you'll get a shape that's different from your pictures.

So I guess you could say that a projection from 4D onto 2D is different from a projection of (a projection from 4D onto 3D) onto 2D. I'm not sure if the projection from 4D to 2D is the same as your picture or not, there are probably lots of consistent ways to do it.

PWrong
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### Re: Discovery About Tesseract Projections

I think both projections are right. just as a cube with one face off can project as a square within a square when seen dead on, from another (third dimension) angle one square appears to be behind the other. Your projection of a cube behind the other cube is just seeing the same object from a different fourth dimensional angle. But this doesn't make the more common way of drawing it wrong, your way just makes it easier for you to understand, because it provides more information, but it could be harder for someone else to understand.
vegangypsie
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### Re: Discovery About Tesseract Projections

The translation you are making when "inventing" another dimension is somewhat arbitrary and depends on your perspective. Some would say the 4th dimension is actually time or entropy, but for the purposes of visualizing an extra dimension neither of these concepts is very useful since our visualization of objects is based specifically around 2D space (with "interpolated" 3D spacial ideas existing only in our head using shading cues, depth of field, dual vision from more than one eye, physical touch, perspective information, etc). So in order to compact 4 dimensions into a 2D or 3D space that we can understand with our eyes (which are only capable of handling 2D) and brains (which only infer 3D information from 2D images and touch), most people would invent a new direction in 3D space to move to so that we can at least get a representative 2D picture of what 4D is. With that said, choosing to simply make an extra copy of the cube and pull it in an arbitrary direction as you are doing is an acceptable method of generating a compacted 2D picture of a 4D object, albeit once again, arbitrary. Just as is choosing a field of view width for a 2D perspective drawing is for a depiction of a 3D object, or as is choosing a camera lens for a fish-eye or zoomed picture. If your decision was to pull the cube away from the first and connect them and call that "the extra dimension", then your illustration is perfectly acceptable and the facing order would make sense.

The reason most models depict a cube within a cube, I'm guessing, is probably due to the fact that most people would naturally rather prefer to consider the 4th dimension as being the "scale" (it is arguable whether scale is useful as another dimension), rather than time, entropy, or a new direction; since I would expect, it seems more instinctive than simply moving the cube in a random direction. Therefore the result is obviously a "cube within a cube" and may help explain why more illustrations that are done this way and are more readily understood and popular. This perspective of a 4D object is also a perfectly valid depiction because the definition of the 4th dimension's "direction" is simply different than yours; it moves smaller and further inside itself and centered rather than away from it with the same scale.

Another valid perspective is merely the animation of a cube spinning and calling the 4th dimension time (with the 3rd dimension compacted into the 2nd as always), with the dimension of time being compacted into a series of 2D/3D images and calling each one a "temporal" snapshot. But we already have a good understanding of how this works on a fundamental and natural level so that isn't very interesting to us since we aren't adding anything new to our understanding, and in most discussions of this sort time is removed from the "stack" for exactly this reason, and thus we look on for a new name and direction for the 4th even though "time" would have sufficed perfectly well. Yet another perspective is dealing with the dimension of entropy which delves into quantum physics and randomness which very quickly becomes increasingly difficult to conceptualize which I don't even feel like going into myself at all since I have a very difficult time comprehending it.

You must not forget that as a species and human being, you are built with some limitations that don't allow for you to comprehend new directions. Your eyes can only image 2D spacial information, your brain is only meant to handle 3D spacial information internally, with additions and amendments that deal with the temporal dimension stacked ontop of that; and anything that you wish to imagine outside of those dimensions must be compacted into that set of space (or less) which is therefore going to carry some arbitrary truncations until humans are more capable of thinking outside of their design. Which dimension you choose to compact from naturally will change the appearance of the object in 2D space drastically but it is still the virtual equivalent of choosing any other definition of the 4th dimension.
gumenski
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### Re: Discovery About Tesseract Projections

gumenski wrote:[...]You must not forget that as a species and human being, you are built with some limitations that don't allow for you to comprehend new directions. Your eyes can only image 2D spacial information, your brain is only meant to handle 3D spacial information internally, with additions and amendments that deal with the temporal dimension stacked ontop of that; [...]

I would argue that this is merely an assumption (albeit an understandable one) about how the brain works. I have heard of experiments where human subjects were immersed in a 4D virtual environment, and, after some period time, started developing facility with manipulating 4D objects in their native space. Unfortunately, I have not been able to find out any more details about these experiments -- how 4D was represented, or how the subjects interacted with the virtual environment. But it does indicate that the human brain is quite a flexible thing, and, given the right stimuli, may be induced to do things we didn't think it could do before.

There's also the question of whether the 2D-ness of the retina is hard-coded in the brain, or whether it is the result of the brain rationalizing the signals it receives from the optic nerves. Remember that the density of light-sensitive cells in the retina is uneven, and one would imagine that the nerve density is also uneven, so for the brain to get any coherent interpretation of the seen images at all, it would have to be able to somehow unscramble the uneven distribution of light signals so that it forms an undistorted, "flat" 2D image. I find it a bit hard to believe that this mapping is hard-coded, since eye shapes do vary between people, yet we all still perceive a circle as a circle and not as an ellipse or some other distortion thereof. So the brain must be somehow internally reconciling the uneven optic signals in such a way that a coherent interpretation is formed. Furthermore, the brain appears to have the tendency of trying to reconcile the two 2D images from either eye in such a way that a 3D model results. In other words, our 3D perception appears to arise from the brain's tendency to try to reconcile things: reconciling the uneven distribution of optic nerves in the retina so that it forms coherent 2D images, and reconciling the differences between two 2D images as seen by either eye, with the result that a 3D model is formed even though there is no 3D at all in the optical signals.

So it appears that the brain's tendency to reconcile inputs is quite general, and it would appear that, if there were a way to feed 3D data directly into the brain, it might begin to "see" 3D directly, and, given stereoscopic 3D feeds that morph in a coherent way, perhaps even infer 4D depth and construct a 4D model that rationalizes the input signals. Whether such a thing is actually possible still remains to be seen, of course, but I wouldn't underestimate the flexibility of that marvelous neural organ of ours up there.
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