Talk:Tesseract

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Contents 1 "Wind"?? 2 Stereographic projection of a tesseract 3 unfolding the tesseract 4 "link cells"? 5 information theory? 6 Tesseracts in Literature/ Media 7 Hypercubes in computer architecture

[edit] "Wind"??

I don't know who added the paragraph on nets being unstable, but it doesn't belong in the Projection to 3 dimensions section. Also, I'm not sure I understand what it's trying to say. What exactly is meant by 'wind' blowing the net over? Can somebody please clarify? I suggest we move it to a more appropriate section, and perhaps re-word it so that it's clearer.—Tetracube 00:54, 23 July 2006 (UTC)

I'm pretty sure it's referring to the fact that a figure built from squares, in 3 dimensions or 4, isn't rigidly self-supporting if the vertices can swivel (this is why trusses are made from triangles). I don't see how this is useful to have in this article, though, so if it's causing confusion, by all means remove it and reword the section. --Christopher Thomas 22:53, 16 August 2006 (UTC)

[edit] Stereographic projection of a tesseract

I haven't built stereographic projections of 4D hypercubes, but it seems that a cube on a stereographic projection should not have any straight lines, as it has in the main picture. Am I guessing right, that only vertexes are stereographically projected, and lines are just drawn straight to connect correspondent vertexes? -- Dubovik 08:59, 1 August 2006 (UTC)

It's not the projection, but the polytope representation itself. Is a polytope a "tiling on a hypersphere", or "flat elements in 4-space" or BOTH? The first interpretation has curved edges (geodesics on the curved surface), and the second straight edges in flat space. Straight lines should map to straight lines in a stereographic projection. Tom Ruen 09:29, 1 August 2006 (UTC)

No, circles (including straight lines as a degenerate case) map to circles. Possibly Tom had in mind something other than stereographic projection. —Tamfang 01:17, 18 August 2006 (UTC)

We should distinguish between the two representations somehow, not sure what's best. My vote is for flat polytopes, and explain the curved models are different, like a cube blown/stretched out into a spherical balloon. Tom Ruen 09:29, 1 August 2006 (UTC)

Um, wait... why is the blue image at the beginning of the article labelled "stereographic projection"? That's not a stereographic projection; that's a perspective projection of the tesseract into 3-space.—Tetracube 16:26, 1 August 2006 (UTC)

Tom found stereographic projections of many uniform tilings of S3, and added them to some other polychoron articles; possibly he considered replacing the image here but changed his mind after changing the label? —Tamfang 18:04, 1 August 2006 (UTC)

Okay, I dropped in the new image as well. I guess I accepted a "stereographic projection" as a special "perspective" projection viewed from a point on the 4-sphere. Maybe it only makes sense to talk of this projection with points/edges/faces/cells ALL only on the 4-sphere surface. I guess "perspective" is "more general" in the sense that the point of projection can be placed anywhere from the n-sphere surface and out to infinity where it becomes an orthoscopic projection. Tom Ruen 18:53, 1 August 2006 (UTC)

Ah, I see. On another note, does anyone know who came up with the blue image? Is it possible to perform the projection from a different viewpoint? I'd like to see projections from other viewpoints. The one represented is cell-first, which is one of the most common, but there is also vertex-first, which yields a rhombic dodecahedral envelope, and is analogous to the vertex-first projection of the 3D cube (hexagonal envelope). Well, it's a regular rhombic dodecahedron if it's an orthogonal projection; otherwise it's slightly distorted. Edge-first orthogonal yields a hexagonal prism, and face-first orthogonal yields a cuboid, but both are more interesting when you use perspective projection (I don't know what to call the shapes, though I can visualize them in my mind).—Tetracube 19:10, 1 August 2006 (UTC)

[edit] unfolding the tesseract

There are 261 ways to unfold a tesseract:

Turney, P.D. (1984), Unfolding the tesseract, Journal of Recreational Mathematics, 17 (1), November, pp. 1-16.

http://members.rogers.com/peter.turney/Unfolding.pdf

Salvador Dali's famous painting Crucifixion ('Corpus Hypercubus') shows one of the 261 unfoldings:

http://lloydsfunds.com/dali_crucifixion.html

- Peter Turney

Sadly, neither of these links work any more. 130.194.13.102 05:25, 16 August 2006 (UTC)

a search for 261+tesseract yields Turney paper —Tamfang 06:34, 16 August 2006 (UTC)

[edit] "link cells"?

On July 17, Tom changed "[[cell (mathematics)|cell]]s" to "link [[cell (geometry)|cell]]s", with the annotation "link cell (geometry)". Assuming that insertion of the word "link" was a mere lapse, I'll remove it, as I can't think what it might mean here. —Tamfang 01:20, 18 August 2006 (UTC)

[edit] information theory?

The section titled "Hypercubes in information theory" is wack. It tosses around layers, dimensions, and precision as virtually interchangeable terms. It contains statements like "hypercubes allow you to reference a number of factors at once," which is really referring to some kind of database, not to a geometrical form. It repeatedly addresses three-dimensional data sets, which are not hyperdimensional. And the whole section seems to stray quite a bit from information theory with discussion of analyzing business data, reference to "end users," etc. This section should be killed, or (gasp!) actually discuss hyperdimensionality in information theory. --Stybn 07:01, 28 August 2006 (UTC)

While you can reasonably think of n-tuples of data as organized in an n-cube, and while things like vector quantization happen in n-space, I agree that the section as-written is whack. --Christopher Thomas 07:11, 28 August 2006 (UTC)

First of all, I agree that I don't quite see what's the geometric connection between tesseracts and information theory (the emphasis being on geometric, since otherwise you're dealing with n-polytopes and not tesseracts specifically). Second of all, even if there is a connection, I doubt it's specifically 4-dimensional, and therefore belongs in the hypercube article, not here. I suggest cleaning up this section and moving it to polytope. Or just junk it.—Tetracube 17:45, 31 August 2006 (UTC)

I second the fact that this section needs a serious cleanup, some source citations and probably should be moved to polytope, as suggested by Tetracube, if not completely removed. It doesn't add anything to the article and as such serves no purpose whatsoever. Jim 10:10, 2 September 2006 (UTC)

Unsourced, unreferenced, and possibly inaccurate. Removed. --Kjoonlee 18:51, 13 September 2006 (UTC)

[edit] Tesseracts in Literature/ Media

I don't know if anyone has pointed this out yet, but the whole concept of tesseracts in popular media is completely wrong. Tesseracts aren't bigger on the inside than on the outside, and they definitely don't allow you to move in four dimensions just because you're inside one. Imagine a cube sitting on a piece of paper. If some two-dimensional being were to look at it, He would see a square. If he went inside, he would see a square. The only thing that would even indicate to him it was different from any other square is that, since it is essentially made up of an infinite amount of two-dimensional squares, it would appear to him to have an infinite mass. If someone would like to clean up and add this paragraph to the article, feel free to. Unless somebody can give me a good reason why I'm wrong. --Aljo 14:29, 13 September 2006 (UTC)

You're not wrong, per se, although the media concept isn't completely off-base either. Imagine, instead of moving under the bottom face of the cube, your 2D being instead crawled up one of the 4 sides of the cube (he would not notice he was doing this from his 2D perspective, of course). All of a sudden, there are 5 extra square areas inside the initial apparent square, so the inside of the square turns out to have an area equivalent to 5 squares. By dimensional analogy, if a 3D being were to "crawl up the side" of a tesseract instead of going under its bottom facet, he would discover that what he thought was a just a cube before turns out to have the volume of 7 cubes inside. Of course, there are other implications of such a geometry, such as unusual behaviour around the vertices, etc., so the media concept isn't exactly accurate either.

Note that this is just one way of rationalizing the media concept... another approach is, instead of being limited to the facets of the tesseract, imagine a very long cuboidal sheet of 4D paper (or a long rectangular sheet of 2D paper for the analogous 3D case) folded up in a zig-zag fashion so that it becomes a tesseract (cubical) stack of sheets. One end of the sheet is attached to 3D space with an entrance. As a 3D being enters through it, he finds himself in a very long corridor (which can be made infinitely long if the sheet is infinitely thin in the 4th dimension) made of cubical sections, which are the "folds". So here you have another way of extracting infinite 3D volume out of a tesseract while still having only a cubical interface to 3D space. Even if you didn't have an infinitely thin sheet, you still get a lot of 3D volume out of it (think, for example, how many pages would fill a cubical book in 3D: probably numbering in the thousands because they are paper-thin).

There are many other possibilities... although I agree that the media concept is a bit overly generalized (4D isn't equivalent to "fit any volume of any shape in a smaller space"). What would be more interesting is exploring the side-effects of such geometries. For example, And He Built a Crooked House by Robert Heinlein is a very nice story exploring the strange behaviour (as perceived by us 3D beings) around the vertices of a tesseract when 3D beings travel on its cubical facets, such that rooms that "should" be 90 degrees apart are paradoxically connected to each other in a seemingly impossible manner.—Tetracube 18:20, 13 September 2006 (UTC)

But then, going back to the cube, if he could walk up the sides from the inside, couldn't he walk up the sides from the outside as well? I assume this would make the outside and the inside the same size from his point of view. Also, if he were able to walk up the sides, that would somehow mean he on his own had the ability to move in the third dimension, and that he just needed a surface on which to do it, unless that's what you were refering to when you said "unusual behaviour around the vertices"? -- Aljo 19:13, 16 September 2006 (UTC)

I was referring to walking up on the outside. Of course, in discussing things like this, we're making a lot of assumptions. From a "logical" standpoint, there is no reason to assume that 4D objects, if they existed, would be made of the same stuff that objects in our 3D world are made of, and so any generalizations of physical behaviour is potentially inconsistent. Having said that, what is "correct" behaviour depends on what assumptions you're making. If a 3D object bumps into a 4D object, it could simply crumple up, or bounce off, or cut through the 4D object. To use the analogous 2D situation, if a 2D piece of paper bumps against, say, a cube embedded in the plane, it could either crumple up, or get displaced upwards/downwards (move up/down along the sides), or, if the 3D cube was soft enough, it could penetrate straight through (and therefore have no realization that it's a 3D object---for all it knows, the cross-section of the object is the entire extent of the object). All I'm saying is, it is possible to have a model of 3D/4D interaction such that the presence of a 4D object can be perceived as bigger inside than outside. This, of course, requires certain assumptions about how such interactions work; if you have different assumptions, then obviously you'll end up with a different set of behaviours.

Anyway, by unusual behaviour around the vertices, I mean how a 3D being would perceive the vertices of, say, a tesseract, if he was confined to its surface. Analogy: if a 2D being could only travel on the surface of a 3D cube, and light travels on the surface of the cube (bending 90 degrees at the edges, etc.), then the 2D being would not see the cube edges at all, it would look like normal, contiguous 2D space to him, except around the cube corners, where you can circle around something by travelling around only 270 degrees rather than 360 degrees. In other words, it looks like there's a "kink" in space around that point. In the analogous 4D situation, there would be "kinks" at the vertices and also along the edges (but not the ridges) of the tesseract's facets.—Tetracube 05:25, 18 September 2006 (UTC)

I still don't think a two-dimensional being can move in three dimensions, therefore he can't move up and down the sides of the cube. However, if he could, it wouldn't be bigger on the inside than the outside. My main point is that most of the examples in the "in literature" section don't belong here. --Aljo 19:10, 12 December 2006 (UTC)

[edit] Hypercubes in computer architecture

OK, surely this section belongs in hypercube rather than here?? I don't see why it should be here, since the description uses general n whereas this article deals exclusively with the 4-dimensional case.—Tetracube 01:44, 8 November 2006 (UTC)