Talk:Tesseract/Archive 2

Tesseract.gif
I've rendered a few versions of a rotating tesseract, and think that it doesn't make sense to put all of them on the tesseract page. However, I am having some difficulty choosing between them. I think that the difficulty is in finding a balance between what is more visually clear and what provides more eye candy. I've provided four versions of the file, from earliest to most recent. Which version would be preferred?

In addition to this image, I plan to render other 4D polytopes, and apply a uniform style to all of them (I recently added an animation to the 24-cell page. Input about what makes one image more desirable than another will help me to create better animations.  Thanks in advance for any comments.


 * Image:Tesseract.gif
 * Image:GlassTesseract.gif
 * Image:Tesseract2.gif
 * Image:8-cell.gif

JasonHise 05:44, 18 February 2007 (UTC)


 * 1) All very nice. I can't decide. It seem like the transparent versions have ray-traced reflections which for me distracts from seeing what apparent edges are real. If so, I'd vote for one with transparent faces and simple depth-sorted blending of direct images only. Tom Ruen 07:23, 18 February 2007 (UTC)
 * 2) Yeah, very nice. I like the second one (GlassTesseract). The third one has too much reflection, which becomes distracting. It would be nice also to use a different color for back-facing edges/faces (those that lie on the "far" side of the cube in the 4th direction). Well, if it doesn't become too confusing.&mdash;Tetracube 23:56, 18 February 2007 (UTC)
 * 3) Yeah, the raytracing is the main thing I am debating... the reflections are simultaneously both delicious eye candy and detrimentally obfuscating. I might be able to find a way to make ana/kata distance correspond to a color spectrum, though I'd need to do a bit of research to figure out how to pull it off.JasonHise 13:33, 19 February 2007 (UTC)
 * 4) (The following is copied from my user talk. — JasonHise 01:58, 24 February 2007 (UTC)) Please take this as just a constructive criticism. The tesseract you created isn't as easy to understand as this one, as yours is rotating in two ways, but this picture is obviously inferior in quality to yours. I wondered if it would be too hard for you to remove the superfluous rotation.  M rug  2  18:33, 23 February 2007 (UTC)[[Image:Changingcube.gif]]
 * 5) I was initially thinking that I needed to strike a balance between the flashy eye candy and something that is easy to understand, but I am now thinking that I should make two separate versions of the image and put both on the page: one that is more fun to watch, and another which would be more useful for building a mental model. For the latter, I think I am going to drop the reflective faces, make the bars thicker, and eliminate the rotation about the z axis (aka the z-w plane).  The only other thing I still need to decide is the camera angle.  Any preferences? — JasonHise 01:57, 24 February 2007 (UTC)
 * 6) My preference has always been for Image:GlassTesseract.gif, but I would have liked it to be slower in its rotation.  As it is, it's too hard for the eye to follow a single point or line.  Thanks.  &mdash;Lee J Haywood 10:34, 24 February 2007 (UTC)
 * 7) You might like some ideas from the java animation with hidden volume elimination I added to the references. This is basically real time 4d ray tracing though I didn't do it like that as microcomputers weren't up to it at the time! There is some documentation attached which explains more. By the way I didn't do it in this old java program but I found that nodding the image up and down when the user wasn't moving it gives me a better 3d feel than a sideways shake.  &mdash;Dmcq 08:36, 29 March 2007 (UTC) I've now found an MSc thesis on the same sort of thing to give it more verifiability Four-Space Visualization of 4D Objects by Steven Richard Hollasch 1991 plus Four-Dimensional Views of 3D Scalar Fields by Andrew J Hanson Pheng A Heng, though I don't know why they make such heavy work of it all.  &mdash;Dmcq 16:27, 3 April 2007 (UTC)
 * 8) A good trick for dealing with the complex internal structures in the 3D image when a 4D object is viewed is to only consider the 3D image as solid where the 4D colour changes quickly and as varying levels of transparency as it changes less rapidly. For very fast changing places like lines in a tesseract the 'solidness' should overflow and spread out a bit - that way you'll automatically get nice tubes for lines pictures like in the image above for a tesseract. Dmcq (talk) 09:01, 8 July 2008 (UTC)

"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.&mdash;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)

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.  &mdash;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.&mdash;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? &mdash;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).&mdash;Tetracube 19:10, 1 August 2006 (UTC)

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 &mdash;Tamfang 06:34, 16 August 2006 (UTC)
 * The tesseract seems pretty interesting, but what the fuck is up with the net?! Is it really six cubes or is it some other fucking optical illusion.  Maybe that should be explained in the article.  Montgomery&#39; 39 (talk) 20:35, 25 March 2009 (UTC)
 * I doubt tesseracts have anything to do with sexual intercourse and none of the secondary meanings in Fuck (disambiguation) seem applicable. Think you could manage to make the question more meaningful thanks? Dmcq (talk) 21:26, 25 March 2009 (UTC)
 * To answer the question as I think he asked it, the net is eight cubes. No, it's not an optical illusion - there are cubes at the back. 4 T C 02:05, 14 February 2010 (UTC)

"link cells"?
On July 17, Tom changed "cells" to "link cells", 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. &mdash;Tamfang 01:20, 18 August 2006 (UTC)

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.&mdash;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)

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)
 * "If some two-dimensional being were to look at it, He would see a square. If he went inside, he would see a square" - actually he would see a line, since he's 2 dimensional - the only way he could actually realize or know (or perceive) that it's a square would be to move around it (which would mean that he is in fact in Time, and thus 2D+1D (of time)=3D total) - law of perceived dimensions - created by: :) BriEnBest 06:27, 25 March 2007 (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.&mdash;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.&mdash;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)


 * Uh-kay I am like totally freaked out by the whole "HCE turns into a tesseract in Joyce's Finnigans Wake" with a (incomprehensible without advanced degrees in puns and etymology) quote, which describes approximately... zero terreacts. The fact that the word "tesseract" appears in the text doesn't seem to have anything to do with a man transforming into a geometric idea as the article currently claims.  Anyways, I'm assuming good faith; can someone check this out? Is this true?  Should it even be in the article? Re-reading that quote is making me confuzled. Dikke poes 16:55, 19 December 2006 (UTC)


 * Well, whether or not a 2D being can move in 3D is really up to one's interpretation, but your point is right: most of the links in the literature section are irrelevant and should be removed. I don't see why it should become the cistern for collecting anything and everything that makes the slightest allusion to hypercubes or 4D in general.&mdash;Tetracube 18:01, 5 January 2007 (UTC)

What about Madeleine L'Engle's book A Wrinkle in Time? The concept of a tesseract is explained much differently from cubes or hypercubes, but isn't entirely off-base, either. It's a way of taking the 4th dimension (accepted, in the book anyway, to be "time"), and "folding" it, thus allowing for time-travel, and travel through the space-time continuum. It uses different language, but the concepts aren't that far off from what the tesseract is described as in this article. Therefore, this book should be included in the list of literary/media features of tesseracts.


 * Eeh, actually, I think she's way off-base. I think in Wrinkle the word tesseract refers to "the act of tessering", tesser being an invented verb for The Interogatives' main mode of travel. And besides, to wrinkle time or space, there needs to be another dimension into which spacetime can wrinkle, for a grand total of five dimensions. Even if cubes were involved (which they aren't), L'Engle's "tesseract" would belong in Hypercube! Proginoskes (talk) 22:40, 18 May 2008 (UTC)

I'm just noticing a bunch of bands and the like who are trying to use this section for some cheap promo. I don't think that sticking the word tesseract into the name of your group really merits an addition to this page, especially when it includes an external link that is completely off-topic. I already removed a few, though I would not be surprised if they tried to made their way back here. Mbruno42 01:30, 8 September 2007 (UTC)

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.&mdash;Tetracube 01:44, 8 November 2006 (UTC)

Tesseract
tesseracts: 4dimantions hypeper dimentional i am 11 so if i am missing somthing put it on. —The preceding unsigned comment was added by Deathclaw11 (talk • contribs) 11:52, 20 February 2007 (UTC).


 * wtf...?

Ionic Greek to Roman letters?
I reverted a change which added the Roman letter equivalence preceeding the Greek letters. The translation looks correct, but it seemed confusing as added. Its pasted below in bold, if anyone thinks its worth keeping or making more clear. Tom Ruen 03:40, 28 April 2007 (UTC)
 * ...the Ionic Greek tesseres aktines, τεσσερες ακτινες” (“four rays”), referring to the four lines from each vertex to other vertices.

Sentence fragments and clarification
The following appears in the article:
 * "A multitude of cubes that are nicely interconnected. The vertices of the tesseract with respect to the distance along the edges, with respect to the bottom point."

I bet the first part is a suggestion for conceptualizing a tesseract, while the second is impenetrable. Can someone who knows make these into sentences? Also, it seems like there are some places where it's unclear whether a tesseract is homeomorphic to a 4-ball, a 3-sphere or maybe even a graph. The article's main definition implies the first, while other descriptions seem to conflate things like whether a square is homeomorphic to a 2-ball or to a circle, leading to my confusion. In other words, is a tesseract "hollow?" Orthografer 06:24, 9 November 2007 (UTC)

Double Rotation
I have a question about the animation of the tesseract performing a double rotation. If you would do something similar to a cube, you would have this: http://img293.imageshack.us/img293/112/rotatingprojectionas0.gif But in reality this is a cube rotating on a single axis, and the animation of that cube rotating is being rotated clock-wise. This can't be considered a cube rotating in 2 axis (or can it?). When we rotate a cube in 2 axis, the shape of the 2d projection will distort and you will see 3 faces of the cube at once at some points. By analogy, I'm guessing you can only rotate the tesseract without afecting the shape of its 3d projection if you only rotate it on a single axis, and if you rotate it in more than 1 axis, this shape would vary between 1 and 3 cells, and not just 1 and 2 like in the animation. Is it right? (i.e, isn't that rotating only the projection of a tesseract performing a single rotation?) —Preceding unsigned comment added by Chronometrier (talk • contribs) 21:27, 1 January 2008 (UTC)


 * I'm sorry to insist on this, but I think this is an interesting question to ask here, and maybe there are more people with the same doubt. I'm still a little confused by the animation of the "double rotation" on the article, but probably is only the therm that is misleading. I'm basing this only on the analogy with the rotating 3d cube, I'm not a mathematician. The cube has 6 faces, and if you rotate it in a way that the top and bottom face dont change orientation, you will only see 4 faces on the "front". The tesseract has 8 cells (the 6 bounding cubes that form the outer cube, the outer cube itself, and the inner cube). In both the animations of the single and the double rotations, only 4 of these cells become the outer cell, and there are 4 cells that remain "reversed" the entire time. Isn't it possible to rotate the tesseract in a way that all cells eventually becomes the outer cell? If it is, what would that rotation be called? I hope you understand why I'm confused. Chronometrier (talk) 04:32, 14 January 2008 (UTC)


 * "Rotation about an axis" is a concept that only makes sense in three dimensions. In higher dimensions the correct analogy is "rotation in a plane". In three dimensions, rather than thinking about rotating something about the z-axis you should think about rotating it in the xy plane. In four dimensions, you can do rotations in the xy, xz, xw, yz, yw, or zw planes, but there is no such thing as a "rotation about the w-axis". Now a 2-plane will intersect the tesseract in four of its faces (since a square has four sides). This is why only four of the cells "become the outer cell" in the animation. Hope that helps. -- Fropuff (talk) 05:10, 14 January 2008 (UTC)


 * I realized my post got too big and I was just repeating myself, so I removed some of my comments. (I'll try to post my questions in a forum in the future instead of here.) Fropuff: Thanks for replying. I didn't think about it this way before. Replace the word "axis" for "plane" in my first comment, on the part about the tesseract, that's what I was trying to ask. Chronometrier (talk) 23:54, 15 January 2008 (UTC)


 * I would say that the best way to understand this is that a rotation in n-dimensional space is specified by a collection of n-2 vectors. For example, in 3-dimensional space, the length of the vector specifies the magnitude of the rotation (you choose the units) and the direction specifies the axis. If k of these vectors are nonzero, this would be the situation of having "k axes of rotation" using the vernacular in your comment. As for what a rotation such that "all cells eventually becomes the outer cell" would be called, I think the word would be transitive.  But I don't think such a thing exists for cubes in n-dimensional space unless n=2 (ie, squares), because any face intersecting the subspace spanned by the "axis vectors" will be fixed by the rotation. Orthografer (talk) 22:33, 19 January 2008 (UTC)


 * Just to wrap this up, here are the answers I found. It is possible to rotate the tesseract freely as described above, and you can see up to four cells at a time. In this case, the shape of the projection is a "rhombic dodecahedron" (this is partially in the article, but it's a little confusing). As for the double rotation, the two rotations are "independent" from each other, unlike in the cube rotation for instance, where two rotations can only be combined into a single rotation (this isn't in the article, and the explanation in SO(4) is not very friendly either). Chronometrier (talk) 16:32, 29 March 2008 (UTC)

Net?
"Unfolding the tesseract: The tesseract can be unfolded into eight cubes, just as the cube can be unfolded into six squares. An unfolding of a polyhedron is called a net. There are 261 distinct nets of the tesseract. The unfoldings of the tesseract can be counted by mapping the nets to paired trees (a tree together with a perfect matching in its complement)."


 * The second sentence of this section says the unfolding of a polyhedron is called a net. I agree, obviously, but I thought that this was a polychoron.  Shouldn't this state that the "unfolding" of any polytope into a lower dimension is called a net?  Or maybe the just lower dimension (would a one-dimensional net of a cube still be a cube?).  Anyways, I don't think this makes much sense and I'm going to change it.  If you wish, edit my edit.  98.27.171.83 (talk) 14:56, 2 January 2008 (UTC)


 * Good catch - I changed polyhedron to polytope to be correct. Tom Ruen (talk) 15:50, 2 January 2008 (UTC)
 * Thank you. I was going to edit the article, but when I switched tot he page, I forgot what I was doing and left it.  :)  98.27.171.83 (talk) 20:07, 4 January 2008 (UTC)

Tessera and tesseractic
Mark Ronan in Symmetry and the Monster1 writes,

"The tesseract is a four-dimensional crystal of type B, and its name comes from the Greek word tessera, meaning four. If you were measuring four-dimensional volumes you would use 'tesseractic centimeters', as opposed to cubic centimeters in three dimensions, or square centimeters in two dimensions."

This illustrates a connection between the words tessera and tesseractic which seem to be noun and adjective forms. In Latin tessera, -ae can refer to a cubic die or square tile and may suggest an alternative usage or origin for the word tesseract.

1


 * In connection with the idea of tiling it should be noted that the Greek word for build is κτιζω and that for bricklayer is κτίστης which would imply that a tesseract is made up of tessera. --Jbergquist (talk) 10:27, 11 January 2008 (UTC)


 * It would make as much sense to say that triumviratal is the adjective for trio. The connexion between tessera and tesser-act is that the latter was coined from two roots one of which was tessera – meaning 'four' (as in Greek), not 'tiles' (as in Latin).  Tesseractic is the adjective for tesseract, nothing else.  —Tamfang (talk) 06:14, 13 January 2008 (UTC)


 * The Greek word for the number "four" is "τεσσερα". But one might also consider the possibility that the root words are borrowed from some other language since there are similar words in ancient Egyptian to the Greek words mentioned above and Alexandria was one of the major centers of learning in the ancient world. "ţes" in ancient Egyptian was a kind of stone possibly cut stone or a stone block. "ţeseru" was the plural for "ţes". "qeţ" was the verb meaning "to build" and "qeţu" meant a "builder" or "mason". Could this be the missing link? Could there be some Masonic influence? --Jbergquist (talk) 23:55, 17 February 2008 (UTC)


 * Well, yes, it's conceivable that tesseract could have been coined from Egyptian roots meaning 'block'+'mason', or from Polynesian roots meaning 'dew'+'laugh'. Those of a less romantic disposition could look up the book in which Hinton introduced the term, and see what he says about the derivation.  —Tamfang (talk) 17:37, 18 February 2008 (UTC)


 * τεσσερα is Demotic (Koine) Greek which dates from the time of Ptolemy II who came to power in Alexandria in 285 BCE. Prior to that Attic Greek was the dominant dialect and the word for the number four was τετταρες, -α. I've read that τεσσερα is used in Homeric and Ionian Greek which would be consistent with an eastern or Egyptian influence. The Oxford Dictionary of Modern Greek translates four as τεσσερα. In German the word for tesseract is tesserakt which is more consistent with Greek. A tesseract is made from tessera would be logically more consistent with a bounded object than "rays" would be. I will have to look at Hinton to see what he says. Did he coin the word himself or is he repeating something? --Jbergquist (talk) 23:39, 18 February 2008 (UTC)


 * In The Fourth Dimension Hilton writes,
 * "If now the cube A moves in the fourth dimension right out of space, it traces out a higher cube--a tesseract, as it may be called."
 * There is no derivation for the word tesseract here. This was written in 1904(?). I haven't seen A New Era of Thought which is out of print. --Jbergquist (talk) 05:43, 19 February 2008 (UTC)


 * In The Fourth Dimension, on p. 37, Hinton writes,


 * "The blocks of stone out of which a house is built are the material for the builder; but, as regards the quarrymen, they are the matter of the rocks with the form he has imposed on them." --Jbergquist (talk) 00:05, 2 March 2008 (UTC)

Tesseracts in popular culture - Television and movies
"The television program Andromeda makes use of tesseract generators as a plot device. These are primarily intended to manipulate space (also referred to as phase shifting) but often cause problems with time as well."


 * This seems misrepresentative of the impact that tesseracts had in the plot of the last two seasons of Andromeda. While there were "tesseract generators" there was a dimensional spacetime phenomena known as "The Route of Ages" which basically a giant tesseract which links a nearly infinite number of universes through a single conduit. While minor, this would be more accurate if it was re-written to address the Route as well as normal tesseract generators. I'm bad at writing objectively right now, so if someone who has an easier time with that could rewrite the quoted entry, that would be great.—24.19.185.181 (talk) 11:03, 15 January 2008 (UTC)

Pictures of the "shadow"
Shouldn't it be stated near the top that the images shown here are not actually what a tesseract looks like, but merely what its shadow looks like in 3D? That's what Carl Sagan said, http://www.youtube.com/watch?v=Y9KT4M7kiSw&NR=1 64.236.121.129 (talk) 19:38, 4 February 2008 (UTC)

Looks like! :-) That's a good one. Depends on what you mean by looks like I guess. I believe blind people see. Perhaps a tactile device would be a good way of seeing 4 dimensions. The two most common representations are as x-ray images or via slices. In section 7 of Talk:Tesseract above I've referenced another representation I set up that is more like what an eye does so you might like that. And there's all sorts of way of showing distance in the 4th dimension. Was it Plato who said all our perception is of shadows? Thanks for the You tube link I hadn't thought of looking there. Really I need more time in my life for everything I'd like to do :) Dmcq (talk)  —Preceding comment was added at 00:06, 5 February 2008 (UTC)

Projected onto a 3-Sphere?
In the gallery section of this article the first image, I believe, says "(Edges are projected onto the 3-sphere)". I was wondering if this is correct. I would believe that it should be a "4-sphere", instead of "3-sphere". My reasoning is quite simple: If the edges are all projected at one time, which is what is implied, than it would be like a square being projected onto a circle (a tesseract is 4-D, a 3-sphere is 3-D. Likewise, a square is 2-D and a circle is 1-D.  There is one dimension of difference between the square-family object and the circle-family object). This is meaningless. However, one may project a 2-D square onto a (2-)sphere. There are zero dimensions of difference between the two objects. So, a cube (3-D) may be projected onto a glome (3-D sphere), but a tesseract (4-D) cannot. It must be projected onto a 4-D sphere.

Also, this is a menial point, but should it not be "a _sphere", not "the _-sphere"? Thanks. 98.27.163.42 (talk) 19:59, 13 May 2008 (UTC)


 * 3-sphere is a 3D surface embedded in 4D space, just like this image is a cube projected onto a sphere (2-sphere) in 3D space. Image:Square on sphere.png Tom Ruen (talk) 23:52, 13 May 2008 (UTC)


 * Yes, the notation is unfortunate. The 2-sphere (ordinary sphere) consists of only the outer surface, which is 2D (being a surface). Your idea of "sphere" includes the interior as part of the sphere, which would be 3D (a 3-ball). Double sharp (talk) 02:23, 15 April 2012 (UTC)

Mimsy Were the Borogoves
Is there any particularly good reason to believe the children construct a tesseract in the "Mimsy Were the Borogoves" short story? My impression, from reading the rest of this article, is that the idea is put forth by the movie, but I'm quite certain no such claim is made in the story. Either way, there's something wrong here. If there is reason to believe the children built a tesseract in the original story, the reference in the movie could not be an homage to "A Wrinkle In Time", which it predates by almost 20 years. Volfied (talk) 21:39, 7 June 2008 (UTC)

Tesseracts in popular culture
For the sake of future edits (since MANY of these will surely be RE-ADDED in the future as long as this cultural section exists), here's the references removed today: Tom Ruen (talk) 16:35, 2 July 2008 (UTC)

Television and movies Business Video Games Games
 * A hypercube is used as the main deus ex machina of Robert J. Sawyer's book Factoring Humanity, even appearing on its North American cover.
 * Piers Anthony's novel Cube Route also features a tesseract.
 * Alex Garland's second book is called "Tesseract: a novel".
 * The DC Comics crossover DC One Million showed a future Earth in which cities occupied extradimensional areas called tesseracts, leaving the planet's surface unspoiled. Similar technology was used for Superman's second most recent Fortress of Solitude, and was used as storage space in the headquarters of the original incarnation (pre-Zero Hour) of the Legion of Super-Heroes.
 * Ian Irvine's Sci-Fi Fantasy 'Tetrarch' Book two of the 'The Well of Echoes Quartet'.
 * David Lubar's 'Sleeping Freshmen Never Lie'. Tesseract meaning "spiraling into another dimension."
 * Umberto Eco references tesseracts in Foucault's Pendulum
 * The television program Andromeda makes use of tesseract generators as a plot device. These are primarily intended to manipulate space (also referred to as phase shifting) but often cause problems with time as well.
 * Gene Roddenberry's "Andromeda (TV series)" used the tesseract to represent a hole in the space-time continuum which could be used to travel forward and backward through time, as well as to different dimensions. Harnessing this fictional tesseract hole was never entirely possible in the show.  It was, however, used to stage the start of a new story arc and the end of two story arcs in the series.
 * A character in the television program Numb3rs shows a model of a tesseract in the second-season episode Rampage, during a discussion of using a 4-dimensional perspective to analyze an event.
 * The TV programme Strange Days at Blake Holsey High has an episode where the school campus transforms into a self-folding hypercube.
 * The movie Cube 2: Hypercube focuses on eight strangers trapped inside a massive cube full of further divided cubic rooms, inside one of which is a "trap" consisting of a rotating tesseract which shifts in the direction of the 8 strangers' movements, and kills one person by slicing them as the object rotated. Tesseracts featured elsewhere in the film, where it was noticed by the group that there were drawings of tesseracts in the rooms themselves, and it is also suggestable that the entire cube that they were inside was a form of tesseract.
 * The movie The Last Mimzy mentions tesseracts in a list of other geometrical shapes when the children are dreaming about the bridge across the universe, as does the short story on which it is based, 'Mimsy Were the Borogoves' (listed above). This may also be in homage to A Wrinkle in Time.
 * Tesseract Books was a prominent publisher of Canadian science fiction books. The company is now an imprint of Hades Publishing Inc.
 * Starflight included a tesseract as an artifact which could be found by exploring planet surfaces.
 * A tesseract forms the basis of the fantasy Advanced Dungeons & Dragons module Baba Yaga's Hut, which appeared in an early issue of Dragon Magazine, with the tesseract existing as the interior of the titular Hut.

Trivia II

 * A "trivia" selection collection deleted, so I also copied it below, for future reference. Well, I'd say perhaps something deserves to stay, like Crucifixion (Corpus Hypercubus), but whatever! Tom Ruen (talk) 23:48, 29 October 2008 (UTC)

Books/print Visual arts Movies
 * In Edwin A. Abbott's novel Flatland, 1884, a tesseract is imagined by the narrator.
 * Madeline L'Engle's novel A Wrinkle in Time uses tesseracts as a way for Meg Murry and her companions to travel to other planets and dimensions; however the description more closely matches a wormhole.
 * Carl Sagan describes the tesseract in great detail using layman's terms in the tenth episode of his mini-series, Cosmos, titled "The Edge of Forever."
 * Robert A. Heinlein mentioned hypercubes in at least two of his science fiction stories. In "—And He Built a Crooked House—" (1940), he described a house built as a net (i.e., an unfolding of the cells into three-dimensional space) of a tesseract. It collapsed, becoming a real 4-dimensional tesseract. Heinlein's 1963 novel Glory Road included the foldbox, a hyperdimensional packing case that was bigger inside than outside.
 * Hypercubes and all kinds of multi-dimensional space and structures star prominently in many books by Rudy Rucker.
 * Lewis Padgett's classic short story, 'Mimsy Were the Borogoves' features two children who construct a tesseract using information from the future. They ultimately disappear into another dimension.
 * Mark Clifton's classic short story "Star Bright" involves two superintelligent toddlers who become accomplished time-travelers through the use of visualized tesseracts.
 * Alex Garland's second novel, The Tesseract, uses the tesseract as a metaphor for those events in life which people cannot, or cannot easily, understand.
 * Robert J. Sawyer's novel Factoring Humanity features schematics for a strange hypercube technology being sent from an alien signal.
 * Alastair Reynolds' 2001 novel Diamond Dogs, Turquoise Days featured higher mathematical problems dealing with the shadows projected by 4-dimensional objects, specifically hypercubes.
 * Clifford Pickover's book, Surfing Through Hyperspace, uses a tesseract to demonstrate concepts of higher dimensions.
 * In William Sleator's novel The Boy Who Reversed Himself, the main characters travel to the fourth dimension and find it difficult to make sense of what they are seeing. The geometry of "4-space" is more complex than that of their own dimension; they see 3-D cross sections of everything. Cubes are hypercubes, knots easily come untied, and humans can be flipped over and "reversed" while in 4-space. Crosscanyon (talk) 23:35, 10 April 2010 (UTC)
 * The painting Crucifixion (Corpus Hypercubus), by Salvador Dalí, 1954, depicts the crucified Jesus upon the net of a hypercube. It is featured at the Metropolitan Museum of Art in New York, USA.
 * Cube 2: Hypercube, from the Cube film series, is based on characters trapped in a 4 dimensional hypercube, forced to figure a way out.
 * Gene Roddenberry's Andromeda sci-fi tv series feature use of 4-space via tesseracts as teleportation technology and the Route of Ages interuniversal portal looking like a Schlegel diagram of a tesseract.

Why were these all removed? Many are notable Dmcq (talk) 23:24, 29 October 2008 (UTC)


 * I'd like to state here that I second the removal of the trivia section. I have always felt that it was just a haphazard collection of random information, with no real coherent value. I do agree that some of the references are worth including; but I argue that they should be integrated into the main text of the article, not kept as a random list of haphazard facts. The latter attracts a lot of non-notable edits, which then has to be painstakingly monitored and maintained constantly to keep it relevant. Instead, we should integrate these references into the main body of the article, elaborating on the significance of each particular reference. E.g., how is it relevant to the article? Does it have historical significance? Does it exemplify the influence of the concept of tesseracts on, say, the novel form? We should not list all references indiscriminately; e.g., there is no need for 50 references to novels that mention tesseracts; rather, stating the impact of the concept on novels by listing a few representative references should suffice. In a nutshell, what I consider a more acceptable form of these references is actual discussion of the reference in question, rather than listing it just for the sake of listing it.&mdash;Tetracube (talk) 23:49, 29 October 2008 (UTC)


 * Wikipedia's manual of style agrees with you that anything worth keeping in an article like this needs to be part of the article text, not a list of miscellaneous items. It might be possible to use some of those examples to argue that tesseracts have had a significant influence on popular culture, but that would be original research, which wikipedia doesn't publish.  We'd need to find a reliable source to cite to back up the assertion that tesseracts have in fact had a significant influence on popular culture.  --Hugh Charles Parker (talk - contribs) 00:12, 30 October 2008 (UTC)


 * Well there's a number of books that go on about the culture and associated aspects for instance Nahin's Time Machines or Rudy Ruckner's books. So it definitely is notable and researched. So the main problem is to put just the important bits into the article in a coherent way. It is a pity I think that some people prefer to just delete things without putting work into them when they are notable but I'm not going to get in an edit war about it. Dmcq (talk) 08:31, 30 October 2008 (UTC)


 * I don't think that's quite fair to me. I'm not in a position to do that work: I'd never even heard of those books until you mentioned them.  I don't think we need to be negative about this, this is how wikipedia is supposed to work:  I've taken out a section that needed to go, and now there's nothing to stop someone with more relevant knowledge than me writing a good, well-sourced section on the influence of the tesseract on popular culture.  Eventualism FTW!  --Hugh Charles Parker (talk - contribs) 11:26, 30 October 2008 (UTC)


 * Yes I see, I do fall into the eventualism camp. I'm more concerned with having relevant information than having it all look professional like a printed work. Dmcq (talk) 11:38, 30 October 2008 (UTC)


 * Fair enough, but relevant is the key word here. The reason I removed it all is because there was no coherent point being made.  Those examples can only be useful to illustrate a point about the influence of tesseracts on pop. culture, and that point isn't made in the article at the moment.   --Hugh Charles Parker (talk - contribs) 12:38, 30 October 2008 (UTC)


 * I just had a look at Fourth dimension and it a some of the most important references. There's a couple like the Crucifixion (Corpus Hypercubus) which quite rightly aren't there and probably should be here, I'll have a think about integrating them and referring to fourth dimension for the wider (deeper, thicker?) picture Dmcq (talk) 19:55, 31 October 2008 (UTC)


 * I like having a list of relevant media somewhere even if it's just here. Crosscanyon (talk) 23:35, 10 April 2010 (UTC)

Question about article consistent with the schlegel images?
"Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP."

This is exactly how I see it, and is 100% consistent with Image:Hypercubecubes.svg

Could someone tell me the corresponding corners on the most popular tesseract image Image:Schlegel_wireframe_8-cell.png

I know that all there are right angles, but could someone ... ahah! nevermind. I think I know the answer.

Is the "large cube" and the "small cube" of the schlegel diagram the "corresponding cubes" which I am used to?

Also, could those 8 lines between corresponding points on the 2 cubes, in the schlegel diagram, not drawn parallel for a reason? I think it is a disservice to the intuitive geometry. Amazing how for 3 months I couldn't "see" the 2D tesseract schlegel image because I thought the correspponding cubes were composed of some of the joining-lines. but 2 minutes into my question, i'm stupid! Sentriclecub (talk) 05:37, 18 July 2008 (UTC)


 * (I took the liberty of standardizing your link syntax.)
 * Any of four pairs of cubes can be considered as the 'parents': in the Schlegel model the inner and the outer are the most obvious, because the others are more distorted.
 * One set of 8 parallel edges is drawn as converging because otherwise the cells would overlap, which the Schlegel model seeks to avoid for clarity. (One cell is represented as the outside of the model.) —Tamfang (talk) 07:30, 20 July 2008 (UTC)


 * Thanks for your response. You're absolutely right, there are 4 pairs of cubes, that could be thought of per my question.  The first cube can be drawn using any 3 of the 4 dimensions, then the complementary cube (inventing terms--sorry) would be dictated and non-ambiguous.  The whole thing that confused me was looking at the schlegel and reading "all corners are right angles" which is true, but doesn't teach you how to learn the thing. Sentriclecub (talk) 11:45, 20 July 2008 (UTC)

history or anciant history
i been looking at the tesseract on the tellevision, one where some space man or other is sucked into universes , through time, and i looked at the shape , and the idea of the way the sci fi programme created a new time frame. then i remembered somthing a long time ago that the pyramid shape was used for communication. to centre the sphyci, so that others would feel the sphycic connection. i just had to look up the maths,. and wanted to place a blogg, hope no one realy minds. but if you put the mian pyrimid with other pyrimids across the globe, right down to wigwam, central point, i figure though a level of maths i dont quite understand , that the shape would be formed with the seperate entitys of the , say elders, sphycis. anyone else see this. that the shape came from age old communication. Daniel Jordan —Preceding unsigned comment added by Dannydynamic (talk • contribs) 18:59, 30 August 2008 (UTC)

L'Engle
Almost certainly A Wrinkle in Time is where most people first hear the term "tesseract." Shouldn't the novel be mentioned in some context in this article? john k (talk) 07:51, 15 March 2009 (UTC)
 * It talks about what it calls a tessseract but if you want a proper tesseract in fiction you'd have to go for something like "—And He Built a Crooked House—". And I think your almost certainly is almost certainly wrong, provide citation if you want to come out with something like that. The lack of some sort of references is I think wrong but it would have to be worked in as something relevant to the subject, for example where it has influenced something, and not just as tacked on trivia with everybody sticking in every single time they ever hear or see the word. See WP:Trivia and especially the intro to WP:Handling trivia Dmcq (talk) 15:20, 15 March 2009 (UTC)
 * I do not think the example of trivia given there is at all comparable to this. Mir was a very well known space station which was talked about a great deal in a variety of widely read media (especially the mainstream news media), and which appeared tangentially as a minor part of a single episode of a television show.  Tesseract is a term for a relatively lesser known mathematical concept which was also used (incorrectly or idiosyncratically, admittedly) as a central element of one of the seminal works of children's literature of the 20th century. I really think it deserves a mention of some sort. john k (talk) 06:03, 16 March 2009 (UTC)


 * A compare of google hits on the two shows "tesseract" is referred to in twice as many pages as "A Wrinkle in Time". I think that indicates your idea of how well they are known is not quite right. Also as I said before it uses the term tesseract but the thing it refers to is not a tesseract. This reference is just trivia, a bit of relevant trivia with some impact would be much better. All this would illustrate is how people misuse terms to sound mathematical. Dmcq (talk) 13:40, 22 March 2009 (UTC)

Should be mentioned earlier why a 4D cube can be depicted in 2D
It should be mentioned in the first paragraph. This isn't obvious and is important for grasping the significance of the illustrations. Cubes are intuitive, 2D depictions of cubes are intuitive. Tesseracts are not intuitive, 3- and 2D depictions of them are not intuitive. Unless it is clarified that a 4cube can be unfolded into 3D just as a cube can be unfolded into 2D (or that a light could be shone on a 4D prism and the 2D projection traced, just as a light could be shone on a cube prism and the 2D projection traced). That this or something similar is mentioned somewhere else in the article is mostly irrelevant: Most of the information in the leading paragraph is mentioned elsewhere in the article.Jackessler (talk) 04:03, 1 October 2009 (UTC)


 * OK. But there is no reason to emphasize unfolding above any other method of visualization, such as projections or intersections (slices). Perhaps the lead should be reworded to indicate the various means of visualization?&mdash;Tetracube (talk) 04:06, 1 October 2009 (UTC)


 * Possibly should just refer to the fourth dimension article about visualizing four dimensional objects. There possibly is a bit too much overlap between the articles on this though I don't mind a bit of redundancy. Dmcq (talk) 07:44, 1 October 2009 (UTC)

New Animations
The animation is a e8Flyer.nb animation of a 4D rotation of a 4-cube tesseract projected into 3D. The somewhat redundant 8-cell animations (one being on a horizontal plane, the other simply rotating) are not the same. Actually, the 4-cube orthogonal projection animation is more accurately representing a true tesseract. This new animation is of the actual 4-cube and looking closely you can see the orthoganal projection envelopes . Jgmoxness (talk) 22:38, 19 March 2010 (UTC)


 * You are trying for a purely orthogonal projection rather than using any depth cuing which make it hard to make things out, if so I'm pretty certain it has gone wrong as the central image has not the correct width. I cannot make anything of your huge attached file, what I read of it seems rather confused. Your picture would be inferior to those already in the article even if you got the maths right. Dmcq (talk) 23:42, 19 March 2010 (UTC)
 * No, you are confused. I am NOT "trying for a purely orthogonal" projection. What this represents is an n-dimensional rotation of n=4 projection basis vectors. It will in some frames of the animation get "close" to the pure orthogonals - but don't assume that they are simply planar rotations (as in the other 8-cell .gif animations).


 * > I cannot make anything of your huge attached file


 * Are you refering to the e8Flyer.nbp ? that is a a Wolfram Mathematica notebook player file - you need to open it in a free player from Wolfram.com. That is a tool that will allow hyperdimensional visualizations of the popular polychora and Lie Group representations. I will remove the reference -as it seems to be confusing with the short descriptions.


 * I can and have done the static 2D and 3D orthogonal projections in many representations throughout Wikipedia's n-cube articles. For example, see my Petrie projection of the 5-cube Penteract for example or my image in the template Lie groups.


 * BTW - All of the n-cube and Lie Group projections use the same data set (as subgroups of E8). The projection algorithms use the Mathematica n-dimensional RotationMatrix function.Jgmoxness (talk) 00:09, 20 March 2010 (UTC)
 * And I've written a java program to do a better representation of a tesseract than yours and I don't bother showing it in wikipedia. It sounds from what you say that your representation is a non-standard one and therefore origfinal research. Can you point to a paper describing your representation? Dmcq (talk) 00:22, 20 March 2010 (UTC)
 * I don't consider proper mathematical representations of well known objects "original research". If you do and don't want to show your stuff, that is your prerogative.


 * BTW - I challenge anyone to do a proper projection of the 4-cube tesseract and obtain the animated .gif (concentric cube) 3D models shown in the article (and give the 3 4D basis vectors that produce it). These are closer to "original research" but are popularized visualizations.


 * My hats off to you for doing that in Java - it is far more tedious than a symbolic mathematical tool purpose built for the job - Mathematica. My tool does ALL hyperdimensional objects - not just the simple tesseract. (I found your stuff in URL above - will check it out)Jgmoxness (talk) 00:40, 20 March 2010 (UTC)
 * If your interested my implementation in java is at . The original of it was on a machine a thousand or more times slower than a modern machine so it takes a few shortcuts. Dmcq (talk) 01:02, 20 March 2010 (UTC)

Very nice - so you must realize those animations on the current page are not "proper" projections like yours and mine!!Jgmoxness (talk) 01:09, 20 March 2010 (UTC)
 * If I switch on hyper and reveal I get essentially the same sort of motion as for those. I can get close to an orthogonal projection by switching on in out aand dragging the mouse from top left to bottom right a few times which increases the magnification at the same time as increasing distance. After doing that I do not get the strange shrinking and expanding of the view that you get. Dmcq (talk) 01:23, 20 March 2010 (UTC)
 * ok - I too can create "the same sort of motion" -but those cube-inside-cube projections are simply WRONG! Pay attention to where the edges meet vertices - they are switched (the article says these are "in principle" the same (this is the problem). Your and my projections are accurate. The "shrinking" you refer to is an artifact of taking 4 dimensional rotation throughout 2Pi. Yes, it looks "different" than others -but it IS accurate.Jgmoxness (talk) 01:27, 20 March 2010 (UTC)
 * Mine has cube inside a cube. The smaller cube is further away. You don't see it unless you switch on reveal otherwise it is hidden by the nearer cubes. The program implements perspective with hidden volume elimination. Dmcq (talk) 01:33, 20 March 2010 (UTC)
 * I think I see what you are suggesting - but that "smaller cube further away" is in fact NOT inside the bigger cube. It is an optical illusion. They are more side-by-side like the proper projection [[Image:Hypercubecubes.svg|150px]]. There is NO WAY you can produce the animations from Jason Hise with Maya and Macromedia Fireworks with a proper 4-cube set of vertices and using dot-products of 3 basis projection vectors. Your code is correct - so why try to suggest it is the same as that which isn't.Jgmoxness (talk) 04:06, 20 March 2010 (UTC)

I modified the video to be more clear in the construction and projection. I hope this helps. If not - please discuss before removing!Jgmoxness (talk) 17:33, 20 March 2010 (UTC)
 * What I think is that in 4D, further vertices are closer to the center when projected into 3D. --3.14159265358pi (talk) 01:08, 12 December 2011 (UTC)
 * The smaller cube in the 3d projection is inside the larger cube. It is further away in 4D. It is the same as looking at the projection of a 3d cube lookex at vertex on into 2d. The nearest vertex is in the middle. The furthest away vertex is not seen. If you make the cube transparent the furthest away vertex is also inside the hexagon formed by the mid distance corners.
 * You have something wrong with your model. Think of a diagonal between two corners which is face on to the viewer in the small picture that looks like a vertex on cube. That diagonal should be the proper length, square roiot of two times a side. However it is shorter than a side in some of the other screens. It is simply wrong and I see no point in keeping it up and so will remove it. Dmcq (talk) 17:59, 20 March 2010 (UTC)

My 16 vertices are generated from 4D {A,B,C,D} permutations of {±1, ±1, ±1, ±1}.

My 32 edges are selected from all possible 120=Binomial[16,2].

They are the 4D Norm unit length edges.

I then dot-product them to 2 (or 3) basis vectors.

In this case these basis vectors are generated from a simple Sin/Cos functions and are:

H(the basis vector for the x axis)= {0.7071067811865475`, 0.5`, 0.`, -0.5};

V(the basis vector for the y axis)= {0.`, 0.5`, 0.7071067811865475`, 0.5};

Z(the basis vector for the z axis)= {0.7071067811865475`, -0.5`, 0.`, 0.5`};

This produces both the 2D Petrie and 3D Petrie shown - compare against the others in the article - they are spot on!. All the code is the same to produce ALL my images that ARE CORRECT! Only parameters that change the image attributes change in the tool.

Now, if you're still having trouble with the 4D rotation - let's discuss that. It seems you're still trying to assume my rotation is trying to emulate something else. It is not. It is as described in the detail on it's wikimedia commons page. The H,V,Z basis vectors above are dotted with a RotationMatrix[Theta,{u,v}] function in Mathematica. Here Theta is a rotation angle, and {u,v} defines an n-dimensional hyperplane. I do that for some number of movie frame steps (this case - steps=90), so in this case, each angle Theta=i*2Pi/90.

The only possible issue for you might be how the {u,v} hyperplane in the RotationMatrix is defined. This will affect the visualization, but whatever projection vectors are used - they are being projected accurately.

So let's tackle my hyperplanes for H, V, and Z basis vectors.

In the e8Flyer tool, the A dimension is selected for the H, B&C for the V, and D for the Z.

This has the effect of defining the hyperplanes as follows:

For H, u=A={1,0,0,0}, and v=B&C={0,1,1,0}

For V, u=B&C={0,1,1,0} and v=D={0,0,0,1}

For Z, u=D={0,0,0,1} and v=A={1,0,0,0}

I hope that helps (but it may not). Jgmoxness (talk) 19:09, 20 March 2010 (UTC)
 * Saying it is generated from your program by sticking in some parameters is not an explanation of how it is produced. I haven't got Mathematica so perhaps somebody else can check your code but the animation output looks wrong to me and I've explained above how I also checked it. And even if it was correct it wouldn't I believe look as good as the animations already in the article. If you think the animations in the article are wrong perhaps you could explain why and then I might think it worth looking deeper into what your animation does. Dmcq (talk) 21:13, 20 March 2010 (UTC)
 * I'll put a request on Wikipedia talk:WikiProject Mathematics and see if somebody else might be able to make sense of what you are saying and you want some other people for consensus. Dmcq (talk) 21:30, 20 March 2010 (UTC)
 * ok - I added some detail above, but that won't help because it simply explains more about something you can't test. But you CAN easily download the free Mathematica player from Wolfram.com and fire up my e8Flyer.nbp - that would be fun for you and very helpful regardless of the outcome of this debate.


 * First off, I am not trying to replace what is there. I am adding to it. So even if it doesn't "look as good" (to you)- what's the problem ??


 * Your explanation seemed to be based on the Schlegel diagram (or maybe your Java tool, not sure). The Schlegal diagrams and the animation of it by Hise is not a "proper" orthogonal projection using 2 or 3 basis vectors on the tesseract vertices. My animation is. Stop trying to assume it is or is supposed to look like what is already in the article. It is different - and THAT is why it is interesting and value added to the current article. If I were simply recreating that - then I would agree it is not needed. You keep saying "it looks wrong" because you don't understand what it is supposed to look like (even though you assume you do).


 * I did suggest that the Schlegel animation was "wrong". That is probably too harsh. It is what it is - an animation of a popular projection type that was used over 100 years ago to "simplify" the visualization of 4D. I prefer proper orthogonal projections using basis vectors against the standard permuations of vertices.

I also believe the public has a right to understand that some of these "oversimplified" visualizations are not the only ways to view hyperdimensional objects. Let's get into the 21st century. Jgmoxness (talk) 21:41, 20 March 2010 (UTC)
 * Somebody with a mission and the truth. I see little point in further discussion, a third opinion is needed as I see little else working and I have neither the tools nor inclination to pore through your code. Dmcq (talk) 22:33, 20 March 2010 (UTC)
 * There are some problems with the new image. First, the size (over 2 megs) seems a bit excessive for what is being illustrated and the quality of the animation. Second, the slide show at the beginning seems to have nothing to do with the rest of the animation and has confusing labels attached. If the still images are meant to illustrate something then separate them out as separate images with their own captions. I don't see that they are illustrating something that isn't in the images that are already there. Third, the animation itself seems like just a variation on the animated GIF we already have but with lower visual quality. Overall I don't see how the new image enhances the quality of the article.--RDBury (talk) 09:49, 21 March 2010 (UTC)
 * If the video is viewed in full screen - the quality is very nice - just click the picture and it takes you to full screen mode (vs. the smaller perspective in columns on the article). This is an artifact of how wikipedia handles .ogv files (one of the few proper video formats allowed). I agree with the first static image in the slide show and will take it out. The second image adds value because it uses the same H,V basis vectors as the spin rotation in 3D (by adding the 3rd basis vector Z. It is there to prove to Dmcq that my projections are in fact correct. Your third point, like Dmcq, is completely wrong. The current animations are from Schlegel projections (not orthogonal) and as such, mine are unique to this page. They show a completely different method of visualization by animating orthogonal projections in hyperplanes. BTW - the next point uses similar methods (albeit in a bit more constrained set of rotation matrices).Jgmoxness (talk) 15:19, 21 March 2010 (UTC)
 * The fact that you have to give instructions on how to view the animation and need to give a detailed explanation of how it was generated shows it's not appropriate for the article. The purpose of an image in a Wikipedia article is to help explain and motivate the article text. If an image needs complicated explanations then it's not helping the reader understand the material and it should be removed.--RDBury (talk) 20:14, 21 March 2010 (UTC)
 * The fact I have to give instructions on how to use standard wikipedia technology is a testament to the technical computer skills (or lack thereof)of the people deleting my content before understanding it.Jgmoxness (talk)
 * I had a look at the Mathematica demonstrations and I see they have a not very wonderful demo of projection of a tesseract at . The maths in that demonstration are correct, please see that the overall size of the figure does not change much during animation. That is how it should be. It doesn't get flattened or become small in the middle like the animation being proposed here. The figures shown at the beginning of the animation are okay but duplicates of what is the article already. My guess is that the animation is leaving out one of the axes and only doing a 3D dot product but of course I have no way of checking and why should I anyway unless the code is presented neatly and simply. Dmcq (talk) 11:57, 21 March 2010 (UTC)
 * I am pleased and assume you are using the free player to view the demo. Great - now you can download the *e8Flyer.nb and play with that (unfortunately, the free version doesn't allow file read/write). Also, you can clearly see how these (as you admit) correct projections can't reproduce the Schlegel animations (or vice-versa). At no time does one cube project completely inside another AND become small (as in a Schlegel stereographic projection).


 * As I explained earlier, the flattening/sizing it is an artifict of doing 4D rotations through x-y-z-w (or more accurately here as {H=x-yz, V=yz-w, Z=w-x} at the SAME time. The demo you reference constrains this to by simply w-x, w-y, w-z. I understand why you feel "more comfortable" with that -as it is more typically seen. It is much more difficult to do multiple hyperplane rotations like mine in one. Let me get some code snippets for you. Also if it pleases you, I will also try to produce a 4D rotation that doesn't happen to rotate through a perspective that flattens in 3D.Jgmoxness (talk) 15:19, 21 March 2010 (UTC)


 * You misread me. The one on the Mathematica demonstrations site does a projection and it doesn't do the funny flattening and shrinking yours does. It works as I expect and produces the images I expect. And I had a quick try at your program but a) I couldn't see how to get anything useful and b) what would be the point of just seeing the bad display again? Your program is wrong. I know it is wrong because I know what to expect. There's no point you just saying it is right. You have to demonstrate it is right by making the display code simple and getting someone to check it. Your stuff isn't peer reviewed and it is contested. Besides which it isn't as nice a display as the ones here. You get a different display because yours in wrong, not because other people are wrong. Dmcq (talk) 18:07, 21 March 2010 (UTC)
 * Also I am not interested in you getting a non-flattened view to 'please' me. You should explain why the flattened one is flattened. The wolfram one does the rotations properly. See if you can get theirs to reproduce what you have. You can get any particular view you like with yjeirs or are you saying there is a way of rotating to a new view that their one can't reproduce? Dmcq (talk) 18:13, 21 March 2010 (UTC)
 * Having views it three times I can't understand this new animation. The two animations on the page now I understand immediately just by looking at them (a simple and a double-isoclinic rotation, about planes of rotation parallel with the tesseract sides). I can't understand yours, i.e. see what rotation is doing, even after 3 viewings and reading the lengthy explanation at the file's page which looks nothing like any description of a rotation I've seen before. -- JohnBlackburne wordsdeeds 21:49, 21 March 2010 (UTC)
 * I just had a look at the last version of the page with it in and it has two additional problems: the long and confusing (it's not even a proper sentence or sentence fragment) caption below, bigger than some of the pictures; and it's been encoded at the wrong size for its use on the page, as per WP:CUM.-- JohnBlackburne wordsdeeds 23:03, 21 March 2010 (UTC)
 * ok - let me try again. It seems we all are (or have been) defining "different=wrong". Different is not necessarily wrong. We have also been debating several different isssues related to the article. So let me re-cap:


 * We all agree the Wolfram demo and code are correct orthogonal projections. I hope we can all agree the Schlegel animations are beautiful and correct stereographic (but not orthogonal) projection anmations. The Schlegel animations are different from Wolfram (and mine) - sorry for implying them being incorrect. My former animations were different from Wolfram due to the my deviating from standard w-x, w-y, w-z projection hyperplanes (sorry for being too creative - and thus correctly assigned the moniker "original research"). I have re-done the video using the same code from my tool(which I will post snipets of shortly) that recreates a more familiar animation (ala Wolfram orhtographic demo - NOT Schlegel stereographic).  Please see the current video that should at least be closer to your familiar view. Again - I insist my work has always been correct -just different (so you incorrectly thought there were errors).



Jgmoxness (talk) 01:16, 22 March 2010 (UTC)
 * Ah, so it's a simple rotation about the hyperplane spanned by (1, 2, 4, 0) and (0, 0, 0, 1). That is it's not doing anything different from the first GIF animation, except it's doing it in such a way that I suspect few could deduce that without looking at the math, so it has little encyclopaedic value, apart from all the other concerns raised. One more thing is it really should be a gif. The move player seems broken so I can only play the movie once: I need to reload the page to play it again.-- JohnBlackburne wordsdeeds 08:36, 22 March 2010 (UTC)

In an edit summary for the article Jgmoxness wrote "Please, I told you to discuss before removing. Let's get consensus from several others." Well, we now have had contributions from several people. (If I have counted correctly I am the fifth to comment.) There is a very clear consensus, with a lone dissenting voice from Jgmoxness. I fully agree with the consensus. My reasons for doing so are already covered by other editors above, so I will not waste time repeating them. Since Jgmoxness was keen on consensus, he/she will presumably now be happy to accept that consensus. The other part of the edit summary I quoted is "Please, I told you to discuss before removing." This is an interesting comment. I shall assume that there was an unfortunate choice of words, and that "asked" was meant, rather than "told". However, even in that case, it represents a misreading of Wikipedia's policies: the onus is on someone wanting to include material to produce justification, not on those wishing to exclude material to justify that. Anyway, that is now of little relevance, since, as I have already said, we have consensus against inclusion anyway. JamesBWatson (talk) 08:47, 22 March 2010 (UTC)
 * To JohnBlackburne, both the (correct) Wolfram demo and my code (shown consistent w/Wolfram demo) clearly show the NEW animation is not the same as the current .gif movie. Also, the first .GIF is a stereographic Schlegel projection (NOT an orthogonal). Please, let's get a mathemetician who understands the difference to comment on this. Therefore, my animation IS new and not "the same". Anyone can see that. As to it's value - animating the static graphs is very much appreciated - so more of it is needed IMHO and therefore adds value. As for the player - it can be run as a Java or Mac Quicktime - try a different plugin or different browser or try reinstalling. They work fine on all my computers/browsers.Jgmoxness (talk) 12:56, 22 March 2010 (UTC)
 * I don't think we will ever agree on this, but I would note that the other simple rotation, which you dismissed as "somewhat redundant", is a featured image, one of the few such mathematical images, here, on commons and a number of other wikipedias. It's encyclopaedic value is that it shows the rotation clearly, in such a way that it also makes the structure of the tesseract clearer, because of the choices of plane of rotation and projection, and it is this value your animation is lacking, aside from the other concerns raised.-- JohnBlackburne wordsdeeds 13:43, 22 March 2010 (UTC)
 * The reason for the "somewhat redundant" point was not to be perjorative. It was to point out the illogic of the argument that my work was too close to what was already there to be considered value add. Well, you can't have your cake and eat it too. The two .gif animations are essentially the same (except one adds a second hyperplane rotation). This is not to impune the quality - but to suggest my work is much different in kind and quality to suggest that it has no value add. My work uses 3 hyperplane rotations of a completely different projection style. "Same is ok" but "different is better". Even if it isn't as intuitive, IMHO it adds more value than two essentially identical animations on the same page (no matter how "popular" they are.Jgmoxness (talk) 02:16, 23 March 2010 (UTC)
 * As for consensus (and the semantics of my argument), I am outnumbered (today) and that is the rules. I will add a few features and beauty to my animations - and resubmit in the future (maybe a new crop of editors will see it differently. It would be great if Hise could jump in and share his source code so we could see the real differences. I encourage anyone else who understands this more than those weighing in above to help explain it to them.Jgmoxness (talk) 12:56, 22 March 2010 (UTC)
 * Please do not put comments like '(the tedium of ignorant pack mentality)' in your comments. This is not conducive to collegiate behviour. See WP:NPA about attacks on other editors.


 * As to your 'triple rotation', a tesseract can only be rotated through two independent planes. It doesn't have enough dimensions for any more. I'm sure one could put in various tumbling motions but I fail to see how that would lead to better understanding. Also projection is not completely different from using a Schlegel diagram. It is the same as if the perspective point was at infinity and in my opinion leads to a less informative or useful result. The only point for animations of orthogonal projections I can see is for people who are more interested in watching the pretty lines in a pass a reefer and blow your mind sense than actually understanding anything. And your animation isn't as pretty for such purposes either. Dmcq (talk) 10:37, 23 March 2010 (UTC)
 * ok, sorry about the insertion of the word "ignorant". From what I can tell - it wasn't completely accurate and wasn't necessary (but debating the topic with this "gang of 4" IS a bit tedious - as I am sure we all can agree at this point.).


 * BTW - I don't recall saying they were 3 "independant" hyperplanes - Wolfram and my rotations are w-x, w-y, w-z. As to your view on projections.... To quote from the Shlegel diagram article (emphasis and parentheticals mine):

"In geometry, a Schlegel diagram is a (stereographic) projection of a polytope from Rd into Rd − 1 through a point beyond (e.g. infinity) one of its facets. At the beginning of the 20th century, Schlegel diagrams were an important tool... "


 * Then to extract from the orthogonal projection article:

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection.


 * Dmcq > It (the two current similar .gif animations of Schlegel Steregraphic projection) is the same as if the perspective point was at infinity and in my opinion leads to a less informative or useful result.


 * WE AGREE, YAY! but then....


 * Dmcq > The only point for animations of orthogonal projections I can see is...


 * I value leaving the object as unchanged as possible - thus my animations value. I don't smoke reefer, so that shouldn't be a factor in MY confusion.Jgmoxness (talk) 13:26, 23 March 2010 (UTC)

The point about rotations is there are only two or three sorts of non-trivial rotations in 4D: simple rotations, double rotations and as a special case of double isoclinic rotations. This is two more kinds than in 2 or 3D, where there are only simple rotations, making 4D rotations more complex and interesting. More practically they are a useful in categorising such rotations, in particular by their planes of rotation where there are one or two depending whether the rotation is simple or not. See e.g. SO(4) or plane of rotation. The rotation you've given is a simple rotation, with only one plane which I described above, which can be calculated using bivectors from the rotations you've given.-- JohnBlackburne wordsdeeds 13:49, 23 March 2010 (UTC)
 * I agree, I think. In the definition - each "act of rotation" is a unique transformation in one hyperplane. If I create a rotation from 3 dotted simple rotations of 3 angles - the result is more complex. but may still qualify as "simple". See the resulting transformation matrix




 * I could select a different set of angles and hyperplanes on each step, but that would be really "reefer crazy". Jgmoxness (talk) 14:01, 23 March 2010 (UTC)


 * Why not fix the business about the image being rather squashed for starters. It looks to be in about the ratio 4:5 to me. It may have something to do with why the previous animation before this one shrunk and swelled. It still wouldn't be good enough but at least you'd have a program that was doing what it was supposed to do correctly which would be a start. Dmcq (talk) 14:46, 23 March 2010 (UTC)
 * ok - looks like the conversion program I used to go between Mathematica's 1:1 output in .avi format to Wikipedia (.ogv) changed the aspect ratio to 3:2. I redid the conversion to 1:1 (notice the black borders as you play it now). Sorry for that oversight

Of course, the flattening you had a problem with in my original video was not aspect ratio related - it was the non-standard selection of my projection operation's hyperplanes. That was fixed by applying the identical hyperplane selection as Wolfram (per above source code). So now the only issue is visual appeal and technical difference value add from what is in Schlegel .gif(s). Sounds like progress is being made here :) Since my tool can do n-dimensional projection to 2 or 3 dimensions (with a huge number of options to play with), please let me know what changes you would like to see in order to provide enough value add to include in this article.Jgmoxness (talk) 15:40, 23 March 2010 (UTC)


 * It is looking better. There's an odd problem with the default video player that renders it too big but switching to QuickTime or the Java player and I see it properly. but it could be improved more I think. First there's a lot of white space around, without which it would be even smaller. If you look at this then it would be good to make it perhaps a bit smaller still, so it's at an appropriate size for the page: unlike images videos don't get rescaled by WP to display at smaller size, and after loading several hundred MB of video users should not have to click through to see it properly.
 * The edges and vertices are rather dark and thin, so difficult to pick out, which makes it difficult to see what's in front of what much of the time. You could make them lighter, more contrasting, or wider to help with this. You could even colour edges based on the axis they are parallel to, as your rotations seem to permute them and this would make it clearer. It looks like you are drawing the axes but they are too small and indistinct to see. I would make the moving axes bigger and clearer and hide the fixed axes, or hide all of them if they cannot be made clear enough at that size.-- JohnBlackburne wordsdeeds 16:05, 23 March 2010 (UTC)
 * ok - I removed the axes and frame (it is cleaner) and zoomed in a bit. I reduced the resolution to 500x500 and file size to 1Mb. I didn't color the edges or vertices (need to work that a bit). WP does rescale .ogv w/the same px parameters as graphics (just need to click on the screen to go to full size):



Jgmoxness (talk) 17:14, 23 March 2010 (UTC)
 * I think your comment 'requested changes' is perhaps a bit strong. The changes are not requested. We've simply advised you on ways to get your own thing working for yourself. Dmcq (talk) 17:49, 23 March 2010 (UTC)
 * Dmcq, you seem to be hard over on keeping this out. Why? Let me try to inform your opinion on this point. I said:
 * Jgmoxness>...please let me know what changes you would like to see in order to provide enough value add to include in this article.
 * To which JohnBlackburne replied directly and without qualificaton:
 * JohnBlackburne>It is looking better. There's an odd problem ...
 * To which I have quickly responded with corrections - IN ORDER TO get to a point where I (or someone else) might want to include my Wikimedia Commons .ogv into this or some other article. Thank you very much!Jgmoxness (talk) 18:03, 23 March 2010 (UTC)
 * The point about rescaling video is detailed here WP:CUM - the bold bit in red box in particular. Videos needs to be uploaded at the size it will be viewed in the article, in this case 200x200 if it's to be shown as above. This should also improve the quality, as it should look better than the pretty poor scaling with very rough edges that the default player does. It can be uploaded at other sizes, with links between the different versions, as discussed there. —Preceding unsigned comment added by JohnBlackburne (talk • contribs)
 * You're right (bandwidth over disk space). I was more concerned about the ability to get it fitting into a column window of the right dimensions. As shown in the referenced instructions, this would be worrisome on videos longer than 90 frames and 6 seconds at higher resolutions than 500x500. I will upload corrected size versions as soon as we get consensus that it can be put into the article at some correct size (it would waste everyone's time and WP resources to do that for each iteration as we debate the merits of it).Jgmoxness (talk) 19:02, 23 March 2010 (UTC)
 * I have a problem with this talk page being used to develop your video when I see no chance of it getting to an acceptable point for inclusion. This talk page is for the development of the article, not to help you fix your problems. You are wasting editors time and wikipedia space on your project. Dmcq (talk) 18:58, 23 March 2010 (UTC)
 * I will let JohnBlackburne comment on what he is trying to accomplish. But it seems you are now the only one NOT trying to help improve the article (just trying to keep me from adding value).Jgmoxness (talk) 19:02, 23 March 2010 (UTC)

Arbitrary break
I would like to get opinions on this latest attempt at comparing the orthogonal to perspective rotations. Jason (author of the current .gifs) was gracious enough to share a bit of his code, so I have made a first cut at that (you can see the first rotation set is fairly close). The change from orthogonal was to divide x, y, and z by w+1 (before rotational projection). The .gif animations also seem to be using some sort of stereographic morphing (that is not seen in this video) where the edges flow over each other at the ends. There are 4 sets of rotations over 2Pi with 30 frames each. I've added edge coloring as well.

Jgmoxness (talk) 19:28, 27 March 2010 (UTC)


 * To me all the rotations seem wrong. The first seems to be rotating about a plane containing the w, i.e. the hidden, axis but there's no foreshortening as it does so. The third is just rotating in 3D, so seems unnecessary. In all of the rotations on the right the small cube is rotated, so you are rotating your axes and the cube which is just confusing. Nothing gets rotated in a plane which contains the axis, the axis gets rotated instead. Or at least that's how I'd explain it, but it's clearly not just rotating the tesseract.
 * I like the colours which help a lot, but it's a bit small and quick to see what's going on. It's an odd shape: make the movie narrower and taller as it's clipping the rightmost animation but there's an area of white space to the right of it, and a smaller space to the left.-- JohnBlackburne wordsdeeds 19:57, 27 March 2010 (UTC)
 * I still don't see how the changes are helping. The article already has enough illustrations and tweaking and massaging something that basically a duplicate of what's already in the article seems like a waste of time.--RDBury (talk) 20:36, 27 March 2010 (UTC)
 * I am trying to convince you guys by producing the comparisons that they ARE NOT the same. You can clearly see they are very different. BTW - These are "frame synchronized", so each left hand frame is identical to the right with the exception of the x,y,z division by w+1 "perspective" change. Maybe making the perspective change after the rotation would help those right hand visuals.Jgmoxness (talk) 21:00, 27 March 2010 (UTC)


 * While I agree with several of the aesthetics (some are simply related to the video conversion and others easily changed in the creation process -will do). As to the critique of the actual rotations, you need to first check out and compare the already confirmed accurate Wolfram demo . As above, I show a proof that my projection code is identical. You can also see the similarity between the left side (orthographic) frames and the Wolfram demo and the "orthographic envelopes" picture at the beginning of this topic.


 * You don't get "foreshortening" unless you do "perspective" projections. All of these rotate in steps through $$\theta=i2\pi/steps$$ on the horizontal (H) projection vector. What is different in each rotation set is how the vertical (V) and (Z) projection basis vectors change. The 4th set rotates all H,V,Z basis vectors through $$(\theta,\frac{\theta}{2},\frac{\theta}{4})$$, this is why it is the least "familiar" of the set. While I agree the third set LOOKS simply as a 3D frame spin rotation, it is actually the same type of 4D projection and rotation as all the others. These are the 4 sets of H,V,Z projection basis vectors:$$((\theta,0,0),(\theta,0,\frac{3\pi}{4}),(\theta,\frac{\pi}{2},\frac{\pi}{4}),(\theta,\frac{\theta}{2},\frac{\theta}{4}))$$Jgmoxness (talk) 20:48, 27 March 2010 (UTC)
 * I'm not convinced by the Wolfram link: it's moving too quick to see anything, but it doesn't look like any of your animations so even if it were correct it doesn't support your animation. I'm not sure why it's "already confirmed accurate" just because someone's uploaded it to wolfram.com: if someone did the same here without references it would be OR.
 * And you're doing a perspective projection as that's what makes one of the cubes smaller. Then as you rotate the back cube "towards" you it changes shape and size: as in the correct animations already on the page. Your small cube doesn't do this and it looks completely wrong.-- JohnBlackburne wordsdeeds 21:11, 27 March 2010 (UTC)
 * No, you have to download the free Wolfram player and open the demo (look at the source), manipulate the demo yourself. That capability is a tool every one who does math should have!! That Wolfram code (per others) is valid and accepted (not OR) -this is not research, it is simply coding up a visualization. It is no more or less valid than the ones on the page animating the .gif from more hand coded source! Mathematica requires significantly less coding because the Graphics3D functions are built in. BTW - I am doing BOTH 4D orthographic (left) and 4D perspective (right) to perspective 3D. As I mentioned above, it may be that I need to rotate the 3D camera angle and/or change when the w+1 divides x,y,z (after the rotation). I will check that out. But I still think these views help see the tesseract in all it's dimensional variation. Jgmoxness (talk) 21:37, 27 March 2010 (UTC)
 * I don't want to download a file, and a player, then try and get them to work. But your rotation does not look like the one on that page, nor does it look like the perfectly good ones here. I've tried to explain it as best I can, but fundamentally your rotations look wrong compared to all the others.-- JohnBlackburne wordsdeeds 22:42, 27 March 2010 (UTC)
 * They don't look like the one on the Wolfram demo, because if you notice the web preview has the w-x, w-y and w-z set at .5 (and then rotates through 2Pi). If you had the tool - it allows precise duplication of what you see here, setting w-y and w-z to zero then using w-x to spin through 0-2Pi (BTW-that is my first set).Jgmoxness (talk) 23:41, 27 March 2010 (UTC)

Ok folks. I think the change of waiting to divide x,y,z by w+1 until after rotation did the trick for the perspective projection (right hand side). It was basically going from a 4D camera that rotated with the image to one that now is stationary.

BTW - I added a rotation set to the beginning which esssentially emulates the default orthographic web preview on the Wolfram demo (left side frame). Specifically, $$(\theta,\frac{1}{2},\frac{1}{2})$$

The second set perspective view (right side frame) now emulates Jason's single rotation animated .gif. Many thanks to Jason Hise (author of the original .gif(s)) for sharing code. You are a credit to the cause.



This video is larger. I am working on the translucent faces for the 8 cubes. Any other thoughts?Jgmoxness (talk) 19:14, 28 March 2010 (UTC) I've added an anaglyph version for depth cueing. Red-Cyan anaglyph glasses needed (Left eye red). I've completed the generalization of n-Dimensional perspective projection, so will be working the penteract to hepteract as well.



BTW- I also improved the non-anaglyphic video (added frames on each set, added the double rotation set like the other animated .gif (and moved them to the front). The 6 sets of H,V,Z projection basis vectors are now: $$((\theta,0,0),(\theta,\theta,0),(\theta,\frac{1}{2},\frac{1}{2}),(\theta,0,\frac{3\pi}{4}),(\theta,\frac{\pi}{2},\frac{\pi}{4}),(\theta,\frac{\theta}{2},\frac{\theta}{4}))$$. Jgmoxness (talk) 14:41, 30 March 2010 (UTC)

In my last post on this I mentioned that Jgmoxness had asked for consensus, and that consensus had clearly been achieved, so presumably Jgmoxness would now accept that consensus. In fact Jgmoxness posted a note acknowledging that he/she was "outnumbered", but would "resubmit in the future". Jgmoxness seems to show a remarkable inability to see the point of view of others or to act cooperatively. It is true that subsequent edits have resulted in one editor taking not so totally negative a view of Jgmoxness's efforts, but the fact remains that there is overall a clear consensus against it. For the editor who wrote "Let's get consensus from several others" to persist in the face of this consensus is unconstructive. JamesBWatson (talk) 13:09, 23 April 2010 (UTC)
 * Fair enough. There were no criticisms of the most recent animations after having made all requested edits and corrections (with only Dmcq absolutely refusing to help improve the article). I had hoped there was opportunity to enlighten article readers to the real difference between orthogonal and perspective views of animated tesseracts (with the unique addition of anaglyphic depth cueing). Since I am uniquely able to create complex math visualizations of these and other E8 subgroup geometries, I ask again: What changes would be necessary to be allowed to add value to this article? (is it really possible the page is already "perfect" ?) Jgmoxness (talk) 23:24, 23 April 2010 (UTC)
 * BTW, just to itemize the current "less than perfect" aspects of article visualizations...


 * There are redundant (therefore no value add / space wasting) representations - with 2 Schlegel diagrams and 3 Hasse/column graphs. Not to mention that 2 of them are actually inaccurate, having been manually altered in order to avoid showing the proper overlap of the central vertex. Jgmoxness (talk) 23:51, 23 April 2010 (UTC)


 * Don't be so quick to assume you're "uniquely able" to create math visualizations. Someone has already taken higher dimensional animations to another level using POVRAY-rendered transparency to reduce the ambiguity of wireframe-based animations by showing 2-faces as well, for example, , , and others.&mdash;Tetracube (talk) 00:02, 24 April 2010 (UTC)
 * Cool! They should be added to corresponding articles. BTW - I also do POVRAY using exports from my tool- NO PROBLEM! Coding is easy when you know what you're doing! Is that what is needed HERE ? Jgmoxness (talk) 00:14, 24 April 2010 (UTC)


 * For that, I defer to the other editors.&mdash;Tetracube (talk) 00:29, 24 April 2010 (UTC)
 * From what I can tell, this is your work - good stuff. Your deference is Wikepedia and the public's loss. BTW - if this IS your work, can you do a frame synchronized orthogonal+perspective anaglyphic tesseract animation with 2-faces ? (surfaces on the penteract to hepteract are less value-add). Another test for "uniqueness" would be to do that for the 6720 shortest sqrt(2) edges and 240 vertices of E8 in 3D. Jgmoxness (talk) 00:38, 24 April 2010 (UTC)
 * Also, on those pages you referenced, there is support for me (and against Dmcq) regarding his objection to the "strange shrinking and expanding" of the hyperdimensional rotations. Hopefully, he will check it out an learn something. Jgmoxness (talk) 00:57, 24 April 2010 (UTC)


 * Thanks Tetracube for the link to eusebeia.dyndns.org, that's very nice site, I've stuck a link to the tesseract section in the external references. I think I'll stick a reference to the visualizing section into the 4d article too if one isn't already there. Dmcq (talk) 22:45, 29 April 2010 (UTC)

Understanding the 4th dimension by models.
Oh my god I am finally understanding the 4th dimension. I see all these models and one kicked the whole thing in gear. I then went back and looked at all the pictures and they made sense. Forgive my vocabulary/language as I am 14 and in love with math, but i can explain the models to most people. You may need to print screen to see what i am talking about though. Also i may say inner, outer, etc., but i am talking about the 3D perceptions of inner/outer.

First, the non rotating picture. This tesseract must have 8 cubes. If you don't i can explain. See the "inner" cube, that should be easy. But look at the "larger" points. Those 8 denote a "bigger" cube. Take the front face of each cube. Don't they form a polyhedron. While it may not seem it, that polyhedron is another cube. Now using the left, back, right, top, and bottom faces you can make more cubes. 8 in total. The inner, outer, front, back, left, right, top, bottom. Not lets move to the single rotation.

Print screen the 1 dimensional rotation and paste it in word. If you got the right frame you will see the first image. Now study how the cubes look, watch the front, back, top, and bottom cubes rotate but still stay in place. Remember, your watching the cube, not the face. Now watch the inner cube very closely, it looks as if it becomes the right cube, the outer cube, and the left cube. Now think back to 3D. Rotate a cube on one axis. Looking at the correct side you will see 4 separate faces. Now back to 4D. Your 4 faces just became 4 cubes. When the inner cube becomes the right cube, the right becomes the outer, the outer becomes the left, and the left becomes the inner. These 4 cubes rotate around each other using the other 4 cubes as "stationary" points. Confused. Take the front cube into the rotating cubes example. While the inner cube rotates around the front cube is simply moving its edges. Watch its edges rotate and watch the inner cube rotate an you will see what I mean.

Now for the net. Ah, the net. An unfolded tesseract that looks impossible to connect, but look back at the model. Now disconnecting a few cubes you can create 7 of the 8 cubes in the net. This i can't explain very well but take away the problem of the outer cube connecting and you have 7 cubes. Now take the bottom part of the net. What's that cube going to be used for.Why don't we say it this way. Connect the bottom face with the top cube's top face. Now connect the bottom left face with the left cube's left face. Do the same with the right cube. Do this with the front and the back cube. Now you have the outer cube. Connect the other logical edges and a tesseract will form itself. (I will phrase this better if I can, remember I'm 14.)

Now if you understood what i said models should look simple. Try to figure the 2 dimensional rotation out. LZ Ravanger (talk) 02:02, 9 September 2010 (UTC) —Preceding unsigned comment added by LZ Ravanger (talk • contribs) 01:45, 9 September 2010 (UTC)

Central projection
Wikipedia doesn't seem to have anything about central projection that I can see and there isn't much on the web. Anyone know where there's something simle that could go in an article, plus is there a convention about what to do about the image of the central point an lines to it? Dmcq (talk) 21:41, 24 November 2010 (UTC)


 * I've not seen that terminology, but agree something is needed for clarity of what point-centered perspective projections mean. (p.s. I reworked the sections, separated parallel and perspective projections, still not great.) Tom Ruen (talk) 21:43, 24 November 2010 (UTC)

A long article
IMHO this article is too long. I'll surely be crucified here for saying this but I can't figure what's the relevance - from the enciclopedic point of view - of tutorial explanations about 'zillions' of ways of projecting and unfolding a tesseract. I understand that, since it is a geometrical entity belonging to a four dimentional (euclidean) space, one has several ways of projecting it into 3-space so one can 'visualize' it. This, of course, worth being mentioned but I'm not sure about the need of exaustive explanations of how actually do it (at least not in this article). —Preceding unsigned comment added by 200.133.215.4 (talk) 14:37, 26 January 2011 (UTC)


 * Lots of people have written about representations and Wikipedia is simply reflecting that. You may not think it interesting but this article has a quite high hit rate. Dmcq (talk) 15:31, 26 January 2011 (UTC)
 * Yes. 4D is generally difficult to visualise. That's true of many maths topics but 4D geometry is also very popular, disproportionately so. And the tesseract is one of the most popular elements of it. And I would classify the many different representations as aids to understanding. Most people can't think in 4D, so unwrappings and projections into two and three dimensions are very useful for understanding and there are quite a lot of of them.-- JohnBlackburne wordsdeeds 15:40, 26 January 2011 (UTC)

How to draw a tesseract
The animation just drawing a tesseract one line at a time says nothing about tesseracts. There is no point in putting in the random drawing. Plus it conflicts with the description of how to construct a hypercube below it. This is clearly against WP:IMAGE. Also see WP:Image_use_policy, animated images should be used sparingly, using them for something which is not discussed inthe text and contributes in no way to the article is clearly against that. Dmcq (talk) 07:48, 8 June 2011 (UTC)

Banchoff and Tesseract
Someone added 'Thomas Banchoff reserves the term tesseract to the central projection of the four-dimensional hypercube' to the lead and cited his book Beyond the Third Dimension – Geometry Computer Graphics and Higher Dimensions. I'm surprised because on a web page of his http://www.math.brown.edu/~banchoff/Yale/project12/math.html he says 'Traditionally, the tesseract is a term used to describe the hypercube' and disagrees with the use in a Winkle in time. I haven't access to the book, could someone check what that statement in the lead is about please? Dmcq (talk) 21:58, 10 July 2011 (UTC)


 * I own the Italian version of the book by Banchoff: Oltre la terza dimensione – Geometria, computer graphics e spazi multidimensionali, Bologna, Zanichelli, 1993.
 * Page 115: [Speaking about the 4d-hypercube] Alcuni scrittori chiamano questa proiezione centrale tesseratto [tesseract], un termine che sembra risalire a un contemporaneo di Abbott, C.H. Hinton.
 * English translation: Some authors call this central projection [of the 4d-hypecube] tesseract, a term which dates back to a person living at the same period of Abbott, C.H. Hinton.
 * I do not own the original English copy but since there are no footnotes saying the text has been altered, I think it's almost the same in the original English version.--Carnby (talk) 22:41, 10 July 2011 (UTC)


 * Well, that doesn't say that Banchoff (or anyone else) reserves the term tesseract; Banchoff here reports how some unnamed others use it, and implies that he does not use it himself at all. — I'd translate it "a term which seems to originate with Hinton, a contemporary of Abbott." —Tamfang (talk) 07:30, 15 July 2011 (UTC)

Projections to 2 dimensions needs an edit
There's a line in the Projections to 2 dimensions section which isn't a full sentence and makes no sense to me, so I have no idea how to fix it: "The vertices of the tesseract with respect to the distance along the edges, with respect to the bottom point." I'm thinking it's got to be related to the sentence which either preceeds or follows it, but I can't tell which. — Preceding unsigned comment added by 173.225.203.115 (talk) 08:11, 19 August 2011 (UTC)

Very Good Article
I started the Tesseract article Nov. 9, 2001. It may make everyone who worked on this smile if they catch the humor in my talk page entry. So... it became more than I ever imagined, even though I never thought there were corners in time 'til I was told to stand in one. BF (talk) 02:44, 21 October 2011 (UTC)


 * Perhaps you mean this one? —Tamfang (talk) 09:32, 24 November 2011 (UTC)