# Talk:Tesseract

WikiProject Mathematics (Rated B-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B Class
 High Importance
Field:  Geometry
One of the 500 most frequently viewed mathematics articles.

## Lost Cube?

During an analysis of the tesseract "folding" .gif, I noticed that only 7 3-Dimensional figures seem to exist in the tesseract seen in the pictures on the page: 6 truncated square pyramids warped from cubes, and one cube in the center. That infers the erroneous "destruction" of one cube, that, at least in my eyes, is impossible and therefore proves the tesseract an invalid 4-dimensional shape.

Could someone clear this up for me? Thanks. - This Is M4dn355 300 (talk) 02:27, 13 March 2012 (UTC)

I don't know which figure you are referring to. If you click on a picture you will see its name. However from your description it sounds to me that you have not included the outside which is another cube just like the inside one. It may look outside in the 3d projection but is a cube face just like the others in 4d. Hope that's what you're talking about. Dmcq (talk) 11:34, 13 March 2012 (UTC)
This is the only "folding gif" I know about, comparing folding a tesseract and a cube. Tom Ruen (talk) 22:03, 13 March 2012 (UTC)

That's the folding .gif I was looking at, however, I've noticed it in every picture I've looked at. Dmcq, does this mean there are two cubes? If so, I think it is necessary to mention in the article, as some users such as myself may be confused. - This Is M4dn355 300 (talk) 00:00, 14 March 2012 (UTC)
There are eight cubes bounding the tesseract. Each cube touches six other cubes and has an opposite cube. In 3d a cube has six faces and each face touches four other faces and has an opposite face. Dmcq (talk) 00:11, 14 March 2012 (UTC)
I understand that from looking at the .gif. My question is what happens after folding it. There are 6 truncated square pyramids formed from the 6 cubes immediately surrounding the one at the intersection of the 3 lines formed by the cubes of the 3D representation of a cross (no better name for that :/ ). Since there only seems to be 1 cube in the apparent center of the larger cube, it seems to me that there're only 7 cubes at the end of the process.
From your explanation, it seems to me that there are two cubes where there seems to be only one, one actually being on the outside. Is this correct, and if not, please explain where the 8th cube (farthest from the "center" on the longest arm) goes.
I said prior that the article needs to elaborate on this, as many users such as myself won't understand this. Can a better 2D representation of this be created or is the outside cube (if what I'm saying is correct) always facing towards the viewer? - This Is M4dn355 300 (talk) 02:40, 14 March 2012 (UTC)
I don't know what 'it' in 'I have noticed it in every picture I have looked at' refers to. I don't know what 'this' in your last paragraph refers to.
You have referred to 'in the centre of the larger cube'. The larger cube on the outside is the eight cube. Can you count the six squares in the 3D picture? If you look at the picture of a cube in 3D you will see that the sixth square is on the outside in that picture. The large cube is the closest in 4D just like the outside square is the closest in 3D. The small cube is the furthest away one in 4D just like the the small square is the back square in the 3D picture. If we had an opaque tesseract we'd only see the big cube, the third picture under perspective projection shows an opaque tesseract projected to 3D and one can only see four cubes in that. Dmcq (talk) 09:02, 14 March 2012 (UTC)
I'm not referring to what it would look like in 3D. Forget what I've said previously. Where does the bottommost cube (from the .gif) go?

It's circled in this rather crappy image (Imageshack's new registration rule -.-): http://desmond.imageshack.us/Himg24/scaled.php?server=24&filename=2000pxtesseract2svg.png&res=medium

- This Is M4dn355 300 (talk) 15:46, 14 March 2012 (UTC)
Aahh... I understand now. Thank you for attempting to help me understand this. - This Is M4dn355 300 (talk) 16:10, 14 March 2012 (UTC)

You may want to see Warner Model under the "Alternative Projections" section. It clearly shows all eight cubes plus how their faces fold to adjoining cubes. As a bonus, it shows the central VOID that other projections obfuscate. It demonstrates that there are no "outer" faces. AWrinkleInTime (talk) 12:13, 12 December 2015 (UTC)

It depends how you want to think about it but I would say that all points, lines, faces and cubes are on the outside and only the four dimensional volume is in the inside. Dmcq (talk) 11:40, 18 December 2015 (UTC)

Slightly different question on the same folding theme: is it accurate to show the cubes stretching/contorting when you fold the 3D cross into a tesseract? What I mean is, if we fold a 2D cross into a cube, a 2D being watching from their 2D universe should see the squares simply vanish as they are folded up and out of its 2D universe, leaving only the bottom square visible since it's the only one still in the 2D universe. If that's the case, then should we not see all but one of the cubes vanish as they are folded into a tesseract? — Preceding unsigned comment added by 192.65.0.118 (talk) 21:07, 19 October 2018 (UTC)

If you make the 2D cross of wires and have a light shining on it and look at the shadow on a piece of paper then you'll see the shadow squares distort and eventually touch along their edges to form a smaller square within the large square with lines between the corners - i.e. the usual image of the edges of a transparent cube.. Dmcq (talk) 21:37, 19 October 2018 (UTC)

## Wrinkle In Time

There was a reference to Tesseract in the children's Sci-Fi book "A Wrinkle in Time". Perhaps worth a mention? Thnx! -HK — Preceding unsigned comment added by 24.1.233.206 (talk) 22:24, 26 June 2012 (UTC)

There's a continual battle on adding and removing "cultural/literary/art" references. It looks like the current battle lines stand at none of them. A while ago I made a copy on the talk page, now at Talk:Tesseract/Archive_2#Tesseracts_in_popular_culture. Tom Ruen (talk) 22:44, 26 June 2012 (UTC)
The 'tesseract' in a wrinkle in time is not the same as the tesseract described in this article. The only connection is someone decided to use a cool word. Too trivial to include unless some secondary source makes some sort of meaningful linkage. Dmcq (talk) 23:52, 26 June 2012 (UTC)
Actually I missed seeing it, but it is listed at Tesseract#See_also. Tom Ruen (talk) 00:24, 27 June 2012 (UTC)

Agreed that should be included in the article. I also have three books published in which the tesseract is a core concept. I am holding off on citing them until #4 is published, in which the tesseract concept comes out of the shadows. AWrinkleInTime (talk) 12:15, 12 December 2015 (UTC)

L’Engle does describe a 4d hypercube (into which she shoe-horns an allusion to the idea of spacetime) on the way to her use of the term “tesseract,”. The real confusion is that the object she ends up calling a “tesseract” is not a tesseract, but rather a 5d hypercube, and that she confounds this concept with what seems like an Einstein-Rosen Bridge. Here’s the relevant excerpt:
Meg sighed. "Just explain it to me."
"Okay," Charles said. "What is the first dimension?"
"Well—a line: —————————————"
"Okay. And the second dimension?"
"Well, you'd square the line. A flat square would be in the second dimension."
"And the third?"
"Well, you'd square the second dimension. Then the square wouldn't be flat any more. It would have a bottom, and sides, and a top."
"And the fourth?"
'Well, I guess if you want to put it into mathematical terms you'd square the square. But you can't take a pencil and draw it the way you can the first three. I know it's got something to do with Einstein and time. I guess maybe you could call the fourth dimension Time."
"That's right," Charles said. "Good girl. Okay, then, for the fifth dimension you'd square the fourth, wouldn't you?"
"I guess so."
"Well, the fifth dimension's a tesseract. You add that to the other four dimensions and you can travel through space without having to go the long way around. In other words, to put it into Euclid, or old-fashioned plane geometry, a straight line is not the shortest distance between two points."
And yes, her insistence on using the term "square," to describe adding another dimension is not mathematically accurate; but any reader can see that she's clearly describing hypercubes in dimension 1-5. There’s an article out of Brown University that describes L’Engle’s use of the term, what she gets right, what she gets wrong, and what’s confusing. [1]
Off hand, I would guess she was inspired by And He Built A Crooked House which describes a tesseract both as a 4d hypercube, and as a means of instantaneous teleportation. But I can find no reference to corroborate that suspicion.
Regardless because she aptly describes a 4d hypercube on the way to defining her fictional concept of a “tesseract”, I think it warrants inclusion; probably citing the Brown University article, and summarizing the fact that while a 4d hypercube appears in the book as part of her explanation of a “tesseract,” the thing she calls a “tesseract” is actually a pastiche of a 5d hypercube and an Einstein-Rosen bridge. Probably also worthwhile to mention that the movie adaptation does not reference a 4d hypercube at any point (at least as far as I recall). If the group's amenable, I would be happy to write up the reference.Peter Jarrett-Schell (talk) 17:23, 30 August 2018 (UTC)

If there is no objection, I will add the following reference to the main article, next week.

• "A Wrinkle In Time", Madeleine L'engle‘s 1962 science features a discussion of hypercubes in dimensions 1-5. However, what is ultimately called a “tesseract” in the discussion appears to blend the concepts of a 5d hypercube and a wormhole. <ref>{{cite web|url=https://www.math.brown.edu/~banchoff/Yale/project12/math.html/|title=Mathematical Analysis|website=math.brown.edu|accessdate=31 August 2018}}</ref> Peter Jarrett-Schell (talk) 16:09, 31 August 2018 (UTC)
I have removed it. As both your contribution and the source show it is not about the 4D cube. The book seems to use it as the name for something, but it unclear what. The source is just someone’s personal musings, not a reliable mathematical source, written 20 years ago. If there was more to it then there would be more coverage in e.g. reliable sources, especially given the book’s recent adaption. But absent better sources it does not belong.--JohnBlackburnewordsdeeds 18:56, 3 September 2018 (UTC)

User:JohnBlackburne by "it" I presume you mean L'Engle's use of the word "tesseract." But the section I added was not about her use of the word, but rather the discussion between the characters Charles and Meg, in which Meg describes a 4d Hypercube by way of analogically extending a cube into 4 dimensions. I mentioned L'Engle's use of the term simply to clarify confusion, given the notoriety of L'Engle's "tesseract." As for reliability of the source, it is a published teaching resource of the Brown University Math Department, and I can find three recent citations of the article in Verge, Variety and Refinery, following the movie's release.Peter Jarrett-Schell (talk) 20:17, 3 September 2018 (UTC)

## Change arrow to point up

In the diagram showing how to get from dimension 0 to 4, just change the green arrow to point up instead of down. That way, we're always moving in a positive direction. Thanks. — Preceding unsigned comment added by 99.254.2.209 (talk) 02:24, 6 July 2012 (UTC)

## Retrieving "Tesseracts"?

At the moment "Tesseracts" redirects to "Tesseract" but there is a series of science fiction anthologies named "Tesseracts". If I were to create a page for it, how would I keep it from redirecting here? Monado (talk) 01:20, 9 April 2013 (UTC)

If you enter "Tesseracts" in the search page, you will be directed to this page, but you will see a small note at the top that says "(Redirected from Tesseracts)". If you click on the Tesseracts link in that note, you will get the redirect page, which you can then edit. It might be best to title your article something like "Tessaracts (magazine)" and make the Tessaracts article a disambiguation page that points here and to your article. Leave me a note on my talk page if you need more help.--agr (talk) 03:20, 9 April 2013 (UTC)
Simpler to put a hatnote on this article: Tesseracts redirects here. For the science fiction anthology series, see Tesseracts (magazine).Tamfang (talk) 20:23, 22 July 2013 (UTC)

## About image in Tesseract article

So may I consider consensus is reached and I may add image again? Rawal of Jaisalmer (talk) 08:40, 4 June 2013 (UTC)

No, we can't so quickly come to that conclusion.

I understand your frustration, as I too have been edited off this island with actual 4D rotation matrix dot product algorithm produced comparative static (and animations) of nD (in this case 4 for the 8-cube Tesseract) to 3D (and 2D as well). I do both perspective and parallel orthographic, normal 3D and stereoscopic 3D projections. I have never seen yours (but will try the Isometric_projection transformation matrix and check it out with my code.

I disagree with your suggestion that the Schlegel projection is more artifical as "2 steps". The way you describe it sounds like you are doing it by hand. If you actually do the math to project nD objects (which I do), it is really all about proper projections from 3 nD basis vectors (for 3D projections), the transformation matrices, and defining of the camera (eye) location.

If this is indeed a true projection (and not simply a hand drawn model - that may be in error), what basis vectors are used on the vertices of the 8 cell in order to project to that image. Do you have the code to do that?

Jgmoxness (talk) 13:58, 5 June 2013 (UTC)

I also still feel the image shouldn't be here, bevause (1) there are already too many images, and I don't understand what this is supposed to bring thatt the others don't, and (2) despite repeated questions on my talk page, I still did not get a clear explanation of exactly what principle was used to construct this image and what it is supposed to mean. The lack of continued discussion on my talk was me getting frustrated and not continuing to ask the same questions that had already been not answered before, it was not agreement. —David Eppstein (talk) 16:00, 5 June 2013 (UTC)
The image in question is File:4D completed.svg. I, for one, can't make sense of it. I would also caution the Rawal that two days' silence by one person elsewhere is not by the wildest stretch a sign of consensus here. —Tamfang (talk) 20:41, 22 July 2013 (UTC)
Count me as opposing that image as well. Wikipedia is not a place for people own ideas, the very least I would require is a citation to some reliable source which describes this. Dmcq (talk) 21:12, 22 July 2013 (UTC)

The movie Interstellar (film) has just come out and shows a tesseract used to provide a dimensional barrier separating a time changed earth to that of a space traveler encountering gravity time dilation. Suggest adding this to the article under the section See Also / Uses in fiction. Tony (talk) 04:01, 12 November 2014 (UTC)

No.
First, this article is about the concept of a tesseract (a four-dimensional generalization of a cube), not about the word "tesseract" — see WP:NOTDICT. So unless the movie actually has a four-dimensional hypercube in it, not just a "dimensional barrier", it is off-topic. For instance, A Wrinkle in Time is more famous for its use of the word "tesseract", but uses it to mean something else, so is not listed here.
Second, we need reliable secondary sources to add material like that, that attest to the significance of the topic within that film. If you just want to list everything that mentions tesseracts, regardless of how trivial or unimportant the mention is, Wikipedia is not the place; try TVTropes. (Yes, the two instances already there, the Dali and Heinlein ones, don't actually have sources, but they probably should be sourced and moved into the main text, with a discussion about how Dali uses the fourth dimension as a metaphor for the metaphysical and sources like this one.)
David Eppstein (talk) 04:17, 12 November 2014 (UTC)
For context, here's a list of previous removed cross-links Talk:Tesseract/Archive_2#Tesseracts_in_popular_culture. I don't have a strong opinion on what should be included. Tom Ruen (talk) 04:18, 12 November 2014 (UTC)
WP:NOTDICT applies to the request. This isn't about all uses of the word 'tesseract' but about 4 dimensional cubes. Users should follow the he line at the top 'For other uses, see Tesseract (disambiguation)." for other notable uses of the word. Dmcq (talk) 14:12, 12 December 2015 (UTC)

## Related polytopes section removed

User:David_Eppstein removed a Related polytopes section as too obtrusive, saying " remove typical Tomruen huge image galleries of things that are almost, but not completely, unrelated to the actual topic of the article. A single line of text with a wikilink and NO IMAGES should suffice"

I reduced it to a compact See also section with links to each template of related polytopes.

User:Dmcq objected, saying templates don't belong and removed it, saying "Shouldn't be linking to templates.They're not user content in themselves. "

I cleaned up the template wikilinks, and User:Dmcq reverted again, saying "Please take this to talk per BRD rathert than edit warring." and again a THIRD UNHELPFUL revert saying "Don't stick templates in like that."

I don't understand why he claims I'm edit warring by improving and he's not edit warring by removing. Tom Ruen (talk) 11:24, 18 December 2015 (UTC)

See WP:BRD about discussing if people give reasoned objections rather than there being some right to stick any stuff one likes in and it being difficult for anyone else to remove it. Dmcq (talk) 11:30, 18 December 2015 (UTC)
Will you please self-revert then discuss as per WP:BRD. Dmcq (talk) 11:41, 18 December 2015 (UTC)
How do you discuss something that you seem to want completely removed? Tom Ruen (talk) 11:46, 18 December 2015 (UTC)
In the discussion below I have said how it could be included. You have to set up a separate article complete with notability reasons and citations and refer to thaat in the see also. Just because a discussion implies that something you wrote should be removed does not mean the discussion is invalid and sticking in the stuff is fine. Now please remove it. Dmcq (talk) 11:55, 18 December 2015 (UTC)
Your objections are not acceptable to me. Either its a navigation table between articles, or its important enoguh as an article in its own right. You don't believe its worthy as an article, and you don't want to accept it as a navigational template. How do you suggest we show related articles? Tom Ruen (talk) 12:03, 18 December 2015 (UTC)
A template is not an article. You need to put the stuff into a separate article, most likely by transcluding the templates, and then refer to that article. What is so difficult to understand about that? Dmcq (talk) 12:07, 18 December 2015 (UTC)

## Template references put in as article material

WP:SEEALSO says 'Contents: A bulleted list, preferably alphabetized, of internal links to related Wikipedia articles.' Only in very good reason should it be used for other things. If the content is worth pointing at then point to a section within another article that has it in. That ensures the content is covered by the normal Wikipedia policies and guidelines like all user content should. Dmcq (talk) 11:28, 18 December 2015 (UTC)

That can be easily fixed. And if you don't like "see also" the links could be added back to an original "Related polytopes" section. Tom Ruen (talk) 11:30, 18 December 2015 (UTC)
A related polytopes section would not fit into the article - they re not directly about tesseracts and too big as a small exception to WP:OR. They should be in another article complete with a lead with discussion and showing notability and citations. Dmcq (talk) 11:33, 18 December 2015 (UTC)
What's too big? It's 4 sentences. The whole purpose is cross-linking related figures so wikipedia users can navigate related articles. Tom Ruen (talk) 11:43, 18 December 2015 (UTC)
Templates are not related articles. They are not articles. As WP:Template says 'Templates are pages that are embedded (transcluded) into other pages to allow for the repetition of information.' Referring to them directly is a misuse of them. Using them the way you have would avoid the policies and guidelines that apply to articles if it were standard practice. They are not in article space. Dmcq (talk) 11:45, 18 December 2015 (UTC)
The too big referred to transcluding the templates inline. Dmcq (talk) 11:50, 18 December 2015 (UTC)
The contents can move moved into articles if they are deemed too large for repeated inclusion. I expanded them with noinclude tag content to work both ways. Tom Ruen (talk) 11:51, 18 December 2015 (UTC)
You are referring to a Template. When a person clicks on it they are taken outside of article space. That is what is wrong. Transcluding templates is fine - but then they would be too biug and irrelevant for this place. You need a separate article to put them in before they are referred to here. Dmcq (talk) 12:04, 18 December 2015 (UTC)
How about templates with a "collapse" option, closed by default? Tom Ruen (talk) 12:07, 18 December 2015 (UTC)
No. Nothing that means a user could end up on a page with 'Template:' at the top. That is like them ending up at a 'Talk:' page or a 'User:' page or one of those other pages in the system. Dmcq (talk) 12:18, 18 December 2015 (UTC)
You misunderstood what I was asking, but anyway, I've done as you requested. I relinked into sections of the related uniform polytopes and honeycombs where the template is more relevant. Tom Ruen (talk) 12:23, 18 December 2015 (UTC)
Thanks, yes that is much better. Dmcq (talk) 12:25, 18 December 2015 (UTC)

Ok, maybe here we can discuss the proposed addition to the image gallery.

Exploded view showing eight 3D cubes wrapping around 4D void (symbolized by black 3D cube).

This view contributes to an understanding of the Schlegel Diagram by revealing detail of the two cubes obfuscated by a hidden lines illusion.

The ellipses show the six paths around the perimeter of the tesseract and how the faces of a given cube align with their adjacent cubes in a different 3-brane.

I would like to address specific questions about this visualization model, so that an understanding of its contribution can be accepted for inclusion in the article.

AWrinkleInTime (talk) 19:11, 23 December 2015 (UTC)

Your image looks to me like the pile of wires at the back of one of my computers, nothing like a Schlegel diagram, and not particularly helpful in understanding this polytope. I'm not convinced its connectivity is correct (I think the K-red connection at the far left should go into the opposite side of the red cube). Using the same shape to represent the three-dimensional cube faces and the four-dimensional cube volume is just confusing. And what's the red glitch between K and the void? —David Eppstein (talk) 19:20, 23 December 2015 (UTC)

Hi David -

Thanks for the feedback. This image is taken from a file for 3D printing of the model. The red connection you mention was described in the original image posting. It is not showing connectivity! That is what the six blue ellipses do. Rather, the red diagonal tube represents the fact that in the Schlegel Diagram they would be collapsed together into the central region of the projection. They are "exploded" out of the center along a symbolic 4D axis of red which is rotated so we are not looking at it head-on as in the Schlegel. Also, the red tube adds structural support in a physical model, allows for other instrumentation, etc. I would certainly make all this clear in the post. It would not hurt to have two different views of the same model, but from different angles and maybe 50% larger, so people can differentiate the six ellipsoid tubes better. It is hard to see their entrance and exit plus the three tubes that deal with inner/outer faces [the skinny ones] versus the three fat tubes that deal with the lateral/up-down faces could be made easier to differentiate--with two larger images from different viewing angles.

There is another way to interpret your comment: "(I think the K-red connection at the far left should go into the opposite side of the red cube)." Are you saying you think the left face of the upper yellow cube should go to the right face of the left red cube, instead of the left face? Or maybe that the left face of the left red cube should go to the right face of the right red cube? I give an example in Tom's discussion below, but will address that aspect here. We can look at the vertices involved and find out precisely.

For this model I assume that the yellow cubes are in the (x,y,z) 3-brane and the lower yellow cube then is in our physical brane, with (0,0,0,0) then representing its near lower-left corner [yes, in the image all the cubes are offset by 1/2 unit in the x,y,z axes to show symmetry; this does not affect relationships]. That means that the (y,z) plane on the left side of the lower yellow cube would have vertices: [(0,0,0,0), (0,0,1,0), (0,1,1,0), (0,1,0,0)]. These are the same vertices used by the inner face of the left red cube, which occupies the (y,z,k) 3-brane by the way, for x=0. Traveling from right to left along the red cube then means traveling from k=0 to k=1. In other words, at the left face of the red cube you now are in the (y,z) plane...but at k=1. You are now adjoining the upper yellow cube's left face with the vertices: [(0,0,0,1), (0,0,1,1), (0,1,1,1), (0,1,0,1)].

This confuses a lot of people, the fact that we talk about x,y,z axes in an image like this...but they do not correspond to our physical world except in the (x,y,z) 3-brane at k=0. So, traveling from right to left in the left red cube is not moving to the left in the physical world. A 90 degree bend in a fourth dimension has occurred and we are moving up in the K dimension which is perpendicular to our 3-brane. The x-y-z coordinate system is then primarily used for the viz image, and should not imply that the cubes all exist in an (x,y,z) 3-brane, as they do not.

Now, one ramification of all this is again hidden in the Schlegel Diagram: coordinate systems. You will see that in the Warner Model the top face of the top blue cube adjoins the top yellow cube's top face then the bottom face of that yellow cube adjoins the top face of the bottom yellow cube. In other words, passage through the ellipsoid tube goes from the top face of the top blue cube, through the top face of the top yellow cube, out the bottom of the yellow and into the top face of the bottom blue cube. Top face of blue cube is indirectly adjoined to top face of its opposite through the intermediary yellow cube. Likewise with its bottom face indirectly adjoining the bottom face of the bottom blue cube. This relationship is valid for all faces: they adjoin to their equivalent peer through an intermediate cube. If you again look at the vertices involved, you will see this is mathematically accurate.

Yet, in the Schlegel this seemingly is not so! But, it too is accurate as long as you include an assumption! Haha what is going on? The Schlegel can be said to presume that the major cube (the composite cube formed by all the six other perimeter cubes) is the "upper" cube and the minor cube (at the center) is the "lower" cube. You will see that mathematically this is in some ways equivalent to having the "outer" vertices bend back on themselves to connect to a second "inner" cube that brackets a central void. I can elaborate on this if desired, know this is not clear but have no time to elaborate now. Doing this does have a ramification to coordinate systems which I will get to shortly. Basically, the "small" inner cube's mating opposite cube (hidden) is using the outer-most vertices as proxies for its own mating vertices since the outer cube is merely a larger scale of itself and since those vertices do in fact attach with the proper segments to the proper other vertices (thanks to orthogonal projection). These "outer" vertices are something of an illusion though, because of the 180 degree bend that is occurring and the hidden line issue. If you look at a circle edge-on, you see two end vertices as an illusion: tangent points. Same thing.

Nevertheless, this proxy business essentially works. If the top cube of the Schlegel then is our upper blue cube (x,y,k) 3-brane, we see the top face of it as being shared with this major cube (our "top" yellow cube): it is also a de facto proxy for the top face of the top yellow cube. If we then pass through the "top" yellow cube (in (x,y,z) 3-brane at k=1, traveling from z=1 to z=0), we come to its own opposite face--the bottom face of the Schlegel major cube at k=1 and z=0. That bottom face (as visualized) then necessarily is mathematically the TOP face of the lower blue cube [(x,y,k) at k=1 and z fixed at 0]! Not the (visualized) top face as in my model. Again, the lower blue cube goes from k=1 to k=0 with z fixed at 0 and adjoins the mating yellow cube (now we have bent 90 degrees into the (x,y,z) 3-brane again, entering at its bottom face (z=0). Note that in the Schlegel, moving from k=1 to k=0 in the lower blue cube means traveling upwards along the z axis for visualization (downward mathematically), instead of downward as in my model.

When the Schlegel arrives at the center minor cube through the lower blue cube, it then enters the lower face as it does in my model. There is symmetry about the six peripheral cubes in this fashion, with inward-outward being inverted. Bottom line is as far as coordinate systems go, they behave something like mirror images on opposite sides of the planet. (Not literally, but it helps me to imagine a gravitational normal so that "up" is always away from the center in the Schlegel.) Even though we are moving upward in the lower blue cube's visualization, to enter its bottom face...remember that we are moving "down" in the K dimension from k=1 to k=0.

Let's break this down a bit more. What 3-brane do the blue cubes represent? It is the (x,y,k) brane, with z=0 and z=1 as the twin cubes. The vertices of the top face of the top blue cube then are: [(0,0,1,1), (0,1,1,1), (1,1,1,1), (1,0,1,1)]. It joins with the top face of the major cube which has those same vertices. The major cube then traverses downward from k=1 to k=0. At k=0 its face has these vertices: [(0,0,1,0), (0,1,1,0), (1,1,1,0), (1,0,1,0)]. It adjoins to the top face of the lower blue cube (z=1)!

The trick for the Schlegel Diagram is that "inward" becomes "outward" and vice versa in going from one cube to its opposite. To tie this diatribe back to your initial comment David ("Outer face of left red cube should adjoin to outer face of right red cube"), yes that is correct for the Schlegel Diagram as long as you understand the mirror effect that happens to the coordinate systems. ...But to represent the face relationships in the Warner Model the same way would be wrong.

There is no way to discern this Schlegel mirror effect from simply looking at the diagram! You can argue the diagram is wrong or you can argue that this fact is obfuscated. Either way, it does not serve the person who is trying to understand the relationships by looking at the Schlegel alone. I would argue that the Schlegel is the most compact projection, but it best works once a person fully understands the relationships.

The Warner Model is complementary to it. If you want one coordinate system and do not use a "proxy cube", then the Warner Model is the way to go. The Warner Model is correct, and without that implied assumption. It makes sense you would think it had an error since it does not conform to the Schlegel Diagram. It has better potential for introducing novices to the subtle nuances of tesseracts, in my opinion, and is mathematically correct though not without projection distortions and artifacts of its own.

So, this is a subtle but significant yet additional contribution of the Warner Model. It normalizes things so that the same coordinate system is used throughout. It also gets rid of the illusion of the "outer" cube being a composite of all the others and being larger.

The black cube is a *symbol* of the 4D void that is bounded by the 3D cubes. The Schlegel does not demonstrate this void, and rather insinuates that the tesseract is solely the collection of 6+1=8 cubes. It essentially obfuscates the void and so misleads unintentionally in that fashion. As the 2D faces of a 3D cube are not the cube without including the bounded 3D void they contain, so the tesseract is comprised of 8 3D cubes bounding 1 4D void to form a holistic 4D tesseract: 8+1=10 haha.

Again I would assert that the fact that this diagram seems alien is due to the limitations of the Schlegel Diagram. Though mathematically correct, it misleads the eye due to hidden lines.

I do need help in fitting this image into the Wikipedia page. I know a lot about tesseracts but nothing about Wikipedia. It seems that the image needs to go in the alternative image section, and this sort of discussion somewhere else...not sure where/how "somewhere else" would be.

If I can convince you and Tom of the merits of this contribution, I will be asking your help to fit this information into the page properly and to make the mathematical language rigorous and precise. Ideally, if it pleases people, I would suggest both rigorous language plus a version that is more approachable to lay people.

Your larger points seem to be that: 1) You do not see the value added; 2) you are not convinced of the mathematical accuracy. I believe #1 has been addressed herein with comments to you and Tom but certainly can pursue to the degree necessary. As for #2 I am happy to do additional virtual "walk-throughs" to demonstrate that there are only 16 vertices, yet 48 faces and 24 shared faces, etc. A big issue that addresses #2 is the one with coordinate systems, described above, which explains why it is outer-to-outer in Schlegel and outer-to-inner in Warner. They are both right, once you understand the intrinsic Schlegel assumption.

Oh! That is another point that I believe was made in the original post. People need to understand that only one of the 8 cubes would exist in our 3-brane. The other 7 all have a 4D component. For example, (x,y,k), (y,z,k), (x,z,k). There are two (x,y,z) 3-brane cubes. Each dimension has two cubes as opposites, one on either side of the central 4D void. However, the other one can be considered offset from our 3-brane in the k dimension and so it is parallel to our 3-brane.

Once it snaps into place, it all makes perfect sense. My image facilitates that. Schlegel obfuscates it due to hidden lines.

AWrinkleInTime (talk) 18:15, 1 January 2016 (UTC)

Tesseract net
It's a pretty presentation, but not clearly easier to understand than a simple net, which could have folding arrows added. Tom Ruen (talk) 19:21, 23 December 2015 (UTC)

Tom - The simple net also suffers the same perceptual illusion as the Schlegel Diagram. How many 3D cubes are visible? 7. Where is the 8th? In the center? That is incorrect. The 4D void bounded by the 3D cubes is in the center. From that network, it is not obvious that there are six ways to traverse the perimeter of a tesseract. That is what the blue ellipses (aka "pile of wires at back of computer") demonstrate.

Please understand that the very fact that this model is difficult to grasp demonstrates the failings of the Schlegel Diagram. The Schlegel does not show the 4 "rings" each with 4 cubes in them, that traverse or bound the 4D void. (There is cube overlap between rings.) There is no point in repeating what was said in the discussion that went with the original image insert...but here goes. From the Schlegel, the two cubes along the "K" axis are hidden because we are forced to look at that axis head-on. There is an "upper" cube, the central 4D void, and a "lower" cube and they are all lined up and so effectively hidden in the orthographic projection. (Not to be confused with the "upper" and "lower" cubes in the "Z" axis, which our brain understands make up two of the six visible cubes apart from the center.)

The upper and lower cube in the k dimension are all lines up with the minor square at the center of the Schlegel, rather than the major square that forms the perimeter of the Schlegel (and defines the "upper" and "lower" cubes along the z axis).

The Schlegel insinuates that the 6 perimeter cubes plus the central cube taken together form the 8th cube. This is not correct. The 8th cube is the same size as all of the others and is not a composite of the 7. Rather, the composite is the tesseract proper.

Actually, the Schlegel does not insinuate--rather our eyes are misled by hidden lines. It is correct that the "outer" faces in the Schlegel do in fact connect with the 8th cube--what is not seen is the 180 degree bend back to the other hidden cube in the center! Since two dimensions are being represented along the Z axis, we have two 90 degree folds occurring: 180 degrees. The original post went through all this--the fact that there are no "outer" faces in a tesseract.

"Outer" is always in the 4th dimension no matter what 3-brane you are dealing with. If you are talking (x,y,z) then "outer" occurs in the k. If talking (x,y,k) then outer occurs in the z. Always the 4th. Think of a face on a 3D cube. If talking the (x,y) 2-brane, then "outer" is in the z direction. Same concept.

Tom & David (this addresses Point #2 of David's about mathematical accuracy). Consider this walk-through, mapping this image to the Schlegel.

Example: In the Shlegel, The left-most cube has an "outer" face [WRONG], an upper face that adjoins the upper cube in z-axis, a lower face that adjoins the lower cube in z-axis, plus faces that adjoin the near and far cubes in the y-axis. Its "inner" face seemingly adjoins the center of the tesseract: WRONG.

In the Warner Model, consider the left-most cube [red]. It also has an upper face that adjoins the upper cube [blue] in z-axis, a lower face that adjoins the lower cube [blue] in z-axis, plus faces that adjoin the near and far cubes [green] in the y-axis. Note that "rings are formed in two planes: [(x,y),(x,z)]. For instance, one ring goes from the left red cube through the upper blue through the far red through the bottom blue, returning to the left red. These two rings pass through 4 faces of the left red cube. What about the other two faces?

Well the "inner" face does indeed go to the center of the Schlegel, but it is not the center of the tesseract. It is either the top or bottom yellow cube along the k-axis. The "outer" face does indeed go to the 8th cube, which is the other cube along the k-axis, either top or bottom. In the image, the "outer" face goes to the left face of the upper yellow cube and the inner goes to the left face of the lower yellow.

Another ring is formed! A third ring. You can see the red cube passed through the upper yellow then through its opposite red cube then through the lower yellow and then back to the left red cube. All through a ring in the (x,k) plane.

In other words, for this to be represented in the Schlegel, I suggested making the length of the segments on the peripheral cubes 1/2 unit, to symbolize the 180 degree bend in the projection. (There is no 180 degree bend in the actual tesseract.)

Put still another way, the inner face of the Schlegel adjoins the lower yellow cube in the K dimension. The outer vertices bend back along their line segments (since the Schlegel is an orthographic projection) and connect to the upper yellow cube. And there is this beautiful 4D void [black cube] that is actually at the center. It is all there in the Schlegel. Not debating that. However, distortions in the projection and hidden lines trick the eye.

AWrinkleInTime (talk) 18:23, 1 January 2016 (UTC)

I don't see that the diagram, adds anything useful . Is there an example of a diagram like that in the literature somewhere? Otherwise I don't think it should be included. It looks like own work that does not convey anything in an obvious way and is not obvious otherwise there would not be such a long text here trying to explain it. Dmcq (talk) 01:17, 2 January 2016 (UTC)

Hi Dmcq -

We are talking math here, not opinion. I have yet to field any questions about the math, other than the one that David asked--its answer formed much of my lengthy response. Look, if the issues were obvious and simple, then the Schlegel discussions would have already covered them. These have not been revealed before. This is new discussion revealing complex concepts. The math is right...and it is fascinating isn't it that the Schlegel Diagram has an assumption about a mirroring of coordinate systems from one cube to its opposite? The Warner Model does not.

The length of the explanation actually demonstrates the need for this illustration. I am sorry you cannot understand its contribution; it is complex. I do not know how to explain its benefits any more than what has already been done.

1. Do you understand how the Schlegel misleads due to hidden line illusions? 2. Or that it has mirroring of coordinate systems? 3. Do you understand that the whole composite cube is not one of the eight cubes? 4. Do you understand that a cube in the center is not one of the eight? 5. Or that there is of course a central 4D void that the eight wrap around? 6. Or that there are no out-facing or in-facing faces on the eight cubes? 7. Or that the "central cube" in the Schlegel is actually two cubes on either side of the void, seen head-on? These kinds of questions is what the Warner Model answers by clarifying relationships.

Regarding peer review publication. It appears a lot of the alternate images have been removed thankfully. If there are errors in the peer-reviewed images, what does that mean? At the very least, there are missing details to the Schlegel. These include the enumerated seven for starters. Put another way, in order to unlock the complexities of the Schlegel...the Warner Model is desired. That is its contribution.

If you want to wait for peer-review publishing before adding such clarifications to the Schlegel, so be it. My expectation was that people with the mathematical expertise managed this page in order to authoritatively recognize the current issues, and then assess any such clarifications and enhancements directly (i.e., be able to do so in a rigorous and defensible way). "I don't see it adds anything useful" is not defensible. If you can show me in the article where these issues have already been addressed, then that would be defensible. Can you show me?

AWrinkleInTime (talk) 20:18, 5 January 2016 (UTC)

It is in fact false that "the "central cube" in the Schlegel is actually two cubes on either side of the void, seen head-on". Also it's "Schlegel diagram", not "Schlegel". The diagram is created as a perspective projection from 4d to 3d (and then depicted by projecting again from 3d to 2d); just as a projected 2d view of a 3d cube does not usually include any separately visible object for the interior of the cube, this is true here. All edges and vertices of the 4d cube are projected to distinct non-overlapping line segments and points in 3d. The square 2-dimensional faces of the 4-cube are also projected to non-overlapping quadrilaterals (not all square) in 3d, and the eight cubical 3d faces of the 4-cube are projected to eight cuboids: the inner cube, the outer cube, and the six frustums connecting a square of the inner cube to the corresponding square of the outer cube. In contrast, your diagram shows more than one copy of each edge, vertex, and square 2-face of the 4-cube. The reversal of orientation of the outer cube is necessary in order for the faces of these eight cubes to be attached to each other in the same way that they are in 4d, and is no different from the fact that when you look at a 2d projection of a 3d cube the back square faces of the cube have a reversed orientation from the front faces. If you don't already understand all of this, you have no business trying to replace the image. —David Eppstein (talk) 21:46, 5 January 2016 (UTC)
If you look at the perspective illustration of the tesseract with hidden volume elimination - the coloured one with the red point showing the closest point, you'll see the analogue of the usual picture of a cube not showing the hidden sides. The extra points and lines in the usual projective views are the analogue of looking at a transparent cube. One doesn't normally show the contents of even 3D cubes. However one could by having the cube transparent and having some points inside the cube e.g. in an array or as a cloud, that might be interesting as an illustration, it would be an intermediate between that hidden volume one and the one with all the points and lines.
WP:OR is the relevant Wikipedia policy about putting in our own ideas. Basically we should not. Wikipedia is an encyclopaedia and we hould just be describing what is already written about out there. This isn't the right place to think about new maths. Dmcq (talk) 00:03, 6 January 2016 (UTC)

There is no "new math" here, only clarification. I appreciate your stance, though you did not address the issues I mention. In other words, rather than new math, this corrects misleading perceptions that lead to incorrect understandings and usage of the tesseract. Such clarification is arguably a part of the Wikipedia mandate.

Additionally, the seven issues I enumerated with the existing images can easily be verified mathematically if one merely takes the time. It is not clear to me what are the citations and sources for the existing images, either. Are they from you three commentators in this talk session? Not asserting that, just pointing out the issue without citations. How can one know?

If they are, then surely you have authority to assess other clarifications and images. At the least, I would suggest that the current alternative images must have citations as to their origins and related mathematical proofs. We cannot be expected to take these images in blind faith. Surely this is something we can agree needs done. Now, maybe--such citations are here and I just am not seeing them. I would have expected them to be at least footnoted in the caption to each image. I do see references below, but see no ready way to map them to each image.

In other words, the current article has issues that remain unresolved. However, I humbly defer to the gatekeepers and trust that should new information change your decision, that you will be open to notifying me and allowing these clarifications to be added [with proper credit to me]. Thank you.

AWrinkleInTime (talk) 19:36, 14 January 2016 (UTC)

The movie "Cube 2: Hyper Cube" (film) shows people within a tesseract which is folding and unfolding. As the tesseract is vitally germane to the plot of the movie, I would suggest adding this to the article under the section See Also / Uses in fiction. Victorsteelballs (talk) 19:09, 6 May 2016 (UTC)

I see no relation to a tesseract except that the rooms are cubical. It is just using the name for something strange that kills people in weird ways rather than anything at all like an actual tesseract. At least Alice in the Looking Glass has some connection with chess even if it didn't quite follow the rules. Dmcq (talk) 19:34, 6 May 2016 (UTC)

## Issues with removal of culture refs when they are not 'exact' to the term

I recognize that, at least for Interstellar, it is not exactly a tesseract as the page describes, but the film uses the term several times, and we have sources that say they were trying to present the notion of a tesseract/hypercube. In this case, what is presented in film is very much related to the concept of a tesseract to the point that a lay person (one without higher mathematics training) will know the difference. We're not supposed to be that technical, particularly when we consider the use in popular culture.

I would totally agree that if Interstellar never said the word "tesseract" in film (or maybe only one and never again), and the producers only refer to it as inspiration, inclusion would not be proper or necessary. Nor are they using "tesseract" in an entire different concept that has zero relationship to a true tesseract, a case that I would agree should not be included just because its the same term. This is a slightly incorrect version of a proper tesseract, but it certainly bears and extremely close resemblance to a casual reader. It seems fully proper to include it here, even if we have to include language that says "While the concept is not a true tesseract as it lacks (X)." (though that might get into some OR/SYNTH if not spelled by sources). An average reader is going to expect to find that here, and its absence is clearly an omission.

I can't speak to the Wrinkle in Time aspect, but similar consideration should be had. ----Masem (t) 01:26, 7 December 2017 (UTC)

Using the term is definitely not enough. The issue is that (per WP:NOTDICT) Wikipedia articles are about specific meanings, not about words. The word "tesseract" has two different meanings, only one of which is relevant here. It is used heavily in A Wrinkle in Time, but with a different meaning: the idea (spelled out in the title) that 3d space can be thought of as like a flexible fabric, that can be folded or wrinkled so that parts of space that we think of as far apart in the internal metric are nearby in some higher-dimensional space, and that this proximity can be used to move from one place to another, shortcutting the usual type of motion through space. The closest topic that fits that idea on Wikipedia is probably wormhole. In contrast, this article is about mathematical structures with 16 vertices, 32 edges, 24 squares, and 8 cubes, connected together to form a four-dimensional polyhedron. That concept is nowhere to be found in A Wrinkle in Time and has nothing to do with connecting far-apart three-dimensional locations. And although the sources suggest that the Interstellar writers did at least look at the polyhedral definition, the description that was included here much more closely resembled the off-topic Wrinkle in Time definition. The four popular culture items that remain (Dali, Heinlein, the Grande Arche, and Fez) use the geometric polyhedron as central concepts, so they can and should stay. If Interstellar is to be included, it should only be included with a description that makes clear how the content of the movie relates to four-dimensional polyhedra (and not to off-topic plot details about traveling through space or time). And I strongly object to any inclusion at all of A Wrinkle in Time, because I am familiar enough with that one to be sure that no polyhedra are involved. —David Eppstein (talk) 01:40, 7 December 2017 (UTC)
Given that description of Wrinkle in Time, I completely that its not appropriate because the concept is too far off the mathematical term. But I disagree that because the Interstellar is not "exact" because it doesn't check off certain boxes that a mathematician would be looking for, but a layperson would consider it. But I will look to see if there are more sources (given that Neil deGrasse Tyson spoke of the scientific accuracy of the film, I would not be surprised there might be more on this). ----Masem (t) 03:42, 7 December 2017 (UTC)
Found this [2] which has a quote from Tyson "In the tesseract as they call it, which is just a higher-dimensional space system—the tesseract actually has a very specific meaning mathematically, but ever since the Thor films and The Avengers series, “tesseract” is any access you’re going to have to a higher dimension and I’m fine with that. It’s a fresh word to most members of the public, so why not give it a fresh definition?" I might think that we should have a article Tesseract (fiction) which represents, as Tyson puts it "a device that allows one to access a higher dimension", to cover its fictional, not-exact uses. This article then at least needs to reference that, says "Tesseract has also been used like..." --Masem (t) 00:24, 9 December 2017 (UTC)
Tesseract_(disambiguation) is a good place for wider cultural references outside the specific geometric object, and that is being done now. Tom Ruen (talk) 01:25, 31 August 2018 (UTC)
I see that there was some recent debate about pop culture references (with regard to the Cosmic Cube) I certainly don’t mean to reopen that can of words (I agree with the consensus that the Cosmic Cube doesn’t belong) However, I feel obliged to note that User:David Eppstein’s assertion that there are no polyhedral involved in A Wrinkle In Time is incorrect. L’Engle does describe a 4d hypercube (into which she shoe-horns an allusion to the idea of spacetime) on the way to her use of the term “tesseract,”. The real confusion is that the object she ends up calling a “tesseract” is not a tesseract, but rather a 5d hypercube, and that she confounds this concept with what seems like an Einstein-Rosen Bridge. Here’s the relevant excerpt:
Meg sighed. "Just explain it to me."
"Okay," Charles said. "What is the first dimension?"
"Well—a line: —————————————"
"Okay. And the second dimension?"
"Well, you'd square the line. A flat square would be in the second dimension."
"And the third?"
"Well, you'd square the second dimension. Then the square wouldn't be flat any more. It would have a bottom, and sides, and a top."
"And the fourth?"
'Well, I guess if you want to put it into mathematical terms you'd square the square. But you can't take a pencil and draw it the way you can the first three. I know it's got something to do with Einstein and time. I guess maybe you could call the fourth dimension Time."
"That's right," Charles said. "Good girl. Okay, then, for the fifth dimension you'd square the fourth, wouldn't you?"
"I guess so."
"Well, the fifth dimension's a tesseract. You add that to the other four dimensions and you can travel through space without having to go the long way around. In other words, to put it into Euclid, or old-fashioned plane geometry, a straight line is not the shortest distance between two points."
And yes, her insistence on using the term "square," to describe adding another dimension is not mathematically accurate; but any reader can see that she's clearly describing hypercubes in dimension 1-5. There’s an article out of Brown University that describes L’Engle’s use of the term, what she gets right, what she gets wrong, and what’s confusing. [3]
Off hand, I would guess she was inspired by And He Built A Crooked House which describes a tesseract both as a 4d hypercube, and as a means of instantaneous teleportation. But I can find no reference to corroborate that suspicion.
Regardless because she aptly describes a 4d hypercube on the way to defining her fictional concept of a “tesseract”, I think it warrants inclusion; probably citing the Brown University article, and summarizing the fact that while a 4d hypercube appears in the book as part of her explanation of a “tesseract,” the thing she calls a “tesseract” is actually a pastiche of a 5d hypercube and an Einstein-Rosenbridge. Probably also worthwhile to mention that the movie adaptation does not reference a 4d hypercube at any point (at least as far as I recall). If the group's amenable, I would be happy to write up the reference.Peter Jarrett-Schell (talk) 17:25, 30 August 2018 (UTC)

## Avengers' Tesseract

Just caught sight of this morning's mini-edit war over the use of the term "tesseract" in the Avengers franchise.

While the anonymous poster's use of the term "snobbery" with regard to the edit by User:JohnBlackburne does seem kind of trollish and uncalled for, they may have a point (regarding the inclusion of the Avenger’s Tesseract in the article, not regarding snobbishness.)

Because most art is either two or three dimensional, pop culture depictions tesseracts will usually, by necessity, be imperfect. With few exceptions, they won't "real" tesseracts.

Dali uses a net of the tesseract. The Grand Arche evokes the image of one particular projection of a tesseract (while not actually being a projection, and not announcing to its viewer in anyway the connection). Avengers uses an actual projection of a tesseract. Therefore, as regards as JohnBlackBurne’s critique, that this section is only for things which are in some sense 4-d cubes; well, the Avenger’s tesseract is, in some sense a 4-d cube, because a cube, is in fact, a 3-d projection of a tesseract.

And while one could make a reductio-ad-absurdum argument regarding this line of reasoning, (that is, to assert that, under this rationale any cube in pop-culture should be included in this section), the fact that Avengers uses the term “Tesseract” to describe an actual 3-d projection of a tesseract, fairly well cements a connection to the mathematical construct in a specific fashion.

Additionally, having once endured two hours of cgi-explosions at my son’s behest, I can attest that Avengers treats the tesseract as an object occupying 3-d space in a peculiar fashion (if I recall one of the character's say something about it appearing in two places at once, or something like that.) This could be considered a nod to higher dimensional spatial construction (in somewhat the same way that the tesseract named “dot” functions in “Fez.”)

Taken together I would argue that the Avengers’ Tesseract has significantly greater mathematical claim to inclusion than the “Grand Arche” (though less so than “Corpus Hypercubus,” and a lot less than “And He Built a Crooked House” or “Fez.”)

And when we consider the criteria of notoriety (which does seem appropriate for the pop culture section,) we would have to allow that, lamentably, the Avengers’ tesseract is a lot more famous than any of the other offerings in this section, with the possible exception of the Grand Arche, which is, again, rather less of a 4-d cube than the Avengers’ version.

Seem like it should be included. ElijahBenedict (talk) 14:13, 24 August 2018 (UTC)

I refer you to the article for it, the Cosmic Cube. There is an image there of it and it is clearly just a cube. Nowhere in the article is there any mention of it being a 4D cube, or a projection of one, or at all related to the shape in this article. The name "tesseract" is only used as a name, and only in a very limited way. None of those uses relates it to the 4D shape. So, no it is not a use of the 4D cube. It is just a use of the name. As such it belongs on the disambiguation page Tesseract (disambiguation), along with other uses of the word, but not in this article.--JohnBlackburnewordsdeeds 14:24, 24 August 2018 (UTC)
A 3-d dimensional projection of a tesseract is, from the perspective of viewer in 3-d space, "just a cube." The iconic tesseract projection (a cube hanging inside a cube), is a kind of x-ray vision, or wire-frame reveal of the same. You wouldn't actually see it from any 3-d space. It seems evident that in calling a cube shaped, space warping, device a "tesseract" the writers are referencing a 4-d cube. We can debate the validity of that reference, but, the reference is clear.
I gather you're taking a hard line on this, which is fine, I gues. But it would seem appropriate, if that is the line you're drawing, to remove the reference to the Grand Arche as well, for the sake of consistency, given that its claim to be a 4-d cube is weaker.ElijahBenedict (talk) 14:34, 24 August 2018 (UTC)
So, I admit, I digging in now. But I can find no evidence that Johann Otto von Spreckelsen ever intended the Grand Arche to be a hypercube projection. So its claim to tesseractdom is, in essence, that some mathematician looked at it and said, "hey that looks like a projection of a tesseract." The reference in this article infers an awareness of that fact, through its use of weasel-words. A similar claim to "tesseractdom" could apply to the object in the Avengers; but in the case of the Avengers, they actually called it a tesseract, and stipulated that it occupies 3d space in an unusual fashion. ElijahBenedict (talk) 15:00, 24 August 2018 (UTC)
"it occupies 3d space in an unusual fashion" means that it is definitely not a 4-dimensional hypercube, because that does not fit into 3d space at all. If it is some other thing, it doesn't belong here. The Heinlein, Dali, and Grand Arche examples are not 4d hypercubes, but they are explicitly 3d unfoldings of 4d hypercubes, according to reliable published sources and not just fancruft. I don't know where you're reading the lack of intention in the Grand Arch example, but it doesn't seem to be in the source, and other sources say that the connection is explicit in Reitzel's "Le Cube ouvert" (1984); Reitzel was one of the designers of the Grand Arch, so he should know. —David Eppstein (talk) 17:19, 24 August 2018 (UTC)
I think a quote misattributed to Abraham Lincoln covers this. "How many legs does a dog have if you call his tail a leg? Four. Saying that a tail is a leg doesn't make it a leg." Dmcq (talk) 18:21, 24 August 2018 (UTC)
Except that a cube, is in fact a 3d projection of a tesseract, a fact that people seem oddly to be overlooking in this discussion. It is, therefore, just as much a representative of a tesseract as the Grand Arche, which is also a 3d projection of a tesseract. So your dog analogy doesn't hold. It would be more like the following conversation:
"How many legs does that dog have?"
"Four."
"How many legs does that table have?"
"Four."
"No, zero, because, as we saw from the case of the dog, legs have knees. And that table has none."
"I'm sorry, I don't think you fully understand the definition of the word 'leg'".
It just seems strange that one 3d projection of a tesseract, which does not explicitly reference tesseracts, is considered a valid reference to tesseracts in popular culture, while another 3d projection of a tesseract, which is actually called "The Tesseract," is not.ElijahBenedict (talk) 19:49, 24 August 2018 (UTC)

The point of this article, and of articles in general on Wikipedia, is not what something is called, but what it is. See WP:NOTDICT. —David Eppstein (talk) 19:52, 24 August 2018 (UTC)

Right, so confining to what things are, the Avengers Tesseract is a 3d projection of a tesseract. The Grand Arche is a 3d projection of a tesseract. Clarify what it is, about what they are, that makes one a pop culture representation of a tesseract, and the other not.ElijahBenedict (talk) 19:57, 24 August 2018 (UTC)
I'm not disagreeing that the "Tesseract" of Avengers and other fictional works do not meet the mathematical definition, but I will caution that this article will continue to see newer editors inserting the pop cultural use of "Tesseract" regardless what we want the article to be. So either we establish a section "Tesseract in popular culture" and make it clear that these typically are not the mathematical construct but based on assessing a type of higher dimension as if it were in space, or have a separate page of the same type linked off this. --Masem (t) 20:00, 24 August 2018 (UTC)
@Masem: that other page and the link to it exist in a very prominent place, the very first line of the article, the one that you probably skipped past that says "For other uses, see Tesseract (disambiguation)." This article itself is only for the 4d polyhedron. Again, see WP:NOTDICT. —David Eppstein (talk) 20:50, 24 August 2018 (UTC)
No that link does not cover other predominate uses where we arent likely going to have a separate for the object itself (like from Avengers, Interstellar, or Wrinkle in Time). --Masem (t) 21:06, 24 August 2018 (UTC)
If they belong anywhere they belong there. Those uses are about other meanings of the word tesseract, not about 4d polyhedra. Again, see WP:NOTDICT and read it this time. Wikipedia articles are not supposed to be about multiple unrelated meanings of specific words, they are supposed to be about specific topics regardless of what those topics are called. So things called "tesseract" do not become relevant here merely by being called "tesseract". 4d hypercubes are relevant here, regardless of whether they are called tesseracts, hypercubes, measure polytopes, or something else. —David Eppstein (talk) 21:44, 24 August 2018 (UTC)
The problem with putting those on the disambiguation page is that that can lead to any random use of "tesseract" in a random piece of pop culture being documented there. Per WP:TRIVIA we don't just include name-drops, we want to make sure third-parties have noticed that, which is true for Interstellar, Wrinkle, etc. But that means they need to be sourced, and you do not source things on disambiguation pages. That's why a better solution is to craft "Tesseract in popular culture" as a seperate page, acknowledging most uses are not true tesseracts, but clearly inspired by the concept (where the 4th dimension is treated as a traversable dimension like the other three). That would then just have to be linked on this page, and that's it. --Masem (t) 21:58, 24 August 2018 (UTC)
That would fail to solve the problem, because the entries included in the corresponding section here (Dalí, Heinlein, La Grande Arche, and Fez) are not included because of being called tesseracts, so they could not be offloaded to the other article you propose, and their continued presence here would continue to attract cruft like the topic of this discussion. —David Eppstein (talk) 05:39, 25 August 2018 (UTC)
If we made a separate article, I would have those four specifically moved to their own section of such an article titled "True tesseract representations in popular culture" and separate it from the "Other tesserects in popular culture" and make sure the distinction is clear that those do not use the exact mathematical definition of a tesseract but are based on the notion of a 4th dimension that can be traversed. Each such entry would require a source that acknowledge that feature (whether its a true tesserect or not) That does not dilute this page, and keeps any cruft to that page. We need to figure out a place for reader that might be searching for the term, and we cannot use disambiguation pages for that purpose. The separate page at least keeps this page clean. Otherwise, we can keep a separate section that the nature of tesserect/hypercubes has also influenced constructs of similar nature but which are not true hypercubes, on this page which keeps it on topic and avoids the claims that it is NOTDICT, but there seems to be hard fought resistance to that. --Masem (t) 06:07, 25 August 2018 (UTC)
Which still leaves the question, why include the Grand Arche, but not the Avengers Tesseract? — Preceding unsigned comment added by ElijahBenedict (talkcontribs) 20:06, 24 August 2018 (UTC)
Because we have multiple reliable independent published sources that say that the Grand Arch looks like and was intended to look like the projection of a 4d hypercube. Whereas for the space-fantasy macguffin-holder we know that until Hollywood changed it around it was called the cosmic cube, it looked like a 3d cube then, it still looks like a 3d cube, it's merely called a tesseract like many other things that are not 4d hypercubes, and your addition of it to this article had no sources at all. —David Eppstein (talk) 20:41, 24 August 2018 (UTC)
Exactly. When I see a cube I don't think to myself - oh that looks like a projection of a hypercube. A cube is just too common an object for people to remark that it looks like a projection of a hypercube. We don't have reliable sources saying the cosmic cube looks like a tesseract or is really supposed to be a projection of a hypercube. Dmcq (talk) 22:06, 24 August 2018 (UTC)

## What about clarifying the projection distortions?

Have posted previously on this and in reading geometry section, it appears some attempts at clarification have been made...yet people are still going to think the "large cube" is the 8th cube, that there is no central 4D void, that there is a "center" cube, etc. Vids are available on TikTok under @RicoWarner. Schlegel Diagram is accurate + misleading if person does not understand the projection distortions. The vids (two series) use a 2nd model, an "exploded Schlegel" to complement it and illuminate the distortions.