I hope you won't be embarrassed if I bring an old thread up to the top. This is in response to
***Sorry Paul, but you're miles from understanding my model:***
To the contrary; I understand your admittedly simple model very well.
***Paul: "(The tesseract) happens to be the simplest and most straightforward, and yet accurate, way of doing so."
Luis: In other words, and the tesseract is the simplest model you know of.***
In the same words (i.e. don't delete the words "and most straightforward, and yet accurate"), yes, the tesseract is the simplest, most straightforward, and yet accurate model I know of.
***The tesseract shows a 3 dimensional cube, and connects it with another, offset & identical 3 dimensional cube. It then attempts to indicate 4D perspective by moving one or both cubes through 2 dimensions. Nonsense. What you get is a very awkward manner of expressing the effect of the fourth spatial dimension.***
I'm not sure what you are expressing with this paragraph, Luis. It seems you are either trying to teach me what a tesseract is, or you are paraphrasing my description of a tesseract.
If the former, then I agree with you that it is nonsense.
If the latter, you need to read my description again, which was, "The most common method of drawing the two-dimensional projection of a tesseract is to join corresponding corners of congruent Necker Cubes which are slightly offset from one another at an angle not aligned with any line of either Necker Cube."
There is nothing in my description which "attempts to indicate 4D perspective by moving one or both cubes through 2 dimensions". As I will try to explain, there is no benefit in attempting to indicate perspective. But, even if there were, "moving...cubes through 2 dimensions" is not a method that will achieve a perspective illusion.
***It's much like representing a 3d cube by drawing a square, and connecting it with another, offest & identical square -- a very common way of drawing a cube, sure, but certainly not the best.***
"Certainly"?? What is "best" in one circumstance may not be "best" in another. Since the simple Necker Cube is so commonly drawn, it must be "best" in many circumstances. Rather than use such an ambiguous term as "best", I alluded that it was the "simplest and most straightforward, and yet accurate" method (I actually said that about the tesseract, but my argument was largely based on the same characteristic of the Necker Cube).
***The third dimension is much harder to perceive without perspective.***
True. That is because the 2D images formed on our retinas are perspective projections because of simple geometry. We are used to seeing those perspective images, so if a 2D image of a 3D subject is drawn in perspective, it is easier for us to visualize the third dimension because it looks like the images we see with our eyes.
But -- the subject at hand is the visualization of the fourth dimension, not the third dimension. We have no natural experience at doing that, so there is no 4D equivalent of perspective that we are used to or could relate to. Any attempt to introduce perspective into an image of a 4D object would be useless and could only complicate the figure.
***Paul: "Your method produces a degenerate case of a cube-within-a-cube tesseract."
Luis: LOL! Did you even read my description? There is no second cube in model.***
Yes, I read your description and understood it very well. From your comment, I take it that you read my response but that you didn't understand it at all.
Now that I am aware of your admitted "incompetence" (your word, not mine) in mathematics, I also suspect that you are not familiar with the mathematical concept of "degeneracy", and instead took it to be pejorative and one of the reasons you thought my response was insulting. I'm sorry for that. I should have explained degeneracy rather than assuming you knew what it is.
Let me explain it in the context of your model:
***Forget the inner cube. Imagine a transparent cube -- say, built of plexiglass, and imagine its centermost point. Like the center, or "core" of a Rubik's Cube, around which everything else turns. Now, connect all eight corners of the cube to that point, and you have six pyramid-like shapes turned inward on one another. The 'floors' of these yramids form the six outer planes of the standard (3d) cube.***
Before we "[f]orget the inner cube" let's start out by imagining a cube-within-a-cube depiction of a hypercube, as you suggested. If we continuously shrink the size of the inner cube down to zero, then, in a sense, we can "forget" it, and we are left with exactly the figure you describe as made of plexiglass.
In mathematics, this is known as a degenerate case. The reason is that the inner cube, which is present all during the shrinking process, disappears at once when the limit is reached. "[I]ts centermost point" is all that is left of that cube after the shrinking. So that "centermost point" can be thought of as a "degenerate" cube. That way, if you count up edges, faces, and vertices by including those contributed by the degenerate cube, you get the correct answer for the number that should be present in a hypercube.
***I'm speaking from my own model, of which I introduced a tidbit, and that we've already established you cannot fathom.***
"We" have not established that at all. You have incorrectly assumed that I did not fathom your model. It is my opinion that I understand it much better than you do.
***Insult it all you want***
I have insulted it all I want. I wanted to insult it not at all, and I did insult it not at all.
***but if you can't even grasp its simplest modicum your criticism of it is obviously a reflection of ignorance, and has no bearing on reality.***
If I agreed with you premise, I would agree with your conclusion. But since I disagree with your premise, your statement has no bearing on our discussion.
***Let me try again. Maybe you'll agree with the following: [here you provided images of a Necker Cube and two tesseracts, or hypercubes.]***
Yes, I agree with those pictures. They showed me nothing I wasn't already very familiar with.
***Here's a "hypercube," sometimes equated with the "tesseract"***
I think it would help this discussion to point out a couple things here. You are correct that the terms 'hypercube' and 'tesseract' are sometimes used interchangeably. It is also true that both terms are used differently by different authors, and, most dictionaries don't list either word (not that that would matter).
A big part of the inconsistency comes from whether we are talking about a thing or its image, and if we are talking about its image, what is the dimension of the medium of the image.
To sort this out, the thing we are trying to visualize is a 4D cube. We don't know if such a thing exists, but we know that even if it does, we don't have access to one in our 3D space. That 4D object is typically what is meant by the more formal connotation of the term 'hypercube'.
Now it is possible to form an image of the 4D object either in 3D space or 2D space. (Of course it is also possible in 1D and 0D, but this is rarely done for visualization purposes.)
In 3D, a projection may be rendered in soldered wire, as I suggested to Mike, or in plexiglass, as you suggested. Either way, those 3D models are sometimes referred to as tesseracts and as hypercubes.
In 2D, a projection may be rendered as in the second and third of the three images you referred us to. Again, these images are sometimes referred to as tesseracts and as hypercubes.
***Of these three, the "hypercube" is the best model out there,***
Here, I am sure you mean the cube-within-a-cube as depicted in your third graphic. But you aren't being clear as to whether you mean a 3D model of one or a 2D model of one as in the graphic. It seems obvious to me that the 3D model would be superior since you only need to imagine one extra dimension and not two. If you use a 2D model, then I agree with you that perspective is helpful in visualizing the third dimension, but the problem of visualizing the fourth is more pronounced than if you used a 3D model to begin with.
***...but still lacking in facilitating an understanding of a fourth spatial dimension.***
If you mean understanding dimensionality in a mathematical sense, you might be right. But if you mean being able to visualize some of the characteristics of 4D structures, then I disagree with you.
***A Necker cube? LOL!***
I'm not sure you see the role Necker Cubes play in these images, Luis. If you look at the second of your images, you will see that the tesseract is formed from two Necker Cubes, just as I described. I don't know what you are asking by, "A Necker Cube?" It may be intended as an insult, since you follow it with laughter. I don't get it, but it doesn't matter anyway.
******Sorry pal, but I really believe you're just a tad out of your visualization league here.***
Really!! Some people have some pretty weird beliefs.