This is in reference to your answer to Scott's question:
Since I believe Scott was sincerely looking for a correct answer to his question, I feel compelled to take the time to point out some errors in your response to him.
***I take issue with using the tesseract in picturing extra dimensions...and I think that's unnecessarily complicating the issue.***
You may "take issue" with this approach, but it is misleading to imply that it is not a good method for picturing extra dimensions. It happens to be the simplest and most straightforward, and yet accurate, way of doing so. If you trade off complexity for inaccuracy, you need to acknowledge that you are doing so, especially if you are passing your inaccurate approach on to someone else, as you did to Scott.
***the hypercube in this case is usually depicted as a cube within a cube,***
While this method of depiction is common, it is not "usually" done in this way. 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.
***What we're required to imagine in such a case is a cube, extended in one extra-dimension, and then returning to its 3d state***
This is a goofy description. Yes, we imagine "a cube, extended in one extra-dimension", but it does not leave or "return to its 3d state". To imagine the process correctly, the hypercube is seen to be generated by the locus of points occupied by the cube as it traverses in the extra-dimension. You are correct in implying (but you described it very poorly) that there is something special about the initial and final positions of the cube. Rather than changing the dimensionality of its "state", those initial and final positions are special in that they form part of the boundary of the generated higher-dimensional figure. I.e. they form the analogs of two of the faces of the cube.
***It's so much easier to represent the entire added dimension, instead of just a hint of it, to wit:***
Your method could be seen as "easier to represent", but it is wrong. To wit:
***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.***
Your method produces a degenerate case of a cube-within-a-cube tesseract. As I already pointed out, this is not the best tesseract to use for most purposes. The degeneracy of your figure does indeed simplify the figure, but you must be aware that you have lost information as a result.
In fact, this particular projection cannot be made except in the limit. (If you doubt this, and think you can understand the explanation, let me know and I will explain it to you.)
Your figure is to the common tesseract as a square with both diagonals drawn in is to a Necker Cube. To illustrate my point, compare a square-with-diagonals with a Necker Cube and make a judgement as to which one gives a better illusion of a three-dimensional figure. It should be obvious that a Necker Cube is superior. Thus it should be obvious that a common tesseract is superior to the degenerate figure you propose.
***IAW superstrings, "strings" might be considered the lines from the corners of the outer cube to the central point***
You might consider them so, but such "strings" would have not much to do with String Theory, or Superstring Theory. The only use for such a "consideration" that I can see is in a specious "explanation" to pass on to Scott to impress him but give him nothing but a false, or at best meaningless, understanding of the concept of strings.
***the "membranes" could be considered all the planar surfaces***
Again, this is almost useless visualization advice. Yes, membranes are surfaces, but the picture of your degenerate hypercube adds nothing.
***ten dimensions might be considered the eight points of the traditional cube, plus one inner (the 'core,' dimensions fully reduced) and one outer (the whole cube, or all dimensional) points.***
This description suggests to me that you have no concept or understanding of what a dimension is in mathematics. From other things you have written, I thought you did have some idea, but this is so off the wall that it makes me wonder.
The only way in which a point can be seen to be connected with the establishment of a dimension is if the direction defined by a line to that point from an established origin is taken as the direction of a basis vector for a dimension. If your purpose were to construct a 10-D tesseract in 4-D or 5-D space, then you could establish the projection of the coordinate system by choosing to use the points you described to establish projections of the basis vectors (i.e. eight of them in the 3-D space and the "inner" and "outer" points you describe in the extra dimension(s) of the 4-D or 5-D space.) But since you don't seem to like tesseracts, I doubt that this was your purpose.
***Depending on how you prioritize the resultant points, rods, and planes, you can arrive at 26 (& many other values) "dimensions." ***
This is error compounded upon error. I think you suspected as much since you placed "dimensions" in quotes. You could just as well "prioritize" the number of letters in the various words you used in your "description" and arrive at 26 and many other values. Such "values" would have as much to do with dimensions as the ones you get by prioritizing points, rods, etc. What you wrote is utter nonsense.
***Inflation putatively occur only in some dimensions,...***
***hence the "curled up" theory.***
Total non-sequitur. The "curled up" theory has nothing to do with inflation.
***Otherwise, we should eventually see these other dimensions 'surface' (unavoidable pun).***
The pun was avoidable by eliminating this, another specious non-sequitur, from your post. In my answer to Scott's question, I argued that this inference is false. If you think I am wrong, please rebut my argument specifically.
***Actually, I think a lot of these ideas represent just how far our scientists still have to go.***
Actually, I think the ideas you presented represent just how poorly you understand science and mathematics.