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Hi Luis,
***And my original post was specifically about visualization, not numbers.***
Yes, I understand. My answer to Scott was also specifically about visualization, not numbers. I don't recall numbers entering the discussion at all except in my response to Mike when he asked me to quantify the error present in your model.
***My suggested links to membranes were not meant as any serious mathematical correlation, just trying to confirm to the reader what he should have pictured.***
For reference, here is your original paragraph:
***IAW superstrings, "strings" might be considered the lines from the corners of the outer cube to the central point, the "membranes" could be considered all the planar surfaces, and 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. Depending on how you prioritize the resultant points, rods, and planes, you can arrive at 26 (& many other values) "dimensions."***
If you assume Scott is already familiar with superstring theory, then this suggestion might be useful. Then again, he wouldn't have asked the question he did.
If you assume that Scott is not familiar with superstring theory, then I stand by my characterization of your paragraph as specious.
***The tesseract may be a great model for number crunching, but I think not the best for visualizing.***
Again, I don't know where the "number crunching" comes in. But, I think the problem here is a misunderstanding of our respective connotations of the word "tesseract". To me, both your second and third diagram examples are tesseracts. They are 2D projections of 3D projections of 4D cubes. I think the most useful visualization aids for imagining 4D cubes are 3D models of 4D cubes, made of wood, wire, or plexiglass. It doesn't matter to me whether or not you call those 3D models tesseracts or not. I am not sure what you mean by 'tesseract' and whatever is is, it shouldn't get in the way of our discussion.
At any rate, I think what you have done is to obscure, or deny, two important features of what makes one visualization aid better than another. Those are,
1) 3D models are better than 2D models.
2) Models depicting all edges, vertices, and faces of the 4D cube are better than models which omit some of them.
Your degenerate model omits 12 edges, 8 vertices (or at least 7 if you want to count the degenerate point as a vertex), and 6 faces. It is on this basis that I say your model is not an accurate depiction of a 4D cube.
***Some of us work better if we can see or imagine what we're talking about, I felt this person shared this crutch with me, and so I was simply offering what works best for me.***
I am one of those people as well. I have put considerable time and energy into investigating methods that work for me, and from what you have said, I think I can offer you a method that might work better for you than what you have tried up to now. That is 3D wire models of 4D cubes with all edges, etc. represented.
As a side note, one thing I have been meaning to try but have not yet done, is to make two such models which are projections of the same hypercube from slightly offset viewpoints, which correspond to the separation of our eyes. Then, with one model in front of each eye, gaze on them so that each eye sees only the model which represents the projection of the hypercube from the viewpoint of that eye. What I hope to achieve is the illusion of depth provided by parallax. Since both models already exist in real three dimensions, with its parallax and perspective appearing naturally, this added parallax effect might be seen as depth in the 4th dimension.
As I said, I have not gotten my soldering iron to work on that one yet. It is among the hundreds of fun projects still in my backlog.
***As for the insults – I apologize.***
No apology necessary. It is I who should apologize for appearing to be insulting to you.
***You know me, a real hothead sometimes.***
I don't feel that I do know you, Luis. My impression of you continually changes. But I guess I feel that I am gradually getting to know you better.
BTW. What on earth does IAW stand for?
Warm regards,
Paul |
