Hi Mike,
You add a lot of fun to this forum for me.
***Would mobility in these other dimensions be especially useful in making molecular machines or something?***
We can only guess at the answer. My guess would be that, yes indeed, mobility in these other dimensions would be especially useful for something.
***Could one travel throughout the universe and/or manipulate matter and energy in transcendent ways without these illustrations?***
Not in these bodies we couldn't. That would be like asking if we could build substantial structures here in our world out of shadows. Shadows are two-dimensional things and you can't use them to build three-dimensional things. They just aren't thick enough. Our bodies, being three-dimensional would have the same problem trying to exist in higher dimensions.
***Do those dimensions exist for sure, or are they mathematical maybes only for now?***
Does anything exist for sure? I don't think so with the one exception of thought. I am 100% sure thought exists, and slightly less certain about everything else. The same goes for those dimensions.
***I also ask your subjective estimate of HOW MUCH the 20-edge example failed to illustrate a higher dimension. If it would not give away too many secrets, would you please rank the information loss you just described (32 to 20 sides) in terms of how it affected the problem of illustrating higher dimensions??***
I'm glad you asked for a subjective estimate. I think that's pretty easy. I described how to get such an estimate in my original post to Luis. Maybe you missed it, or maybe I wasn't clear, or maybe you aren't familiar with the term 'Necker Cube'. I'll give it another shot.
If you imagine a "cube" made from 12 wire edges soldered together at the corners, and then imagine the shadow it would cast, you can imagine a Necker Cube. It is nothing more than a line drawing of a cube where you can see all the edges.
Imagine rotating that wire model and noticing the various shadows it can make. One degenerate case is where it is lined up so the shadow is only one square (the light source has to be very far away for this trick). Another one would be two adjacent rectangles. Another one is three adjacent rectangles. And so on.
Now, try to imagine how you would manipulate that cube so that it would cast a shadow that looks like a square and its diagonals. You can't really do that with shadows, but you can get sort of close if your light source is very close to the center of one face but still outside the cube, the screen on which the shadow is cast is in the plane of the opposite face, and you are looking at the screen from 57 miles away in the direction of the light source.
This is the 3D to 2D equivalent of Luis' 4D to 3D degenerate projection. From 57 miles away, you will barely be able to make out the faint shadow of the front square. (It is very large, but very faint and you are very far away.) The shadow of the back square will be actual size, but from 57 miles away, it looks like a dot. Then you have the shadows of the other four edges each making half a diagonal as they go from the outside corners to that dot in the middle.
So now, when you gaze at a square with two diagonals, is it easy to visualize a cube? Is it not a lot easier to visualize a cube by looking at one of the non-degenerate shadows of that wire model? Your subjective judgement of this question will provide you with the subjective quantification you seek.
I gotta go to bed.
Warm regards,
Paul |