pat wrote:I like your document alot.
Thanks :-)
I don't think I ever saw the original.
Oh? I was pretty sure you did... it was the one that talked about Tetra the 4D lady. It was a nice approach for describing 4D, but after a while it just gets really difficult to proceed without adequately covering the basics, so I rewrote the thing from scratch. I'm not sure whether or not Tetra will make it back in the story yet. :-)
In the part about dimension analogy as it applies to retinas, you mention that only the front set of cells would get light. This is true. But, equally important, I think, is that a 2-D array is enough to get all of the light coming toward you. You are mentioning that your cells would block incoming light from reaching the other cells. But, even if the cells were semi-transparent, you'd gain no new information. The scene is only sending you enough light to make a 2-D picture.
That is true. I'll have to rephrase that part.
One dimension is "spent" getting the light from the object to you. I think I would say something like "consider the line from your eye to a point that is reflecting light. If there is another point along this line that is also reflecting light, then either it blocks the light from the other, the other blocks the light from it, or (if the nearer is somewhat transparent) you see a combination of the colors still in that one point.
Yeah, I think the crux of the argument is that to transmit n-dimensional visual information, the light must have n+1 dimensions to travel in, otherwise the information will overlap in a way that cannot be decomposed back to the original. I still have to try to work out alternative wordings to see which one is better.
Did you get to the cross-sections chapter yet? I'm particularly proud of the diagram with a sequence of 3D intersections of a 4D object. (It's a tetracube, btw; I purposely didn't mention this in the text.) I didn't want to bother writing a real polytope intersection routine, so instead I printed out 9 copies of the rhombic dodecahedral projection of the tetracube, and manually traced out the different intersections on paper, deriving parametrized equations for them in the process. The diagram you see is the rendering of these parametrized equations by my geometric calculator program. It was really tedious, and probably rather silly in retrospect, but in the process I acquired my first experience of computing hyperplane intersections using diagrams on paper. :-)
One of these days, I should work out a general scheme for how to draw diagrams of intersecting 3-hyperplanes and visually deducing their intersections. At first I could not imagine how anyone could do this visually, but recently I had an inspiration: when we draw diagrams of intersecting planes in 3D, what we're
really drawing aren't planes at all, but
square sections of these planes. What we learned to visually deduce really isn't the intersection of the planes themselves, but the intersection of the
squares, which we then extrapolate to cover the planes. I am confident that it must be possible to generalize this to 4D as well: it's just a matter of taking the right cubical sections of 3-hyperplanes and drawing them in 3D; then it should be relatively easy to visually pick out their intersection. We just have to learn the various orientations of cubes in 4-space, and the kind of intersections they make with each other.
A corollary of this is that hyperplanes aren't
that hard to visualize: we just imagine a cubical section of it, the foreshortenings and angles of which would give us an idea of the orientation of the hyperplane, and then just extrapolate that to cover a 3-dimensional region.
