PatrickPowers wrote:Weaving in 4D is a challenge. The resulting cloth has to be 3D. One possibility is using ordinary thread -- coarser I think -- and have the thread go three directions, WX, WY, XY. So each intersection of threads would have three threads instead of two. Damn, that won't work. It would block crossings in one direction but not the other. [...]
Yes, and that's precisely the problem with making 3D cloth from 1D threads. The threads just don't have enough dimensions to hold together.
One way to think about this is to take the weave pattern of 2D cloth and extrude it in the 3rd direction. Each thread becomes a sheet, and the cloth becomes 3D. Of course, this is not ideal because tugging the cloth in that same 3rd direction eventually will pull the "thread-planes" apart. But this is a good start, because now we can take this 3D weave pattern and add a similar crossing weave pattern along that 3rd direction by bending the 2D sheets into a wavy pattern so that they interlock with adjacent sheets and block each other from slipping along that direction and coming apart. The result is a weave that covers a 3D hyper-area. I didn't work it out in detail but I think this would work.
However, it's still not ideal; there still remains a preferential direction of the resulting cloth, which may make it easier to tear / come apart along that direction.
The ideal would be 2D wavy sheets with 3 perpendicular orientations, that somehow interlock with each other, taking advantage of the behaviour of 2D sheets in 4D. I haven't yet worked out how exactly this interlocking would work, but if it could be achieved, it would give us a weave pattern that would hold together in 4D and could be used generally as cloth.