Sewing in 4d. Curvature

[DonSoreno]

How would the sewing patterns of 4d pants look like?

4d beings would probably have spherindicral legs. Therefore we could use a map projection, and extrude/taper it into the 3td dimension, to get a sewing pattern.

map projection.

hVBEb.gif (23.66 KiB) Viewed 2251 times

Furthermore in sewing they have these "dart stitches" so the fabric can have gaussian curvature. Generalizing that to higher dimensions should be straightforward.

[quickfur]

The problem with sewing in 4D is that 1D threads do not knot in 4D. Meaning your clothes would unravel into its component threads as soon as you tug on it.

In order to make a knot in 4D, which is equivalent to having threads hold together in a fabric, you would need 2D threads. Or rather, sheets rather than threads. Which makes sewing a lot more complex than in 3D.

[PatrickPowers]

I figure weaving will work as long as the diameter of the thread is more than the amount the thread can slip.

[quickfur]

That's like saying a wall made of stacked marbles won't collapse when you push it.

The presence of the extra dimension makes 1D threads inherently unknottable.

[PatrickPowers]

Consider friction knots. Not the closed curve knots that show up in mathematics but not in real life. Even here in 3D friction knots depend on the diameter and compressability of the string. Adding more dimensions makes it trickier to tie a friction knot in the first place, but it is friction restricting the motion of the string that maintains such a knot.

[PatrickPowers]

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.

Having 2D thread is pretty weird. You can't then have perpendicular threads as that would be 4D when you need 3D. You have to let the sheets flop over to get three D. Then the cloth is going to be stiff in some directions. Would it be breathable? Is weaving feasible? But it might be the way to go.

I figured it was possible to have 1D thread and 2D tape but never felt certain about it. The tape has to be wide enough that it doesn't flop over. I'm not sure that will work.

I guess I can work at it and maybe get an answer.

As far as 4D clothing patterns, that's pretty mind boggling. But such patterns would be 3D so it seems possible to actually make them here in 3D. And mind boggling is the whole point of this exercise, yes?

[quickfur]

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.

[PatrickPowers]

I'm getting interested in 6 thread weaves to get 3D cloth. Consider 3D sphere packing. Replace each sphere with the intersection of six threads. The angle between any two threads is 60 degrees.