I read Ian Stewart's Why Beauty is Truth: A History of Symmetry.
Triangular numbers (p. 21) look like they should be applicable to patterns of holes in sprang, which often run in triangular shapes. That is, I think you could use them to predict how many holes you'd have per row after braiding so many rows on the loom and increasing the width of the triangle of holes as you go. The series begins 0, 1, 3, 6, 10, 15, 21, 28. You should be able to calculate outlines as well, not just solid triangles.
Galois' symmetries (p. 118-123) look like they should be applicable to tablet weaving, because they have to do with rotation. The illustrations look much like those for triangular tablets. I think you could use the symmetries and a series of transformations (turning the tablets) to plot out the final position of a tablet and thus which coloured strand rises to the surface of the woven strap.