Three-strand braiding and anyam gila

Three-strand braiding works like this:

A B C,
B A C,
B C A,
C B A,
C A B,
A B C,
etc.

All six possible permutations of three letters are generated in this periodic process (period = 6) where every letter spends one move in the center and two consecutive moves on the left and right sides. Weavers in anyam gila perform the same movements but through the thickness of the fabric: two triangles on the front of the fabric, one triangle in the middle, two on the rear, one in the middle---a period of six.

In anyam gila braiding there is sometimes a single strand that covers every triangle, i.e., one of the three classes of strands has just one member. Identifying with this strand, we can focus on its 2-1-2-1 movement. On the counts where it stays in place, the other two classes exchange places.