This is very much a rough note. Expect lots of updates.
There is a pair of string figures, Milky Way and Flock of Birds, which seem to cancel each other out. One forms Opening A, performs the figures one after the other, and arrives back at Opening A. It is quite the miracle, though it is hard to appreciate unless you are a giant string figure nerd. The paper
Eguchi, M. & Sato, T. (1996) “On the Relationship between the String Figure Series ‘Milky Way’ and ‘A Flock of Birds’.” Bulletin of the International String Figure Association 3:83-88.
ends with the following tantalizing remark
Although a formal mathematical analysis is beyond the scope of this paper, the reciprocal relationship between Milky Way and A Flock of Birds suggests that their weaving sequences are inverses of each other, a relationship worthy of further analysis.
Our 2026 Bridges paper, Of Loops, Braids, and String Figures: The Loopy Calculus of Cat’s Cradle provides a framework for carrying out this analysis.
One first attempt to get a handle on the equivalence is to check $(FoB)(MW) = e$ in the braid group. This involves writing out the full braid word for each figure and checking that the figures do indeed cancel out. To verify the cancellation, we use the algorithm
Dehornoy, Patrick. “A fast method for comparing braids.” Advances in Mathematics 125.2 (1997): 200-235.
and plot the resulting braids using Sage.
Sage doesn’t like to draw the trivial braid. One can visually check that the final braid shown above is indeed trivial. And there you have it! The figures are indeed equal in the braid group. This is an algorithmically generated human-verifiable proof that the figures are inverses of each other.
It will need a lot of work to make it more palatable and “interpretable.”
Published: Aug 24, 2026 @ 09:33.
Last Modified: Aug 24, 2026 @ 12:03.
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