On the formality of the calculus

My perspective on the string figure calculus is that it is a semi-formal domain specific language for notating string figures. Much like the notation used in knitting, the string figure calculus is well-suited to describing the manipulations involved in the production of most string figures. It is not a total and all encompassing formal system.

The world of string figures is so varied and diverse that there are things which are difficult or impossible to annotate in the calculus. When necessary, I've used English to describe some of these manipulations. I have also taken the liberty of re-working some figures to make them easier to annotate. Re-working figures is one of the great joys of playing with string figures; it is similar to giving constructions for figures known only from their final positions but with the added advantage that one has a construction on hand. It is a pleasant, methodical, sort of tinkering.

Even at its most formal, the string figure calculus is rich and expressive. There are often multiple ways to write down a manipulation. One can include or omit information to make the calculus more or less informative. To pick a concrete example, in the Navaho Many Stars complex below, I notated dropping both thumb loops as $\ensuremath{\Box}1\infty ^{(2)}$ even though the default interpretation of $\ensuremath{\Box}F$ is to release all the loops on $F$. One is free to write $\ensuremath{\Box}1$ or $\ensuremath{\Box}1\infty ^{(2)}$. Throughout this paper, I've preferred annotations which are more informative as they can be used to check the validity of a string figure construction “on your hands”. If there are not two loops to drop, and one encounters $\ensuremath{\Box}1\infty ^{(2)}$, then something has gone wrong.

As I worked on these figures, I found it helpful to freely add line breaks to the calculus. As in poetry, line breaks can change the rhythm and flow of a string figure construction. Adding line breaks was both a stylistic choice to improve readability, and a technique for highlighting “phrases” or “passages” (to use Braunstein's term [2]) in the constructions.

I hope that this paper inspires others to learn the string figure calculus and use it for analysis. It is a wonderful framework for writing, storing, and comparing figures. I feel that it is under-utilized in the string figure literature because it has a significant learning curve and is difficult to typeset. However, it is now possible to typeset the calculus in L^ATEX. This paper is intended to help ease the learning curve of the string figure calculus.