Re-Working Caroline Islands Diamonds

“A good notation has a subtlety and suggestiveness which at times make it almost seem like a live teacher.” — Bertrand Russell [6].

The point of this final section is to show how the calculus can facilitate the analysis of string figures. We turn our attention to the figure Caroline Island Diamonds (p. 260) from Chapter VI of Jayne. Throughout this chapter, there are many openings other than $\underline{O}. {1}$ and $\underline{O}. {A}$ . Generally, it is difficult to render openings using the string figure calculus. The manipulations involved usually do not lend themselves to annotation. However, it is often possible to give an alternative construction of openings.

Caroline Island Diamonds is a wonderful example of a figure with a simple re-construction. The manipulations below are entirely fabricated, and bear little resemblence to moves used in the original figure. The figure begins with a unique opening shown in Figure 606 (p. 262) of Jayne. One can construct this opening from $\underline{O}. {A}$ as follows:

$% latex2html id marker 2976 $\displaystyle \underline{O}. {A} : \ensuremath{\Box}5 :\ >\!1\infty \to W: \overrightarrow{2\infty } \to 5 $$

The original construction then continues $\Circled{2}\ W\infty \to 1\ \textrm{(over)}$ . Let us call this position $\underline{O}. {X}$ .

$% latex2html id marker 2982 $\displaystyle \underline{O}. {X} \equiv \underli... ...nfty } \to 5: W\infty \to 1\ \textrm{(over)} \equiv \Circled{1}\ \Circled{2} $$

In this construction, the $W$ moves feel extraneous. We move a loop to $W$ only to immediately move it back. Notice that the position $\underline{O}. {X}$ is closely related to $\underline{O}. {A}$ . We would like to make some moves to get “closer” to $\underline{O}. {A}$ . To do so, we lift the $5\infty$ through the $1\infty$ and return it to the $5$ . This is closer to $\underline{O}. {A}$ but the $1\infty$ is twisted. Correcting for this twist gives the following equivalence:

$% latex2html id marker 3004 $\displaystyle \underline{O}. {X}: \underleftarrow{... ...iv \underline{O}. {A} : \ensuremath{\Box}{5} : \overrightarrow{2\infty } \to 5 $$

This is an equivalence of manipulations, none of which involve $W$

. Moreover, the left hand side of the equivalence consists entirely of loop manipulation moves. Storer calls this kind of loop manipulation construction is called a heart sequence construction. An important property of loop specific manipulation moves is that they are formally invertible. Thus, we can formally write:

$\displaystyle \underline{O}. {X}$	$\displaystyle \equiv$	$\displaystyle \Circled{1}\Circled{2}$
	$\displaystyle \equiv$	$% latex2html id marker 3017 $\displaystyle \underline{O}. {A} : \ensuremath{\Bo... ...5 : [\underleftarrow{5\infty }\!\uparrow\!(1\infty ) \to 5 :\ <\!1\infty ]^{-1}$$
	$\displaystyle \equiv$	$% latex2html id marker 3021 $\displaystyle \underline{O}. {A} : \ensuremath{\Bo... ...} \to 5 :\ >\!1\infty : \overleftarrow{5 \infty }\!\downarrow\!(1\infty ) \to 5$$

Thus, we have a constuction of $\underline{O}. {X}$ which consists almost entirely of loop manipulation moves. The original construction continues $% latex2html id marker 3025 $ \Circled{3}\ \overleftarrow{1}(\underline{5n})\ \ensuremath{\char93 }\ \ \Circled{4}\ \mathbb{P}\ $$ . Adding these steps, we get an alternative (re)construction of Caroline Island Diamonds.

$% latex2html id marker 3027 $\displaystyle \underline{O}. {A} : \ensuremath{\Bo... ...!\downarrow\!(1\infty ) \to 5 : \overleftarrow{1}(\underline{5n}) : \mathbb{P} $$