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Heart Sequence

1 Intention

The idea of this modeling tool–introduced by mathematician Thomas Storer (1988), and further developed as part of the ETKnoS Project—is to focus on the movements of the "loops" without taking into account the way the fingers operate on them. Storer points out that many string figures all over the world can be seen as the result of sequences of operations implemented on the "loops", such as the insertion of a loop into another, or the rotation of a loop. In other words, if one had the opportunity to perform a string figure in the dark with a fluorescent string, the movements of the string could be summarized in a certain number of such operations on the loops. By focusing on these movements during the process, and by converting them into a mathematical formula, the heart-sequence gives, in that sense, a "topological" view of a string figure algorithm. This conceptual tool turned out to be of fundamental importance in shedding light on certain phenomena which occur frequently in the corpora of string figures.

In the following, we illustrate the symbolism used for writing down Heart-Sequences, through the example of the Solomon Islands string figure Niu (star), collected by Honor Maude (1978).

2 Heart-sequence of Niu

2.1 The first steps of Niu

Niu begins with Opening A, encoded \(\underline {O}.A\) (using the same symbolism introduced the Symbolic Writing of String Figures). The four first steps of Niu can be described (using the ISFA Nomenclature) as:

Step 1: Opening A
Step 2: 1 pick up 2 and return (picture 1a).
Step 3: 3 pick up proximal (lower) 1 far and return (picture 1b).
Step 4: Release 1 and extend (pictures 1c and 1d).

1a

1b

1c

1d

In steps 2-4 taken together (pictures 1a to 1d), focusing on what happens to the previous thumb loops (\(1\,\infty \)), it can be seen that these loops pass from above through the index loops (\(2\,\infty \)), and are transferred to the middle fingers. Pictures 2a to 2i show this passage, displaying the movements of the right hand loops. We can distinctly see that the operations performed by \(R1\) and \(R3\) on the string causes the passage of \(R1\) loop (yellow one) from above through \(R2\) loop (black one).

2a

2b

2c

2d

2e

2f

2g

2h

2i

2j

The release of the thumbs in step 4 (pictures 2c and 2d) entails that the yellow thumb loops (\(1\infty \)) are finally transferred to the middle fingers (pictures 2i and 2j). So, we observe that the previous \(R1\) loop (yellow) has been passed from above through \(R2\) loop (black) and it is carried by and is transferred to \(R3\) at the end of the process. The yellow loop’s motion can be summarized by the diagrams in pictures 3a and 3b:

3a- Initial Position

3b- Done

To convince the reader that such is the case, let us pass \(R3\) loop (yellow one) from below through \(R2\) loop using \(R1\), as shown in pictures 4a to 4e, thus showing that it is the inverse operation, in the sense that it will take us back to the position following Opening A.

4a

4b

4c

4d

4e

The movement of passing \(1\infty \) from above through \(2\infty \) is noted:\(\ \overrightarrow {1\infty }\downarrow \left ( 2\infty \right )\), using a similar symbolism than the one introduced above for the insertion of a finger (functor) into a loop. In the formula, the functor has been replaced by a loop (\(1\infty \)) which operates on another loop (\(2\infty \)). Furthermore, the arrow pointing right over the symbol \(1\infty \) will mean that loops \(1\infty \) pass "away from" the practitioner and "over" all intermediate strings (none here). Moreover, the arrow pointing down indicates the insertion from above of \(1\infty \) through \(2\infty \).

To indicate that \(1\infty \) is finally transferred to the middle fingers, Storer notes \(\underrightarrow {1\infty }\rightarrow \,3\) which is defined as follows: "pass \(1\infty \) away and under all intermediate strings (if any) and place it, as a loop, directly upon 3". The arrow pointing right under the symbol \(1\infty \) is chosen here since just after the insertion through \(2\infty ~\)(black), \(1\infty \) (yellow) have to pass under (proximal to) the far index strings \(2f\) before being transferred to the middle fingers. So, focusing on the motion of \(1\infty \) the four first steps of Niu can be summarized as: \[ \overrightarrow {1\infty }\downarrow \left (2\infty \right ):\underrightarrow {1\infty }\rightarrow \,3 \]

2.2 The three next steps of Niu

With the same point of view, the fifth and the sixth steps show a displacement of little finger loops \(5\infty \). It can be seen that the loops carried by the little fingers go through the index loops (this time from below), then is transferred to the thumbs.

Step 5: Distally, insert 1 into 2 loop. 1 pass proximal to 3 loop. Proximally, insert 1 into 5 loop and pick up \(5n\) and return (picture 5a).

Step 6: Release 5 (pictures 5b and 5c).

5a

5b

5c

Let us focus on what happens to the original little finger loops \((5\infty )\) (red one - pictures 6a to 6g).

6a

6b

6c

75d

75e

75f

75g

We can see that the loops carried by the little fingers \((5\infty )\) pass towards the practitioner under all intermediate strings (here \(3n\) and \(3f\)) and go from below through the index loops. This will be encoded: \(\underleftarrow {5\infty }\uparrow \left ( 2\infty \right )\).
In the latter formula, the arrow pointing left under the symbol \(5\infty \) indicates that both \(5\infty \) have to move towards the practitioner and under all intermediate strings (if any). Furthermore, the arrow pointing up indicates the insertion from below of \(5\infty \) through \(2\infty \).
After this passage, the loops \(5\infty \) are finally transferred to the thumbs. Storer gave the notation \(\overleftarrow {5\infty }\longrightarrow \,1\) which means: "pass \(5\infty \) towards you over all intermediate strings (if any) and place it, as a loop, on 1." Here, the arrow pointing left over the symbol \(5\infty \) indicates that after the insertion of \(5\infty \) (red) into \(2\infty \) (black), \(5\infty \) pass over ("distal to" or "distally") the near index strings \(2n\) before its transfer to the thumbs.
Finally, focusing on the motion of \(5\infty \), the steps 5 and 6 of Niu can be summarized as: \[ \underleftarrow {5\infty }\uparrow \left ( 2\infty \right ) :\overleftarrow {5\infty }\longrightarrow \,1 \]

2.3 Last step of Niu

At this point, in order to reach the final figure, the indices are released and the figure is extended gently (step 7 - pictures 7a and 7b).

7a

7b

As with the notation for openings, we use the same symbolism of our "Symbolic Writing for String Figures": the operation "releasing the index loops" and "Extending the string" will be coded \(\square 2\) and \(\mid \) respectively.
The heart-sequence of the procedure Niu is then given by the following formula: \[ \underline {O}.A:\,\left \{ \begin {array}{ c c } \overrightarrow {1\infty }\downarrow \left (2\infty \right )\,:\,\underrightarrow {1\infty }\rightarrow \,3 \\ \underleftarrow {5\infty }\uparrow \left (2\infty \right ):\overleftarrow {5\infty }\longrightarrow \,1 \end {array} \right \}:\square \,2\mid \]

A Heart-sequence formula always begins with an opening (\(\underline {O}.A\) in the case of Niu) that aims to create a certain number of loops on fingers. This leads to the first "normal position". It is from this normal position that the analysis of loops’s movements can be written.
The presentation in columns has been chosen by Storer to indicate that sub-procedures in which the heart-sequences are \(\overrightarrow {1\infty }\downarrow \left (2\infty \right )\,:\,\underrightarrow {1\infty }\rightarrow \,3\) and \(\underleftarrow {5\infty }\uparrow \left (2\infty \right ):\overleftarrow {5\infty }\longrightarrow \,1\) respectively could be performed simultaneously. Of course, in practice, it is difficult to do so. In other words, we can visualize theoretically these simultaneous movements of loops without usually being able to perform them with our fingers.

3 Summary: Heart-Sequence Symbolism

REFERENCES

Storer, T. (1988). String Figures. Bulletin of the String figures association, Special issue 16 , p. 1-212.

Maude, H. (1978). Solomon Island String Figures. Canberra : Homa Press.