Symbolic writing
A New Symbolic Writing
for Encoding String Figure Procedures
1 Intention
A string figure process can be analyzed as a series of "elementary operations" (or "simple movements")—in the sense that the making of any string figure of a given corpus can be described by referring to a certain number of these operations. A string figure can thus be regarded as the result of a procedure consisting of a succession of elementary operations
With the goal of carrying out a comparative and statistical analysis of the occurrence of the elementary operations—involved in different string figure corpora, we have been led by the idea of creating a symbolic system for encoding string figure procedures permitting to easily isolate (within the whole process) each of these elementary operations.
Furthermore, rewriting string figure corpora using a mathematical/symbolic notation should enable us to better identify and analyze the different ways in which the string figure practitioners have combined these "elementary operations" in what I have suggested calling "sub-procedures"—i.e any succession of elementary operations either shared, or used in the same way in several string figure procedures, or iterated in the same one. In that perspective, a digital tool should be elaborated in order to identify and automatically bring to light the various (encoded) sub-procedures of a given corpus. This would allow a comparative analysis—on that aspect—from one string figure corpus to another.
2 Description of the Symbolic Writing
2.1 Labeling the Functors
As in Thomas Storer’s "Systemology" (1988), Fingers are numbered from 1 (thumb) to 5 (little finger). \(R\) and \(L\)
indicate "right hand" and "left hand" respectively. In this way, \(R1\) is the notation of the right thumb, whereas \(L2\)
denotes the left index.
The ten fingers are sometimes used with the teeth (\(T\)), a big toe (\(G\)) or the wrist (\(W\)). All of these have been termed
"Functors" by Storer.
Summary table - Functors
| |
Symbols | Definition |
\(1\) | Thumb |
\(2\) | Index |
\(3\) | Middle finger |
\(4\) | Ring finger |
\(5\) | Little finger |
\(Ri\) | \(i^{th\text { \ }}\)finger of the right hand |
\(Li\) | \(i^{th\text { }}\)finger of the left hand |
\(R,L,B\) | Right Hand, Left Hand, Both Hands |
\(T\) | Teeth |
\(G\) | Great toe |
\(W\) | Wrist |
2.2 Labeling the Objects
The Functors operate on what Storer calls "Objects". The Objects are separated into two groups: "strings" and "loops". A loop is formed when the string passes around a finger. Picture 1a shows a loop made on the left index \(L2\).
2.2.1 Strings or loops carried on fingers
A loop is denoted by using the symbol "\(\infty \)". When \(i\in \{1,2,3,4,5\}\), \(Li\,\infty \) symbolizes the loop made on the \(i^{th}\) finger of the left hand (for example, the loop on pictures 1a-b will be noted \(L2\infty \)). In the same way, \(Ri\,\infty \) will define a loop made on the \(i^{th}\) finger of the right hand.
The "string" making the loop is divided into three parts. The one which lies on the "dorsal" side of the finger—cf. (?, ?)—is written symbolically \(Rid\) or \(Lid\) for a loop made on the \(i^{th}\) finger. The "near" string of a loop (made on a finger) is the string closest to the practitioner, when the finger is pointing up and the palm is perpendicular to the practitioner’s chest. The third one is denoted as the "far" string (picture 1b).
The notations will be the following:
- \(Rin\): right near string on the \(i^{th}\) finger
- \(Rif\): right far string on the \(i^{th}\) finger
- \(Lin\): left near string on the \(i^{th}\) finger
- \(Lif\): left far string on the \(i^{th}\) finger
It frequently happens that several loops are carried by the same finger. In such a case, Storer uses the following notations: "If, in a given string position the generic finger, \(F\), has the generic natural number \(n\) loops we name these—beginning at the base of \(F\) and proceeding to the tip—as follows:" (?, ?, p. 21).
\(n\) | \(F\infty ^{\prime }s\) |
\(1\) | \(F\infty \) |
\(2\) | \(lF\infty ,uF\infty \) |
\(3\) | \(lF\infty ,mF\infty ,uF\infty \) |
\(4\) | \(lF\infty ,m_{1}F\infty ,m_{2}F\infty ,uF\infty \) |
\(5\) | \(lF\infty ,m_{1}F\infty ,m_{2}F\infty ,m_{3}F\infty ,uF\infty \) |
\(.\) | |
\(.\) | |
\(.\) | |
\(n\) | \(lF\infty ,m_{1}F\infty ,m_{2}F\infty ,...,m_{n-2}F\infty ,uF\infty \) |
1c- Three loops on \(L2\)
The symbols l, u, m are used as the abbreviation of lower, upper and median respectively. However, I did not find, either in the anthropological literature or in my own fieldwork findings, a configuration in which a finger carries more than 3 loops (picture 1c.).
The same symbols l, u and m will be used to differentiate the different strings that run from one given finger. For instance, when the index finger \(L2\) carries two loops, two different near strings start from this finger. They will be noted \(Ll2n\) and \(Lu2n\) (picture 1d).
- \(x\infty ^n\) denotes the \(n\) loops carried by the fingers \(x\). In the example above, the two loops carried by the indices can be encoded \(2\infty ^2\).
Remark: It sometimes happens that a particular loop—after having been released—is temporally carried by another part of the body or the string configuration itself. In this case, we will use the code \(ex.\), followed by the last coding of the loop in question. For instance, \(ex.Ri\infty \) will mean: the loop which was previously carried by the \(Ri\) finger. See the example in section 3.2.
2.2.2 Transverse strings
It happens sometimes that a string cannot be easily defined as carried by a finger. When such a string crosses the configuration from one hand to the other, perpendicular to both palms (when they face each other), it is called a "transverse" string and encoded \(tv\). When several such a transverse strings are created, they can be differentiated by using the terminology l, u, m and n, f defined above.
2.3 Labeling the Starting Positions
"Position I"–that we will encode \(P.I\)–is the initial position obtained when loops are formed on the thumb and little finger of both hands. In this case, the left and right palmar strings (a string which lies on the palm of the hand) are then created (picture 2). These two palmar strings will be denoted as \(Lp\) and \(Rp\).
As shown above, the string connecting \(L5f\) to \(R5f\) and the one between \(R1n\) to \(L1n\) will be noted simply \(5f\) and \(1n\) respectively.
2.3.1 Summary
Summary table - Objects
| |
Symbols | Definition |
Loops | |
\(Li\infty \) | Loop held by \(i^{th}\) finger of the left hand |
\(Ri\infty \) | Loop held by \(i^{th}\) finger of the right hand |
\(W\infty \) | Loop on the wrist |
Strings | |
\(Lif\) | Far string of the loop held by the finger \(Li\) |
\(Rin\) | Near string of the loop held by the finger \(Ri\) |
\(Lid\) | Dorsal string of the loop held by the finger \(Li\) |
\(if\) | Entire string encompassing the connected \(Lif\) and \(Rif\) |
\(in\) | Entire string encompassing the connected \(Lin\) and \(Rin\) |
\(Rp\) | Right palmar string |
\(Lp\) | Left palmar string |
\(tv\) | Transverse string |
2.4 Encoding Elementary Operations
2.4.1 Operation "Inserting"
The operation "inserting" (a finger into a loop) is encoded by letter "\(i\)" and either an "upper bar" or a "lower
bar" in order to indicate whether the insertion of the functor into the loop has to be performed from below or
from above respectively :
- \(\underline {i}F\left (x\infty \right )\) means: insert the functor \(F\) from below into the \(x\) loop.
- \(\overline {i}F\left (x\infty \right )\) means: insert the functor \(F\) from above into the \(x\) loop.
In Picture 3a, the left thumb is inserted from above into the left index loop: \(\overline {i}L1\left (L2\infty \right )\). Whereas Picture 3b shows the right middle finger is inserted from below into the right thumb loop: \(\underline {i}R3\left (R1\infty \right ):\)
Convention: The (elementary) operations implemented by the functors on the string are often performed simultaneously and symmetrically on both hands. When the mention of Left or Right (\(L\) or \(R\)) is omitted, it will mean that the same operation has to be done simultaneously on both hands. In such a way, \(\underline {i}1\left (2\infty \right ):\) means that both thumbs are inserted symmetrically from below into the index loops, considering that "1" indicates both right and left thumbs operating simultaneously and symmetrically.
2.4.2 Operation "Moving a functor over or under several strings"
When a given functor \(F\) has to move either over or under several strings, we will adopt the following
symbolism:
- \(\overrightarrow {m}F(s)\) means: move the functor \(F\) away over all strings up to and including the object \(s\) (either string or loop).
- \(\underrightarrow {m}F(s)\) means: move the functor \(F\) away under all strings up to and including the object \(s\).
- \(\overleftarrow {m}F(s)\) means: move the functor \(F\) towards you over all strings up to and including the object \(s\).
- \(\underleftarrow {m}F(s)\) means: move the functor \(F\) towards you under all strings up to and including the object \(s\).
In pictures 4a and 4b, the right thumb \(R1\) passes under all strings up to and including \(R3f\). This operation is encoded \(\underrightarrow {m}R1(R3f)\).
2.4.3 Operation "Releasing"
When the finger "\(x\)" of the right or left hand carries a single loop, the operation of "releasing" this loop is symbolized by \(r~x~(x\infty )\), or \(r(x\infty )\), considering implicitly that it is the functor "\(x\)" which operates.
In the example below, the right index releases its loop. This operation is encoded \(r~R2~(R2\infty )\) or \(r~R2\).
When the \(i^{th}\) finger of a hand carries more than one loop, the release of a specific loop is written \(r~Ri~(yRi\infty )\) or \(r(yLi\infty )\) with \(y \in {l,m,u}\). Moreover, the notation \(r~F\) will be used when a functor \(F\) (including Teeth, Great toe, Wrist) has to release all of its loops. So, \(r~F\) and \(r(F\infty )\) are equivalent when \(F\) carries exactly one loop.
2.4.4 Operation "Extending"
The bar \(\mid \) indicates that the hands have to move apart in order to absorb the slack on the string. This movement corresponds to the (elementary) operation "extending" (the string).
We will use a double bar \(\mid \mid \), when the string is extended, palms facing each other (cf. pictures 2 & 6b above). This operation is known in the literature as "Return to normal position".
2.4.5 Elementary operation "Picking up"
The operation "Picking up" is encoded by the symbol "\(p\)":
- \(p~F(s)\) means: \(F\) (a finger) picks up the string \(s\) from below, with the back of the finger.
An operation "Moving" generally precedes the operation "Picking up". Furthermore, the functor generally returns to its initial position after having performed the latter operation. In the present symbolism, we will implicitly consider that the latter movement is performed after picking up a string.
In the example illustrated below, the left thumb \(L1\) moves over both the left thumb far string \(L1f\) and the left index near string \(L2n\) and picks up from below the left index far string \(L2f\). Then, the left thumb \(L1\) returns to position (pictures 7a and 7b).
This sequence is encoded: \(\overrightarrow {m}L1\left (L2n\right ):p~L1(L2f)\).
Like Storer, we use a colon to connect two consecutive operations.
As mentioned above, this operation "picking up" is frequently implemented simultaneously and symmetrically by the right and left hands. In pictures 8, the strings \(R2f\) and \(L2f\) are picked up symmetrically by the left and the right thumbs.
This sequence will be simply written as follows \(\overrightarrow {m}1\left (2n\right ):p~1(2f)\).
2.4.6 Operations "Hooking up" and "Hooking down"
The operations "hooking" will be encoded by the symbol \(h\), and either an "upper bar" or a "lower
bar" to differentiate the operation "hooking up" from the operation "hooking down" respectively
:
- \(\underline {h}~F\left (s\right )\) : means that the functor \(F\) (a finger) hooks down the string \(s\).
- \(\overline {h}~F\left (s\right )\) : means that the functor \(F\) (a finger) hooks up the string \(s\).
In this example illustrated by the pictures 9a-9b, the operation "hooking up" is preceded by the operation "moving a functor under several strings" described above. The sequence is thus encoded as : \(\underrightarrow {m}1(5n): \overline {h}~1\left (5f\right )\). Notice that, as in the case of operation "picking up", the movement "return to position" is implicit in the latter symbolism.
In the pictures 9c-9d, the operation "hooking down" is implemented on the \(1f\) strings by the little fingers: this is encoded \(\underline {h}~5\left (1f\right )\).
Remark: After performing the operation "hooking down", the finger stays where it is, without returning to its initial position. If necessary, when not implicit, we will use the symbol # to emphasize this.
2.4.7 Operations "Seizing" and "Grasping"
Seizing an object A string can be seized between either two fingers or the teeth. We will denote this elementary operation by using the letter "\(s\)".
- \(s~L\underline {x}\overline {y}(s)\) means: seize the string \(s\) between the left fingers \(x\) (going under "\(s\)") and \(y\) (going above "\(s\)").
- \(s~L\overline {x}\underline {y}(s)\) means: seize the string \(s\) between the left fingers \(x\) (going above "\(s\)") and \(y\) (going under "\(s\)").
In the pictures 10a-10b below, the string \(L2f\) is seized between \(R1\) and \(R2\). It will thus be encoded: \(s~R\underline {1}\overline {2}(L2f)\).
When a string is seized with the teeth. It will be encoded \(s^*~T (s)\). In the picture 10c below the teeth have seized the string \(5f\): this is encoded \(s^*~T (5f)\).
Grasping several strings It happens that several strings or loops are grasped simultaneously by several fingers. We will use the letter "\(g\)" to denote this operation. In the example illustrated in the pictures 10d-10e the fingers 2345 (of both hands) grasp simultaneously the string \(1f\) and the loops \(2\infty \) and the string \(5n\). This is encoded: \(g~2345 (1f, 2\infty , 5n)\).
2.4.8 Operation "twisting" a finger (or a loop)
A finger can be rotated in order to close the loop carried by this finger i.e. to make its two strings cross each other.
- \(tw^+ x\) means: rotate the finger \(x\), 180° anticlockwise—for an observer located on the left side of the practitioner.
- \(tw^- x\) means: rotate the finger \(x\), 180° clockwise—for an observer located on the left side of the practitioner.
For instance, pictures 11 illustrate the operation \(tw^+R2\):
2.4.9 Rotations
A horizontal rotation of a functor \(F\) (generally the hands or wrists) will be encoded by using the symbol \(hr\). More precisely,
- \(hr^+ F\) if the rotation is performed 360° anticlockwise—for an observer located on the left side of the practitioner.
- \(hr^- F\) if the rotation is performed 360° clockwise—for an observer located on the left side of the practitioner.
The pictures 12a-12c show the horizontal rotation of the right hand (or wrist) 360° anticlockwise. This is encoded \(hr^+ RH\).
Remark: when a horizontal rotation is implemented with an angle less than 360°, we will encode this using a fraction. For instance, the horizontal rotation of the right hand (or wrist) 90° anticlockwise will be encoded: \(\frac {1}{4} hr^+ RH\). See the example in section 3.2.
A vertical rotation of a functor \(F\) (generally the hands or wrist) will be encoded by using the symbol \(vr\). More precisely,
- \(vr^+H\) means: rotate both hands anticlockwise for an observer located underneath the practitioner’s hands pointing up i.e. positioning the palms away.
- \(vr^-H\) means: rotate both hands clockwise for an observer located underneath the practitioner’s hands pointing up i.e. positioning the palms towards the body.
A vertical rotation \(vr^+H\) occurs on the right hand, for instance, at the end of the "Caroline extension" (cf. section 2.5.4) (pictures 12f-12g).
2.4.10 Summary
Summary table
| ||||||
Operations | Symbols | Description | ||||
Inserting | \(\overline {i}F\left (x\infty \right )\) / \(\underline {i}F\left (x\infty \right )\) | \(F\) is inserted from above/below into \(x\infty \) | ||||
Moving |
|
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Releasing |
|
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Extending | \(\mid \) | Extend the string, pulling hands apart | ||||
Picking up | \(p~F(s)\) | \(F\) picks up the string \(s\) | ||||
Hooking up | \(\overline {h}~F\left (s\right )\) | \(F\) hooks up the string \(s\) | ||||
Hooking down | \(\underline {h}~F\left (s\right )\) | \(F\) hooks down the string \(s\) | ||||
Seizing | \(s~L\underline {x}\overline {y}(s)\) / \(s~L\overline {x}\underline {y}(s)\) |
|
||||
Seizing with the teeth | \(s^*~T (x)\) | \(T\) seize the object \(x\) | ||||
Grasping | \(g~xyz(s_1, s_2, s_3)\) | \(F\) grab the strings (or loops) \(s_1, s_2, s_3\) | ||||
with the fingers \(x\),\(y\), and \(z\) | ||||||
Twisting a finger | \(tw^+x\) / \(tw^-x\) |
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Horizontal rotation | \(hr^+F\)/\(hr^-F\) |
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Vertical rotation | \(vr^+F\)/\(vr^-F\) |
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Not returning | \(\#\) |
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Restoring | \(\mid \mid \) |
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2.5 Encoding Sub-Procedures
2.5.1 Openings
An Opening can be defined as a sub-procedure aiming at creating a certain number of loops on the hands, starting from the original loop of string. The openings are noted \(\underline {O}\). For instance, Opening \(A\) is encoded \(\underline {O}.A\) (pictures 13a-13c).
2.5.2 Transfer
Transferring the loop carried by the finger \(x\) to the finger \(y\) consists in inserting the finger \(y\), either from above or below into the loop of the finger \(x\), then releasing the latter finger \(x\). In the first case, we will say that the loop \(x\infty \) is distally transferred to \(y\), whereas in the second case, it is proximally transferred to \(y\).
- \(\underline {TF}\left (y, x\infty \right )\) means: transfer proximally \(x\infty \) to \(y\).
- \(\overline {TF}\left (y, x\infty \right )\) means: transfer distally \(x\infty \) to \(y\).
In pictures 14a-14c, it is the right index loop which is proximally transferred to the thumb. This will be encoded : \(\underline {TF}\left (R1, R2\infty \right )\).
2.5.3 Navajo
When two loops lie on the same finger \(x\) (picture 15a, left thumb) the "Navajo" operation is implemented on this finger by passing the proximal loop over the distal one, and then, over the fingertip where it is released (pictures 15b-15c). We use the term "Navajo" as a verb, saying "Navajo finger \(x\)". Finally, it is encoded \(N~x\), which in this case appears as \(N~1\).
2.5.4 Caroline Extension
When a thumb carries a loop, the Caroline extension consists in picking up the far thumb string (picture 16a), while pressing the thumb against the index in order to seize the latter string (picture 16b), and, finally, rotating the hands outwards (pictures 16b and 16c). This sub-procedure is encoded \(\underline {E}.C\).
2.5.5 Exchange
The sub-procedure "Exchanging two loops" consists in exchanging a loop with the same loop on the opposite hand, after passing one of these loops either from above into the other.
- \(EX~x\infty ~[R>L]\) means: exchange the \(x^{th}\) loops, inserting (from above) the right loop into the left one.
- \(EX~x\infty ~[L>R]\) means: exchange the \(x^{th}\) loops, inserting (from above) the left loop into the right one.
Pictures 17a to 17e show the "exchange" of the index loops, \(R2\infty \) passing into \(L2\infty \): this will be encoded \(EX~2\infty ~[R>L]\).
2.5.6 Twist of a finger
The "twist of a finger"—is a succession of two elementary operations implemented by a finger (generally the indices). The finger \(x\) first picks up a string \(s_1\) and second, hooks up the string \(s_2\). This will be encoded \(TW~x(s_1, s_2)\).
Pictures 18a to 18c show the Twist of the right index, picking up a diagonal string \(s\), then hooking up \(R1n\). This will be encoded \(TW~R2(s,R1n)\). Note that the picked up diagonal string is released as the right index returns to an upright position.
2.5.7 Summary
Summary table
| |||||
Sub-procedures | Symbols | Description | |||
Openings | \(\underline {O}\) | \(\underline {O}.A\) means Opening A | |||
Transfer | \(\underline {TF}\left (y, x\infty \right )\) / \(\overline {TF}\left (y, x\infty \right )\) | Proximally (resp. distally) transfer \(x\infty \) to \(y\) | |||
Navajoing | \(N~x\) | Navajo the finger \(x\) | |||
Caroline extension | \(\underline {E}.C\) | Perform the Caroline extension | |||
Exchanging | \(EX~x\infty ~[R>L]\) / \(EX~x\infty ~[L>R]\) |
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|||
Twisting | \(TW~x(s_1, s_2)\) | \(x\) picks up \(s_1\) then hooks up \(s_2\) | |||
3 Examples
3.1 Niu (sun), Solomon Islands
\(\underline {O}.A :\overline {i}1\left (2\infty \right ):p1\left (2f \right ):\overleftarrow {m}3(2n):\underline {i}3\left (l1\infty \right ):p3\left (l1f\right ):r1 \mid \overline {i}1\left (2\infty \right ):\underrightarrow {m}1(3\infty ):\underline {i}1\left (5\infty \right ):p1\left (5n\right ): r 5 : r2\mid \)
3.2 Katagjuk (the entrance), Iglulik, Eastern Canada, Arctic
\(\underline {O}.A: \overrightarrow {m}2\left (5\infty \right ): \overline {h}2\left (2f, 5\infty \right ):\underline {i}2\left (1\infty \right ):\overline {h}2\left (1n\right ): r1 : \underline {h}1\left (l2n\right ): \underrightarrow {m}1\left (5n \right ): p1\left (5f \right ): \underline {h}1\left (u2n\right ): r\left (ex.1\infty \right )^{*}: \underrightarrow {m}1\left (2\infty ^{2},5f\right ): \overrightarrow {m}1\left (5n\right ):p1\left (5n\right ): \underrightarrow {m}1\left (ltv\right ):p1\left (ltv\right ) : r2 \mid \mid \frac {1}{4} hr^+ H\)
(*) while the operation \(\underline {h}1(u2n)\) is implemented the loop previously carried by the thumbs (noted \(ex.1\infty \)) are finally released.
3.3 Dakuna (Magic Stones), Trobriand Islands
- Figure 1
\(\underline {O}.A :\overrightarrow {m}1(2\infty ): \underline {i}1(5\infty ): p1(5n): \overrightarrow {m}2(u1f): \underline {i}2\left (l1\infty \right ):p2\left (l1f \right ): r1\mid \overline {i}1\left (u2\infty \right ): p1\left (u2f \right ):\overleftarrow {m}5(l2\infty ): p5(u2n): r(u2\infty )\mid \overrightarrow {m}2(u5n): \underline {i}2\left (l5\infty \right ): p2\left (l5n\right ): r5 \mid \underline {TF}\left (5, u2\infty \right ): \overrightarrow {m}1(2\infty ): \underline {i}1\left (5\infty \right ): p1\left (5n\right ): \underline {E}.C : r2 : \underline {E}.C \mid \)
- Figure 2
\(\underline {O}.A :\overrightarrow {m}1(2\infty ): \underline {i}1(5\infty ): p1(5n): \overrightarrow {m}2(u1f): \underline {i}2\left (l1\infty \right ):p2\left (l1f \right ): r1\mid \left \{ \begin {array}{ c c } \overline {i}L1\left (Lu2\infty \right ): pL1\left (Lu2f\right ) \\ \overrightarrow {m}R1(Rl2\infty ): pR1\left (Ru2f\right ) \end {array} \right \}^{*}: \overleftarrow {m}5(l2\infty ): p5(u2n): r(u2\infty )\mid \overrightarrow {m}2(u5n): \underline {i}2\left (l5\infty \right ): p2\left (l5n\right ): r5 \mid \underline {TF}\left (5, u2\infty \right ): \overrightarrow {m}1(2\infty ): \underline {i}1\left (5\infty \right ): p1\left (5n\right ): \underline {E}.C : r2 : \underline {E}.C \mid \)
(*) we use such a column bracket to encode sequences of operations that can be theoretically performed simultaneously, but can also be performed one hand after the other.
- Figure 3
\(\underline {O}.A :\overrightarrow {m}1(2\infty ): \underline {i}1(5\infty ): p1(5n): \overrightarrow {m}2(u1f): \underline {i}2\left (l1\infty \right ):p2\left (l1f \right ): r1\mid \overrightarrow {m}1(l2\infty ):p1(u2f): \overleftarrow {m}5(l2\infty ): p5(u2n): r(u2\infty )\mid \overrightarrow {m}2(u5n): \underline {i}2\left (l5\infty \right ): p2\left (l5n\right ): r5 \mid \underline {TF}\left (5, u2\infty \right ): \overrightarrow {m}1(2\infty ): \underline {i}1\left (5\infty \right ): p1\left (5n\right ): \underline {E}.C : r2 : \underline {E}.C \mid \)
REFERENCES
Storer, T. (1988). String Figures. Bulletin of the String figures association, Special issue 16 , p. 1-212.