String Figure Mathematics (or Trivial Knot Theory)
String Figure Mathematics featured in 19 March 2001 Math Forum newletter
Part I

Last revision: 3 Feb 2001

by Martin Probert

String Figures and Reflectible Motifs

In 1920 the historian of mathematics W. W. Rouse Ball wrote of string figures that "What ... is striking, is the immense variety of well-defined patterns". Yet no mathematical result that applies equally to all string figures (two dimensional and three dimensional, symmetric and asymmetric) appears ever to have been published.

The aim of this article is to encourage work on the theory and mathematics of string figures and to offer some preliminary results that, whether trivial or significant, have the merit of being applicable to all string figures. Suggestions for improvement, or notification of further results applicable to all string figures, are welcome. The attention of the reader willing to devote time to developing string figure theory is drawn to the conjectures in Part IV. Counter-examples or proofs relating to the conjectures would be of particular interest.

The reader wishing to pluck some practical fun out of the mathematics is directed to those examples where we show how, by merely unravelling a string figure, similar-looking figures can be identified without recourse to cutting the loop, complex algebra, or computer software. Of particular interest will be Part II, Theorem 1, Examples 1 and 2; Part II, Theorem 2, Examples 1 and 2; and Part IV, Examples 1 and 2. The 'homeoform generator' at the end of Part III may also be of practical interest.

Spot the difference

‘Spot the difference’ is a familiar game from childhood: the challenge is to spot the difference in two similar pictures. A similar game may be played with string figures. What is the difference in the following two string figures?

string figure string figure
Fig. 1 - Stone Money
Caroline Islands 1902
Fig. 2 - Star
Guyana c.1924

The labelling of the figures indicates the points at which the string figures differ. E, F, G and H on the left differ from e, f, g and h on the right. We shall study such similar looking figures in detail.

What is a string figure?

From the viewpoint of knot theory every string figure—each one of the 2000 known figures and every figure yet to be invented—is the ‘trivial knot’, an unknotted loop of string. To differentiate between one string figure and another we shall take account of the effect of tension as the string figure is extended between a number of suspension points.

A string figure was defined by Caroline Furness Jayne in her 1906 classic String Figures (Dover reprint ISBN 0-486-20152-X, 1962) as a pattern formed by ‘weaving on the hands a single loop of string’; she adds that, in the finished figure, the strings show ‘the effects of strain and of deflection produced by crosses, knots and twists’.

We emphasise the following characteristics of string figures:

  1. A string figure is a supported structure (usually on the hands).
  2. A string figure is a structure displayed under tension. Limp designs, such as those formed by arranging a loop of string on a horizontal surface, while examples of trivial knots, are not here considered as examples of string figures. At times we will talk of unravelling and reforming parts of a figure: it is to be understood in every case that the intermediate and resultant figures are also under tension (possibly with additional points of support introduced as necessary during the intermediate stages).

Reflectible motifs

The substructure efgh (fig. 2) is the reflection of EFGH (fig. 1), the reflection taking place in the plane of the substructure. Such substructures will be called reflectible substructures. A pair of reflectible substructures such as EFGH and efgh will be called substructures of opposite parity. We will also write that a substructure such as EFGH is capable of assuming one of two enantioformic states (the states in this case being EFGH and efgh).

(The desirable term with the meaning ‘opposite shape’ is ‘enantiomorph’ but this term is already used in mathematics with reference to an entire structure. ‘Enantioform’ has been coined for use in this article.)

The substructure ABCD (fig. 1) is also a reflectible substructure. This is evident by considering the reflection of fig. 2 in the plane of the string figure, and comparing the result with fig. 1.

We define a reflectible substructure as one with the following three characteristics:

  1. A reflectible substructure is two-dimensional (allowance being made for inconsequential irregularities due to the thickness of the string). The string figure of which the substructure is part may however be three-dimensional.
  2. A reflectible substructure is connected to the remainder of the string figure by free single segments of string.
  3. The sillouette of a string figure in a plane parallel to a substructure is unaltered by the replacement of the substructure by one of opposite parity.

Any one or two of the three string contacts occuring at D (fig. 1) constitute a substructure that is not a reflectible substructure: attempting to replace such a substructure by its relection causes a change of 'sillouette' (i.e. the string figure collapses). A three-dimensional substructure is unable to qualify as a reflectible substructure since the strands connecting the substructure to the remainder of the string figure would no longer connect after reflection.

Stone Money (fig. 1) may be divided into reflectible substructures in other ways apart from ABCD and EFGH. At one extreme the entire figure ABCDEFGH may be considered as a single reflectible substructure. At the other extreme an examination of the figure shows that A, B, C, D, E, F and G are all reflectible substructures. There are no smaller reflectible substructures than those labelled A to G. Such minimal reflectible substructures will be called reflectible motifs, or simply motifs. If a motif is denoted by an upper case letter, then the substitution in a string figure of that motif by one of opposite parity will be denoted by a lower case letter (compare fig. 1 motif G with fig. 2 motif g). A set of motifs will be denoted by a succession of upper or lower case letters where each letter is the label of a motif (e.g. fig. 2 contains the set of motifs ABCDefgh).

Motifs occur at each point of support of a string figure but, in the case of Stone Money where a single string encircles each point of support, these motifs are trivial (as will soon be appreciated) and have not been labelled.

The motifs of string figures may contain complex clusters of string contacts. Two or more string segments may, for example, twist and intertwine ropewise around one another. We shall not attempt to represent such clusters in terms of distinct string crossings, nor shall we make use of the Reidermeister moves of knot theory. Instead, to justify certain processes involving the unravelling of a string figure, we make use of the Deconstruction Theorem introduced in Part II.

We are not here concerned with the identification of the reflectible motifs. Identification of the motifs in a given string figure is merely a process of establishing the minimal substructures with the reflectible properties listed above. We assume that the motifs have been identified and pursue the following questions:

Changing the parity of one or more motifs will result in one of two outcomes: a similar looking string figure, or an impossible figure (i.e. one that may not be constructed from an unknotted loop). How may the similar looking string figures be identified, and what relationships exist between them?

Examples of motifs

Four motifs we shall meet frequently are illustrated in fig. 3. From left to right the four motifs will be referred to as a single-crossing, a double-crossing, a spiral triple-crossing and a triangular triple-crossing.

string figure
Fig. 3

Infinitely many different motifs exist. For example, two segments of string might twist around one another in spiral fashion through any multiple of 180 degrees to create a motif.

Each motif of fig. 3 above together with the corresponding motif of fig. 4 below make up a pair of motifs of opposite parity.

string figure
Fig. 4

An example of motifs in a three-dimensional string figure

String figure glossary

string figure
Fig. 5 - The Worm
North America 1904

Construction: Insert the thumbs into the loop and separate the hands. Pass the index fingers into the thumb loop from above and pick up the near thumb string. Pass the right thumb to the left and, from below, pick up the string running from the left thumb to left index. Extend. With the teeth pick up, from the back of the right thumb, the lower loop, pulling the loop up over the upper loop and off the right thumb, but maintaining hold of the loop with the teeth. Adjust the figure so that the mouth loop lies symmetrically between the left and right hands. Pass the little fingers under the strings and up into the thumb loop from below, then push the far thumb string and near index string away from you and under the far index string, then with the little fingers hook the far index string to the palm. Release the mouth loop and extend.

The 3D framework of a three-dimensional string figure

The 3D framework is the set of all those contacts in a three-dimensional string figure in which the single segments of string diverging from the contact do not all lie in a single plane. The 3D framework in the Worm is made up of the contacts P1, P2, P3, P4, P5 and P6.

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