String Figures and Knot Theory
- mathematics of the unknot under tension

by Martin Probert

Part IV
String Figures Analysed

Opening A

string figure
Fig. 15 - Opening A and look-alikes

  • The set of look-alikes of Opening A
    = {(A(B))}
    = {AB, Ab, aB, ab}.
  • There are two shards, A and B,
    with catalogues {A, a} and {B, b}.
  • The sequences of 'unravellings by motifs' that relate the four look-alikes are given on the right. For example, to transform AB to Ab, unravel B from AB to leave A, then form b on A to give Ab.
 - - - - - .. - - - - - -
'                        '
'                        '
'    AB - - A. - - Ab    '
'    '      '      '     '
'    '      '      '     '
 - - .B     ..     .b - -
     '      '      '
     '      '      '
     aB - - a. - - ab

Jayne’s Tipstaff

string figure
Fig. 16 - Tipstaff or Truncheon
Great Britain c. 1860

  • The illustration (fig. 16) shows Jayne’s Tipstaff. The original is, however, considerably elongated compared with our illustration.
  • The set of look-alikes of the Tipstaff
    = {((A)BC)}
    = {ABC, Abc, aBC, abc}.
  • There are two shards, A and BC,
    with catalogues {A, a} and {BC, bc}.
  • The sequences of 'unravellings by motifs' that relate the four look-alikes are given on the right.
                        aBC
                         '
                         '
ABC - - A.. - - ... - - a.. - - abc
         '
         '
        Abc

Jayne’s No Name

string figure
Fig. 17 - Jayne's No Name (one half unravelled)
Caroline Islands 1902

  • The set of look-alikes of fig. 17
    = {(((ABC)D)EF)}
    = {ABCDEF, ABCDef, ABCdEF, ABCdef, abcDEF, abcDef, abcdEF, abcdef}.
  • There are three shards, ABC, D and EF,
    with catalogues {ABC, abc}, {D, d} and {EF, ef}.
  • The sequences of 'unravellings by motifs' that relate the eight look-alikes are given on the right.
           ABCDEF                abcDEF
             '                     '
             '                     '
ABCDef - - ABCD..                abcD.. - - abcDef
             '                     '
             '                     '
           ABC... - - ...... - - abc...
             '                     '
             '                     '
ABCdEF - - ABCd..                abcd.. - - abcdEF
             '                     '
             '                     '
           ABCdef                abcdef

Moon Rising over Whale Carcass

string figure
Fig. 18 - Moon Rising over Whale Carcass
Alaska 1913-14

Jayne’s Worm

string figure
Fig. 19 - A Worm
North America 1904

This figure, Jayne’s (Second) Worm, is illustrated again to demonstrate the application of the analysis to a three-dimensional figure.

Hermaphrodites

string figure
Fig. 20 - Hermaphrodites
Hawaii 1915-17 (Dickey)

String figures can be constructed with as many shards as desired. Hermaphrodites (collected in Hawaii by Lyle Dickey in 1915-17) is an asymmetric figure based upon a repetitive weaving process capable of endless repetition. The weaving process generates an additional two shards at each iteration.

Jayne’s Pygmy Diamonds

string figure
Fig. 21 - Pygmy Diamonds
Congo Kasai Valley 1904

Pygmy Diamonds has symetrically placed motifs. The look-alikes of such a 'symmetric' string figure may be grouped by rotational equivalence, one look-alike of a group being transformable into another by rotating the figure in space.

Look-alike generator for Jayne’s Pygmy Diamonds

Each of the sixteen rotationable look-alikes may be constructed by adopting none, one or more of the alternatives at steps 1, 2, 3 and 4 in the ‘look-alike generator’ below. Step 1 affects the parity of motif A; step 2 affects that of motif B; step 3 affects that of motif C; while step 4 affects the parity of the set of motifs DEFG. The digits in the binary representations (fig. 22) refer, from right to left, to steps 1, 2, 3 and 4 in the look-alike generating procedure below: a 0 indicates that the first movement in the step is to be used, a 1 that the alternative movement [given in square brackets] must be adopted. Thus 0100 indicates the adoption of the alternative at step 3.

String figure glossary

  1. Insert little fingers into the loop. Insert the left [right] thumb from above into the right [left] little finger loop, then hook up the near right [left] little finger string by rotating the left [right] thumb towards you and up. Insert the right [left] thumb from below into the left [right] thumb loop. Extend.
  2. Give the right thumb loop a full turn (360 degrees) away from [towards] you.
  3. Give the right little finger loop a full turn (360 degrees) towards [away from] you.
  4. Pass the right little finger loop up [down] through the right thumb loop and return to the right little finger, then pass the right thumb loop up [down] through the right little finger loop and return to the right thumb.
  5. With each index remove the thumb loop from above. Pass each thumb under all intervening strings and from below up into the little finger loop, and then from below up into the index loop. Release the index fingers. Hold each thumb against the base of the index finger to prevent the far thumb strings from slipping during the next action. On each side one far thumb string runs towards the centre of the figure. Pass each index down on the far side of that far thumb string and pick up the string on the tip of the index. Raise the upper frame string high on the tips of the index fingers.
string figure string figure string figure string figure
0000 0001 0010 0011
string figure string figure string figure string figure
0100 0101 0110 0111

Fig. 22 - Eight of the sixteen rotational look-alikes of Pygmy Diamonds

String Figure Mathematics - Part V
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