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

by Martin Probert

Part III
Identifying String Figure Look-Alikes Manipulatively

Partial unravellings

We now introduce an original manipulative process which, if it may be performed, is guaranteed to transform one look-alike into another. Our starting point is once more a string figure in which the motifs have already been identified.

LEMMA F is a string figure implies that the mirror image of F is a string figure.

LEMMA - Proof If a string figure F is constructed in view of a mirror, a mirror image is constructed by the person in the mirror. The mirror and the person in the mirror may be replaced by a second person exchanging movements of left and right hand throughout the construction. Thus the mirror image of F is constructable.

THEOREM 1 (Deconstruction Theorem). If a two-dimensional subset S1 of motifs of a string figure F can be completely unravelled leaving a subset S2 of unaltered string contacts, then the unravelled subset S1 can be reformed as the reflection of S1.

THEOREM 1 - Proof Suppose a two-dimensional subset S1 of motifs is completely unravelled (leaving no string contacts) to leave a subset S2. S2 is a string figure, so the mirror image s2 of S2 (which exists by the lemma above) can be formed. The two-dimensional set of unravelled string segments resulting from the unravelling of S1, apart from the orientation of the remainder of the figure in relation to it, is identical in S2 or s2. The subset S1 is now reconstructed on s2. The resultant string figure is SÈ s2.

The mirror image of SÈ  s2, that is, sÈ  S2, exists by the above lemma. Each motif of s1 is in the same position in relation to S2 as was the corresponding motif of S1 but is of opposite parity. Hence s1 is the reflection of S1.

Suppose that the action described above takes place in front of a mirror: as S1 is reconstructed on s2 to give SÈ  s2 an image in the mirror is forming s1 (the mirror image of S1) on S2 to give sÈ  S2. Similarly, given S2 in a loop of string, s1 may be added to it. That is, given any string figure, if a two-dimensional subset S1 may be unravelled to leave S2, then s1 may be formed on S2 to give sÈ  S2. Thus any such two-dimensional subset of two-dimensional motifs that can be unravelled can be reformed in the reflected state.

Example 1. The subset AB of the Brown Bear (fig. 7) may be completely unravelled to leave CDEF: hence - by Theorem 1 - the unravelled subset AB may be reformed as the reflection ab to give the Brown Bear look-alike abCDEF. The reader is encouraged to work this example with a loop of string.

Example 2. After an unravelled subset of motifs is reformed in the reflected state it may be possible to repeat the process with a different subset. We illustrate with Jacobís Ladder. ABCDEFG of Jacobís Ladder (fig. 9) may be unravelled to leave HIJKLM (fig. 10), then the unravelled motifs may be reformed in the reflected state to give abcdefgHIJKL (fig. 11); next gHIJKLM may be unravelled and reformed to give abcdefGhijklm; then abcdefGhij may be unravelled and reformed to give ABCDEFgHIJklm; and so on. Other look-alikes are revealed by unravelling different subsets at different stages of the procedure: for example, from the original figure (fig. 9) it is also possible to unravel ABCDEFGHI, DEFGHIJKLM or BCDEFGHIJKLM. Each fresh unravelling indicates a new pair of look-alikes. We note that the unravelling operations that may be performed at any one time depend upon which look-alike is available for the process: for example, the seven motifs on the left may be unravelled from ABCDEFGHIJKLM but not from abcdefGhijklm. The reader is encouraged to work this example with a loop of string: thereís a touch of magic at the very end of each reforming process as a tangle of string untangles and the last motif snaps into place.

String figure glossary

string figure
Fig. 9 - Jacobís Ladder
Worldwide

Construction: Opening A; release the thumbs; pass the thumbs under all strings and pull the far little finger string towards you; pass the thumbs over the near index string and pick up the far index string; release the little fingers; pass the little fingers over the near index string and pick up the far thumb string; release the thumbs; pass the thumbs over the index loop and pick up the near little finger string; pass each thumb from below into the index loop close to the index; navaho the thumb loops; pass each index finger from above into the string triangle near the thumb, then rotate the index fingers down and away and release the little finger loops; extend.

string figure
Fig. 10

string figure
Fig. 11

THEOREM 2 (Discoverable subsets of look-alikes). If a string figure may be completely unravelled by unravelling U two-dimensional subsets of motifs in succession (and in addition, in the case of a three-dimensional figure, of three-dimensional subsets of string contacts consisting of entire motifs and/or part or all of the 3D framework of the string figure), then

  1. the U two-dimensional subsets can be reformed in reverse order, each subset being reformed in one of two states of opposite parity (three-dimensional subsets being reformed at the appropriate points in the sequence in their original state)
  2. a subset of 2U look-alikes is discoverable by this process.

THEOREM 2 - Proof

The unravelling and reforming in the same or the reflected state of a subset of motifs is a procedure that affects only the string contacts that constitute the motifs of the subset, not the segments of string left without string contacts by earlier unravellings. Hence the successive unravelled segments of string left by a sequence of unravellings may each in turn, by theorem 1, be reformed in the original or the reflected state (or, in the case of three-dimensional subsets of string contacts consisting of entire motifs together with part or all of the 3D framework of the string figure, in the original state).

Since there are two choices for how each of the U two-dimensional subsets may be reformed, the procedure can lead to any one of 2U look-alikes.

Notation

A notation of nested brackets will be used to represent the process of 'unravelling by motifs' as applied to a particular string figure. Each stage in the unravelling of a string figure is indictated in this notation by the stripping away of the outer brackets and their immediate content. Thus ((A)BC) implies that BC may be unravelled, leaving (A), and then A may be unravelled. Square brackets will be used to surround a non-reflectible substructure (see the Worm in part IV). An expression such as {((A)BC)} is to be understood as a shorthand for the discoverable subset of look-alikes {ABC, Abc, aBC, abc}.

Example 1. The Brown Bear (fig. 7) may be completely unravelled by unravelling EF, then D, then BC, then A: hence {((((A)BC)D)EF)} is a discoverable subset of 16 (24) look-alikes. The look-alikes are ABCDEF, ABCDef, ABCdEF, ABCdef, AbcDEF, AbcDef, AbcdEF, Abcdef, aBCDEF, aBCDef, aBCdEF, aBCdef, abcDEF, abcDef, abcdEF and abcdef. Alternatively, instead of unravelling BC from ABC, AB may be unravelled to leave C: hence another discoverable subset of 16 (24) look-alikes of the Brown Bear is {(((AB(C))D)EF)}. The reader is encouraged to perform the unravellings with a loop of string. Eight look-alikes are common to both subsets, those in the subset {(((ABC)D)EF)}. There are no other look-alikes apart from the twenty-four (16 + 16 - 8) distinct look-alikes revealed through these unravellings.

Example 2. The Bed (fig. 12) can be unravelled by unravelling AB, then CD, then E, then F. Hence {(AB(CD(E(F))))} is a discoverable subset of 16 (24) look-alikes. ABCDeF is a look-alike of this subset (created by forming F, then e, then CD, then AB). ABCDeF can now be unravelled by unravelling AB, then DeF, then C. Hence {(AB((C)DeF))} is also a discoverable subset, this time of 8 (23) look-alikes. Four look-alikes are common to both subsets, those in the subset {(AB(CDeF))}. There are no other look-alikes apart from the twenty (16 + 8 - 4) distinct look-alikes revealed through these unravellings.

String figure glossary

string figure Fig. 12 - The Bed
Torres Straits 1898
(AC Haddon 1912)

Construction: Opening A; pass each thumb into the index loop from below, over the far index string, then pick up the near little finger string and return through the index loop; pass each little finger through the index loop from above, then pick up the far thumb string and return through the index loop; release the index fingers; extend.

Impossible string figures masquerading as look-alikes

The number of impossible string figures which differ from a given string figure in the parity of one or more motifs is given by the difference between (i) 2 raised to the power of the number of motifs and (ii) the number of look-alikes. Some of these impossible string figures may turn out to be identical knots under a topological transformation, but, in a context where we are interested in the look-alikes of a string figure formed upon the hands, all are to be considered distinct.

Example 1. There are 40 (26 - 24) impossible string figures masquerading as look-alikes of the Brown Bear. One of the forty impossible string figures is ABCDeF (fig. 13).

string figure
Fig. 13 - An impossible string figure masquerading as a Brown Bear look-alike

Example 2. For an example of an impossible string figure masquerading as a look-alike in the string figure literature see Jayne 1962, page 45, fig. 94, a faulty illustration of the intended figure (Bagobo Diamonds). The illustration and the intended figure differ in the parity of a single motif.

Compact expressions for the set of all look-alikes

THEOREM 3. The set of all look-alikes of a string figure is a union of discoverable subsets of look-alikes.

THEOREM 3 - Proof

Every look-alike, through some unravelling process, generates a discoverable subset of look-alikes. (Even the trivial string figure generates such a subset, a subset containing 20 = 1 look-alike, namely the trivial string figure itself.) Each discoverable subset so generated includes the generating look-alike: hence the union of the discoverable subsets generated by all look-alikes of a string figure is equal to the set of all look-alikes. In practice a union of a smaller number of discoverable subsets than those generated by all look-alikes will be sufficient.

Example. The Brown Bear set of look-alikes

= {((((A)BC)D)EF)} È {(((AB(C))D)EF)}

= {ABCDEF, ABCDef, ABCdEF, ABCdef, ABcDEF, ABcDef, ABcdEF, ABcdef, abCDEF, abCDef, abCdEF, abCdef, abcDEF, abcDef, abcdEF, AbcDEF, AbcDef, AbcdEF, Abcdef, aBCDEF, aBCDef, aBCdEF, aBCdef, abcdef}.

Shards

"In shards the sylvan vases lie"
(Herman Melville: The Ravaged Villa)

The valid states of a subset S of motifs in a string figure are the various appearances that S can assume in the different look-alikes of that figure.

Example. Motifs E and F in the Brown Bear (fig. 7) might conceivably be EF (as drawn), Ef, eF or ef. In fact only EF and ef are valid states: Ef and eF are only found in impossible string figures. The valid state ef is shown in figs 8a and 8b.

A shard is a subset S of motifs in a string figure such that

  1. for each valid state assumed by the remainder of the figure, S can independently assume any of its valid states
  2. no non-empty proper subset of S has the property in (i) above.

The catalogue of a shard S is the set of all valid states that can be assumed by the shard S.

Shards are not easy to identify. To identify the shards of a string figure it is first necessary to identify and analyse all the look-alikes of the given string figure. But, having identified the shards, an alternative way is available of efficiently listing all look-alikes.

Example 1. The Brown Bear (fig. 14) consists of three shards, ABC, D and EF, with catalogues {ABC, ABc, Abc, aBC, abC, abc}, {D, d} and {EF, ef}. Every Brown Bear look-alike is composed of a valid state from each of the three catalogues. The valid states in each catalogue have been determined by first identifying the various Brown Bear look-alikes. See the example following Theorem 3 for a complete listing of all 24 look-alikes.

string figure
Fig. 14 - Brown Bear and its three Ďshardsí

Example 2. There are 20 look-alikes of the Bed (fig. 12). The Bed is composed of two shards, AB and CDEF, with catalogues {AB, ab} and {CDEF, CDEf, CDeF, CDef, cdEF, cdEf, cdeF, CdEf, cDeF, cdef}.

THEOREM 4. If a non-empty proper subset of motifs is a shard, then the remainder of the string figure is composed of shards.

THEOREM 4 - Proof

We assume there is at least one non-empty proper subset which is a shard. Let this shard be called S with catalogue {S1, S2, ... Sm}. Let the remainder of the string figure be a member of the set {T1, T2, ... Tn} where each Tj is a distinct valid state. We shall show that the remainder of the figure is composed of shards.

S is a shard, and so (by the definition of shard), for each valid state assumed by the remainder of the figure, S can independently assume any of its valid states. Hence for each Tj (that is, for each valid state of the remainder of the figure) we can construct look-alikes {S1, Tj}, {S2, Tj}, ... {Sm, Tj}. By listing the look-alikes for each Tj we obtain {S1, T1}, {S2, T1}, ... {Sm, T1}; {S1, T2}, {S2, T2}, ... {Sm, T2}; ... ; {S1, Tn}, {S2, Tn}, ... {Sm, Tn}. Rearranging we can list the look-alikes as {S1, T1}, {S1, T2}, ... {S1, Tn}; {S2, T1}, {S2, T2}, ... {S2, Tn}; ... ; {Sm, T1}, {Sm, T2}, ... {Sm, Tn}. Thus, for each Si, Tj can assume all valid states. Hence either the remainder of the figure is a shard, or contains a shard. Continuing the argument in this fashion the entire string figure is shown to be composed of shards.

 

String Figure Mathematics - Part IV
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