Pattern Extension Schemata, Part 1: Replication and Inversion · 2 November 19, 2001; last revised...

1 November 19, 2001; last revised December 6, 2004

Pattern Extension Schemata, Part 1: Replication and Inversion

The Replication Schema

Most weave structures are based on aninterlacement motif that is repeated horizon-tally and vertically as necessary to form thecomplete pattern. See Figure 1.

Figure 1. A Motif Repeated

This repetition schema is so common andfundamental that it often is taken for granted.

The Inversion Schema

There are other ways that a foundationmotif can be used as the basis for larger pat-

terns. One of the most interesting of theseways uses a motif and its inverse — the faceand back of a pattern. In a drawdown, theinverse is obtained by changing black cells towhite and white cells to black. See Figure 2.

Figure 2. A Pattern and Its Inverse

The inversion schema juxtaposes a founda-tion motif and its inverse horizontally andthen juxtaposes the result with its inverse ver-tically. See Figures 3 and 4.

Figure 3. Step 1 in the Inversion Schema

Figure 4. Step 2 in the Inversion Schema

This process is then iterated until a suffi-ciently large pattern is obtained. The seconditeration in the inversion schema for the ex-ample above is shown in Figure 5.

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Figure 5. Iteration 2

The foundation motif for the inversionschema can be very small and still produceinteresting results. For example, a single blackcell (not an interlacement pattern per se) devel-ops as shown in Figure 6.

Figure 6. Development from a Single Cell

The iterations shown in Figure 6 repre-sent stages in the development of the infiniteMorse-Thue Carpet, a two-dimensional exten-sion of the Morse-Thue sequence, one of themost important of all integer sequences [1].

Incidentally, if the fundamental motif is asingle white cell, the result is the back of theMorse-Thue Carpet.

Patterns produced by the inversionschema have several interesting properties:

• Successive iterations of the inversionschema double the size of the pattern.This limits the number of iterations thatare practical to use, and in any iterationthe size of the pattern may jump fromsmaller than the desired overall size tomuch larger than the desired overall size.An overly-large pattern can, of course, betrimmed.

• Successive iterations of the inversionschema appear to produce periodic pat-terns but the patterns are not periodic.That is, there is no smaller motif that canproduce them using the repetitionschema. The appearance of periodicity isvisually attractive, but the departure fromit adds interest — even if the reason is notreadily evident.

• Patterns from the inversion schema are,in fact, finite portions of infinite fractalpatterns. Such patterns are self similarand an infinite pattern contains infinitelymany copies of itself within itself.

• The loom resources required for weavingpatterns from the inversion schema (shaftsand treadles, for example) are the samefor all iterations from the first on. TheMorse-Thue carpet, for example, requiresonly two shafts and two treadles, regard-less of the number of iterations. The rea-son for this is fundamentally the same asthe reason why large patterns formed bythe repetition schema require no moreloom resources than the fundamentalmotif does: All the structural variety ex-ists in the first iteration of the inversionschema; beyond that, the complexity ofthe pattern is obtained by the varyingthreading and treadling sequences.

• The upper-left three-quarters of any it-eration of the inversion schema past thefirst, when used as the fundamental mo-

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tif for the repetition schema, matches theentire inversion schema pattern. That is,extension by repetition from 3/4ths tothe dimensions of the full inversionschema pattern matches the inversionschema pattern exactly. See Figures 7-9.Therefore, the inversion schema can becarried out until a pattern of the desiredappearance is reached and then contin-ued by the replication schema. The resultof repeating the 3/4ths motif of an inver-sion pattern is, of course, different fromthe next inversion iteration, See Figures10 and 11 on the next two pages. Switch-ing from inversion to repetition is usefulbecause repetition is easier to do thaninversion and size is more readily con-trolled.

Figure 7. Second Iteration (As Figure 5)

Figure 8. Upper-Left 3/4ths of Figure 7.

Figure 9. Extension by Repetition of Figure8 to Size of Figure 7.

The Appendix shows the development ofpatterns from several foundation patterns.

Relation of Inversion to CombinationWeaves

Some combination weaves use combina-tions of the face and back of a motif as the basis

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Figure 10. Repetition of the 3/4ths Motif

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Figure 11. The Next Inversion Iteration

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for developing a larger motif [2,3]. Many check-erboard patterns are based on this idea. Figure12 shows an example.

face-back motif

replication inversion

Figure 12. Replication Versus Inversion

The difference between this method andthe inversion schemata is that the face-backcombination is used only once to form a motifthat is then repeated. Compare Figures 10 and11 on this basis.

Other Pattern Extension Schemata

There are many possible schemata thatcan be used to produce interesting patterns.Obvious examples are geometric transforma-tions, such as successive rotations.

Different schemata also can be used insuccessive iterations.

Some of these ideas will be explored inmore detail in future articles on the subject.

References

1. The Morse-Thue Sequence, Ralph E. Griswold, 2004: http://www.cs.arizona.edu/patterns/weaving/webdocs/gre_mt.pdf

2. A Handbook of Weaves, G. H. Oelsner, 1915, Dover Reprint, 1952.

3. Textile Designing, C. J. Brickett, International Textbook, 1924.

Ralph E. GriswoldDepartment of Computer Science

The University of ArizonaTucson, Arizona

© 2001, 2002, 2004 Ralph E. Griswold

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Appendix: Examples of Inversion Schema Development

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