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The Fundamentals of PatternStructure: Part II: The Counter-change ChallengeM. A. Hann aa University of Leeds , Leeds, LS2 9JTPublished online: 30 Mar 2009.

To cite this article: M. A. Hann (2003) The Fundamentals of Pattern Structure: Part II:The Counter-change Challenge, The Journal of The Textile Institute, 94:1-2, 66-80, DOI:10.1080/00405000308630620

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The Fundamentals of Pattern StructurePart II: The Counter-change ChallengeM.A. Hann

University of Leeds, Leeds LS2 9JT

Received 10.2.2003 Accepted for publication 3.11.2003

This paper recognises the significant contrihution made hy H. J. Woods to the development ofconcepts as.sociated with the systematic colouring of border and all-over patterns. The vitalrole played hy other scholars (including Schattschneider, and Washbum and Crowe) inmaking these concepts accessible to a wider audience is acknowledged. Original illustrativematerial, developed within the framework of relevant symmetry rules, is provided for each ofthe seventeen two-colour counter-change horder patterns and each of the forty-six two-colourcounter-change all-over patterns.

1. INTRODUCTION

The principles of geometric symmetry have been well rehearsed in the relevantmathematical literature, and were concisely stated for a non-malheniatical audience inPart I of this paper. Accepted wisdom among mathematicians states that all regularrepeating pattems (of which there are only seventeen distinct primary classes) are formedby the repeated action of one or more of the four symmetry operations (translation,rotation, reflection and glide-reflection) within a lattice network of equidistant points.Details of these underlying principles were first made accessible to a non-mathematicalaudience in the mid-1930s through the endeavour of H.J. Woods, a physicist working inthe Department of Textiles Industries at the University of Leeds. In his four-paper seriesThe Geometrical Basis of Patiem Design, Woods (1935 a. b. c. 1936) made the flrsttentative steps towards breaking down the inevitable barrier of unfamiliar symbols andobscure terminology (the staple diet of many of his mathematically aware contem-poraries). Eminent mathematicians such as Doris Schattschneider (1978, 1986 and 1990)and Donald Crowe (1971, 1975 and 1982) enhanced communication with the non-mathematical audience, some years later.

Woods was the first to present the complete and explicit enumeration of thetwo-colour, one- and two-dimensional patterns (i.e. the two-colour counter-change borderand all-over patterns). Woods was keen that his work would empower textile designersin the creation of original pattems. With this in mind, it is the intention of this paperto show the results of taking up this challenge. The illustrations presented in this paperwere constructed by adhering closely to the symmetry ingredients identified initiallyby Woods.

2. COLOUR COUNTER-CHANGE PATTERNS

In Part I. the fundamental geometrical mles governing pattem structure were outlinedwith an emphasis only on symmetry operations which did not involve colour change,i.e. colour was preserved following each symmetry operation. It may however bethe ease, following certain symmetry actions, that colour is systematically changed

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The Fundamentals of Pattern Structure. Part ll: The Counter-change Challenge

in a continuous way. Such designs are called 'counter-change designs', terminologyused by Christie (1910. ch.lO of 1969 edition), and Gombrich (1979, p. 89), as well asWoods (1936).

An array of mathematical literature is available on the subject of colour symmetry.Schwarzenberger (1984). for example. lists at least 100 research papers, reports and otherpublications concerned with the subject. Senechal (1979). Loeb (1971), Lockwood andMacmillan (1978). Grunbaum and Shephard (1987), Schattschneider (1986), Washbumand Crowe (1988) and Wieting (1982) probably produced the most significant work,subsequent to Woods' four-paper series.

2.1 Counter-change Border Patterns

By introducing systematic colour change (of two colours) on the seven primary border-pattern structures, a total of seventeen two-colour counter-change border patterns arepossible. The notation used in the classification of such designs is a modification of the'pxyz' notation used for the seven primary border-pattern classes. This is the inter-nationally accepted notation with which a prime (') is generally associated with one of thesymbols, if the corresponding symmetry operation introduces an interchange of colour.Washbum and Crowe (1988. p. 69) give a readily understandable description of the rulesof the notation. Examples of the seventeen possibilities based on the seven primarystructures are given in Fig. I.

When three colours are involved in systematic colour change, only seven possibilitiesunfold. Table I provides an enumeration of colotir symmetry for border patterns, using upto eight colours.

2.2 Counter-change All-over Patterns

It was stated previously that only seventeen distinct classes of all-over patterns may beproduced using combinations of the four symmetry operations. By introducing systematicchange of colour (using two colours) on these seventeen primary structures, a total offorty-six two-colour counter-change possibilities unfolds. As pointed out by Washburnand Crowe (1988, ch. 5), although there is no universally accepted international notationfor the forty-six two-colour counter-change pattern classes, the notation proposed byBelov and Tarkhova (1964, p. 211) appears to be the most widely adopted. The notationis an adaptation of that used to classify the seventeen primary all-over pattern classes, butis slightly more complex. Washburn and Crowe (1988) gave an easily understandableexplanation, accompanied by relevant schematic illustrations. Woods himself presented aseries of two-colour illustrations, seemingly taken from a variety of historical sources,and used a rather unwieldy notation.

By introducing systematic change of colour, using two colours, on the four primary all-over-pattern classes that lack rotational characteristics (i.e. classes pi, pigl, plml andc I m 1) a total of eleven two-colour counter-change possibilities is realised. From the fiveprimary classes with two-fold rotation (i.e. p211, p2gg, p2mg, p2mm and c2mm) a totalof nineteen two-colour counter-change possibilities unfolds. With the three classesexhibiting three-fold rotation, systematic change of two colours is restricted. Class p3cannot facilitate systematic two-colour change, but both p3ml and p31m can eachprovide one possibility. The three classes exhibiting four-fold rotation (p4, p4gm andp4mm) yield ten two-colour counter-change possibilities. There is only one way tosystematically colour a class p6 pattern and that is to alternate the colour round thecentres of six-fold rotation. Class p6mm all-over patterns may be systematically

./. Tfxi. Inst.. 2(HKl 94 I'art 2. Nns 1/2 © Texlile

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pill

plai

p112

pmi 1

plml

pma2

p1a1

p112

p112'

pm'11

p'm11

p'1a1

p1itT1

pm'a2'

pma'2'

Fig. 1 The seven primary and seventeen iwo-colour counter-change border piittem classes (Continuedopposite)

68 J. Text. Inst.. 2003. 94 Part 2. Nos 1/2 © Textile Institute

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The Fundamentals of Pattem Structure. Part II: The Counter-change Challenge

pm'a'2

pm'm'2

pmm2

p'ma2

pmm'2*

p'mm2

Fig. 1 Continuation

Table IAn Enumeration of Colour Symmetry Tor Border Palterns, Using up to Eight Colours

'Jumber of Colours(Denoted by K)

Class pi 11Class plalClass pmll \ , ., •Class plmlClass pi 12Class pma2Class pmm2

Total number of colour clas.ses

K = l

1111111

7

K = 2

1123235

17

K = 3

1111111

7

K = 4 K - 5

11r-l

3237

19 7

K - 6

1123235

17

K - 7

I1I1111

7

K = 8

1123237

19

coloured (using two colours) in three distinct ways. Examples of the forty-sixpossibilities, based on the seventeen primary structures, are given in Fig. 2.

Twenty-three possibilities unfold when three colours are changed systematically, andninety six possibilities are achievable with four colours. Table II summarises thepossibilities for ail seventeen primary classes using up to four colours.

3. IN CONCLUSION

Following the pathway laid by Woods nearly seventy years ago. it has been seen that thesystematic colouring of pattems results in a finite number of combinations when using agiven number of colours. Woods's two-colour counter-change possibilities were revisited

./. Text. Inst., 2003, 94 Part 2. Nos 1/2 © Textile Institute

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Fig. 2 The seventeen primary and forty-six two-colour counter-change all-over pattern classes iContinuedopposite)

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The Fundamentals of Pattem Structure. Pan II: The Counter-change Challenge

c'm

clml

Fig. 2 Continuation (Continued overleaf)

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Fig. 2 Continuation (Continued opposite)

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The Fundamentals of Pattem Structure. Part II: The Counter-change Challenge

pm g

pm g

Fig. 2 Continuation (Continued overleaf)

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p2min c mm

^fc

pm m

Fig. 2 Continuation (Continued opposite)

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The Fundamentals of Pattem Structure. Part II: The Counter-change Challenge

Fig. 2 Continuation {Continued overleaf)

J. Text. Inst.. 2003. 94 Part 2. Nos 1/2 © Textile Institute 75

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Honii

p4gm

Fig. 2 Continuation (Continued oppi}site)

p4g'm'

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The Fundamentals of Pattern Structure. Part II: The Counter-change Challenge

p4mm

Fig. 2 Continuation (Continued overleaf)

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Hrnm

p6'mm'

Fig. 2 Continuation (Continued opposite)

?a / Text. Inst.. 2003. 94 Pan 2. Nos 1/2 © Textile Institute

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The Fundamentals of Pattern Structure. Part II: The Counter-change Challenge

p6m'm'

Fig. 2 Continuation

IIAn Enumeration of Colour Symmetry for All<over Patterns, Using up to Four Colours

Number of Colour K = l K = 2 K = 3 K = 4(Denoted by K)

Class plCla.ss plglClass plmlClass c 1 m IClass p211Class p2ggClass p2mgClass p2nimClass c2nimClass p3Class p3mlClass p3lmClass p4Class p4gmClass p4mmClass p6Class p6mm

all-over patternall-over patternall-over patternall-over patternall-over patternall-over patternall-over patternall-over patternalt-over patternall-over patternall-over patternall-over patternall-over patternall-over patternall-over patternall-over paltemall-over pattern

1253225550I1235I3

1222112,iI22200022

24

10734

11

1111157

1312

Total number of colour classes 17 46 23 96

and revised for both border patterns and all-over patterns. The paper presents originalcounter-change illustrations, conforming to the now familiar symmetry rules developedinitially by Woods (1935 a, b, c, 1936) and developed further by Schattschneider (1986),and Washburn and Crowe (1988). From the viewpoint of the visual artist or designer, anawareness of the hasie rules of symmetry can stimulate avenues of systematic visualdevelopment. Obvious areas for further research attention include: the systematiccolouring of three and higher colour effects; the interchange of textural qualities ratherthan simple flat colour: and the development of the relevant symmetry principles to aidthe classification and subsequent construction of two-sided structures such as double ortriple and other compound-woven fabrics.

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Hatm

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