Woven Rope Friezes Frank A

Santa Clara University Scholar Commons

Mathematics and Computer Science College of Arts & Sciences

2-1999 Woven Rope Friezes Frank A. Farris Santa Clara University, [email protected]

Nils Kristian Rossing

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Recommended Citation FARRIS, Frank A. with Nils Kristian Rossing. "Woven Rope Friezes."Mathematics Magazine, 72, No. 1 (1999): 32-38. (This cites a link to an MAA-hosted web site with a JAVA applet I wrote to permit readers to experiment with friezes.)

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WovenRope Friezes

FRANK A. FARRIS Santa Clara University Santa Clara, CA 95053

NILS KRISTIAN ROSSING SINTEFTelecom and Informatics N - 7046 Trondheim Norway

Introduction The rope mat shownin FIGURE 1 was wovenby Nils KristianRossing followingequations that produce frieze symmetry. To analyzethe pattern,think of the curvethe rope followsrather than the rope itselfand recognizethat, since rope knows more physicsthan mathematics,the rope does not quite follow the beautifully

FIGURE 1 A wovenrope matwith G2 symmetry symmetriccurve shown. (See FIGURE 2.) This curve, meant to be interpretedas continuingindefinitely in both directions,has translationalsymmetry and rotational symmetryin the formof half-turnsabout the pointsindicated, leading one to identify its symmetrygroup as the frieze group,generally called G2.

2-center

Basicmathematical Actualshape Severalbasic figures shape FIGURE 2

32 VOL. 72 NO. 1, FEBRUARY 1999 33 Here we presenta completeset of recipesshowing how to constructsmooth curves withany desired frieze symmetry; we provideexamples woven by Rossingfor many of the patterntypes, and invitereaders to make others. We reviewthe conceptof friezesymmetry, develop the formulasfor parametric equations with given symmetries,and pose some open questions raised by our analysis.We also providea Java applet to allow easy experimentation;this is now available on the World Wide Web along with high-resolutionphotographs of the ropes; please see http://math.scu.edu / -ffarris / frieze.html or http://www.maa.org/pubs/mathmag.html.

Frieze symmetry For us, a frieze(or friezepattern) will be a set of pointsin the plane invariantunder a translation,and hence an infinitecyclic group of translations. To relatethis concept to the wovenrope above,we idealize the path of the rope as a smoothcurve, treating places wherethe rope passes overitself as self-intersectionsof the curve.We recognizethat this treatment misses important features of the physical objectpictured, such as whetherthe rope goes overor underitself at crossings;these considerationsare expandedbelow and relegatedto futurediscussion. The set of all Euclidean motions,or isometries,that leave a friezeinvariant is necessarilya group,called the symmetrygroup of the frieze.We say thattwo friezes have the same symmetrytype if their groups are isomorphic.It is well-known(see, for instance,Cederberg [1]) thatthere are exactlyseven isomorphism classes of symmetry groupscontaining a single infinitecyclic group of translations.Without going into detail,we assemblea fewfacts about these groups. Suppose G is the invariantgroup of a frieze,which by definitioncontains an infinite cyclic subgroupof translations.Call the generatorof the translationgroup T, a translationof length K along a line 1, so that K is the smallestdistance one can translatethe patternand findthat it falls into coincidencewith itself.We findit convenientto thinkof I as the horizontaldirection. The line I maynot be uniquely definedif G has no elementsother than multiplesof T, in whichcase we call the groupG1, or if the onlyother elements are reflectionsabout linesperpendicular to 1, when the groupis called G4. Every other type of isometrypossible in a frieze group leaves invarianta line parallelto the directionof translation,which we call the axisof the frieze,and name 1. There are limitedpossibilities: The groupG2 containsa half-turnabout a pointof 1,as well as all the half-turnsgenerated by composingit withtranslations, but no other symmetries.We call the fixedpoint of a half-turna 2-centerof the frieze.G3 contains no half-turns,but a reflectionthrough 1, which we will call a horizontalreflection. G5 has half-turnsas well as the horizontalreflection; since these togethergenerate verticalreflections, this group has G2, G3, and G4 as subgroups. A glidereflection is the compositionof a reflectionand a translation.The groupG7 containsonly a glidereflection along I in additionto itstranslation subgroup. G6 has a glide reflectionand half-turns.Examples of friezeswith each typeof symmetryare shownin FIGURE 3.

Frieze curves: general theory The rope frieze above was constructedby first findinga continuouscurve in the plane, using a formulalike c(t) = (x(t), y(t)), -00 < t < 00. We suppose the curveis invariantunder translation along the x-axisof a distance K, and thatK is the smallestsuch distance.We assumethat the curveis parametrized consistentlywith the translationalinvariance, so that c(t + 2L) = ( x(t + 2L), y(t + 2L)) = ( x(t) + K, y(t)). 34 MATHEMATICS MAGAZINE

GI Translation

G2 Half-turn

.gH reflection G3 ~ XHorizolntal

G4 / Verticalreflection

G5 q Half-turn-alnd horizontal reflection- Half-turnand horizontal G6 Q6 / glide reflection

G7 >, Horizontal glide reflection

FIGURE 3

This can always be achieved by a reparametrization,using the equation above, recursively,to definea new parametrizationoutside a base interval[- L, L]. Note that we name the period 2 L for a convenientmatch with the period 2 IT in most of our examples. Beforecarrying out the Fourieranalysis for such a curve,we make some observa- tionsabout symmetry. There are severalnatural ways to decomposethe functionsx(t) and y(t). We definethe even and odd partsof thesetwo functionsas usual,but also identifyparts that we call the glide-positiveand glide-negativeparts:

x(t) = Xe(t) + = x(t) + x( -t) + x(t) -x(-t) xo(t) 2 + 2

= + + y( + y(t) -y( -t) Y(t) Ye(t) Yo(t) _y(t) 2 -t) + 2

x(t) =xg+(t) x(t) + x(t + L) x(t) -x(t + L) 9 +xg-(t)= ~~~~~2 +2

= + y(t) + y(t + L) y(t) -y(t + L) y(t) yg+(t) yg (t)= 2 ++2

It turnsout thatevely type of symmetrypossible forcurves of this typecan be achievedby settingone or moreparts of these decompositionsto zero. To elaborate, we name some particularEuclidean motionsof the plane and give theircoordinate formulas.

TA B LE 1. Someuseful symmetlies

Symbol Descliption Equation Ir translation r(x, y) = (x + K, y) p half-turnabout oligin p(x, y) = (-x, -y) OJV verticalmirror (JV(X,y) = (-x, y) O'l2 lholizontalmirror 21j(X)y) = (X, -y) glidereflection y(x,y) = (x + K, -Y) VOL. 72 NO. 1, FEBRUARY 1999 35 Note thatour decompositionsand choiceof symmetriesfavor the origin.We comment on thislater, but thisis done withoutloss of generality. The decompositionof x(t) and y(t) intoeven and odd partsmakes identification of G2 symmetryeasy: if the even part of each functionis zero, the curve c(t) will be symmetricabout the origin.Since it is also translationinvariant, it will have G2 symmetry. G4 symmetryis also easyto recognizein thisset-up. If the evenpart of x(t) and the odd partof y(t) both vanish,G4 symmetrywill result.We elaboratethis point with the equation ( x(-t), y(-0)) (-x(t), y(t)) = 0'( x(t), Y(0) G6 and G7 are also simplecases. For G7 simplyrequire that xg- and yg, both vanish.Then

(x(t + L), y(t + L)) = (x(t) + =:, -y(t)) = y(x(t), y(t)) G6 symmetryresults when the additionalconditions for G2 invarianceare imposed. A problemarises when we look for G3 symmetryby this method.The desired equationwould be: ( x ( -t), y(t 0) (x(t),x -y (t)) = 0-7(x (t), y(t)), and one mightexpect to satisfythis by choosingthe odd partof x(t) and the evenpart of y(t) to be zero. However,a simplecomputation shows that x(t) cannotbe an even function;this is inconsistentwith the translationequation. Intuitively,the curve cannotsimultaneously move periodically to the rightand appear equallyon bothsides of the horizontalaxis. We have solvedthis problem for woven ropes by usingtwo strands to makea frieze. To create a patternwith a horizontalmirror, we introducea second strandthat mirrorsthe first.This can be done in severalways; to achieve G3 symmetry,the originalcurve can have eitherG1 or G7 symmetry.Our classificationof symmetries by the shape of the paththe rope followsdoes notpermit a distinctionbetween these two types. We arriveat G5 realizingthat a pair of curveswill be necessaryto createa frieze witha horizontalmirror. We wouldlike the patterncreated by bothpaths together to have G5 symmetry,which means we need invarianceunder p and a-> Since p = uZ 0(h and uj,= 01,p, as long as we have one of these symmetries,we musthave the other. Since there are two strands,we may constructeither one to have one of these symmetriesand the compositepattern will have G5 symmetry.Thus G5 patternsmay be made frompairs of curveshaving G2, G4, or G6 symmetry. We summarizethese results in Table 2. It is naturalto ask whetherwe have leftsomething out. We have not considered everypossible pair of conditions.For instance,what if xg-0 and yg- O? One can

TAB L E 2. Syinmetiyrecipes for frieze curves Type Conditionson components min# strands GI no componentsvanish 1 G2 Xe-Ye--? 1 G3 GI or G7 conditionsin mirroredstrands 2 G4 nXe-O, yoO 1 Gs G2,G4, or G6 conditionsin mirroredstrands 2 G6 xe =Ye Xg YgY9 1 __ __ _ xg _=Yg + =0 ______36 MATHEMATICS MAGAZINE easilycheck that such a functionwould have period L, ratherthan 2 L. The smallest translationof such a patternwould have length2. If one wantedto constructfriezes of thatperiod, presumably one would have startedwith that condition from the start. Also, some of the componentscannot vanish. As noted above xo 0 contradictsthe translationequation. The same is truefor xg+.

Frieze curves: Fourier analysis As a firststep in the Fourieranalysis, note thata curve satisfyingthe equation of translationalsymmetry can be thoughtof as having one componentcausing it to move to the right,and another,more interesting componentcausing decorative twists and turns.We sortout the interestingpart by subtractingthe knownmotion to the right(at a rate of K unitsof distanceper 2 L unitsof time),and define:

m(t) = (x(t), y(t)) - ( ,0) = (h(t), v(t)), where h(t) and v(t) indicatethe horizontaland verticalcomponents of the decorative portionof c(t). Now m(t) is continuousand periodicof period 2 L, and so has a convergentFourier seriesof the form

m(tj = a . sin L ) nCos Lf cL Ecmsin( L t ) dmcos(L))

It becomes simple to translatethe requirementsabove into conditionson the coefficientsof the Fourier series. For example,x(t) and y(t) willboth be odd functionsif onlysine terms appear; the recipe for constructing G2 friezesis simply statedas bn= dm=0. Thefrieze in our first example has formula

c(t) = (13t sin(2t)-5 sin(3t)-5 sin(4t),sin(t) + 2 sin(2t)).

HereL = wrand K = 13.This was found by experimentation, oncewe knew that only sineterms could be included. Forsymmetry type G4, we see thatone needs sines in thehorizontal terms and cosinesin thevertical terms. The ropefrieze in FIGURE 4 wasdesigned from this

o.\ ~ ~ .

'~~~~~~~~~~~~~~~

FIGURE4 A wovenG4 frieze VOL. 72 NO. 1, FEBRUARY 1999 37 recipe,using only one termof each Fourierseries for simplicity. The equationis

c(t) = +cos(2t), -0.8 sin(t))

Types G6 and G7 are onlyslightly more tricky. One mustuse even values of n (the coefficientsin the horizontalterms) and odd values of m (thosein the verticalterms). Restrictingboth series to sine termsonly will add the symmetryof the half-turn, yieldinga friezecurve of typeG6. One thingthat may disturb the readerabout G6 patternsis thatthey have vertical mirrorswithout obeying the G4 recipes.The verticalmirror in G4 recipeswas set up throughthe origin,through our privilegedchoice of coordinates.The verticalmirror in the G6 patternsis about the line x= L2 FIGURE 5 showsa patternwoven from two ropes,with G6 symmetryin each strand, producingG5 symmetryoverall. One thingto note at thispoint is thata formulafor

FIGURE 5 A G6 patternwoven from two ropes the patternabove could be detectedby spectralanalysis, using softwarelike MAT- LAB. The equationsused to producea singlestrand are as follows:

x(t) = -3t - 1.5 sin(2t) -3.5 sin(4t);

y(t) = -0.3sin(t) -3.2sin(3t).

Table 3 givesa summaryof all theserecipes. We hope readerswill experimentand make friezesof theirown.

TAB LE 3. Symmetryrecipes for frieze curves

Type Strands Recipe and remarks G1 1 No additionalrequirements; use generalparity G2 1 sine termsonly G3 2 No additionalrequirements; use generalparity G4 1 sinesin x(t), cosinesin y(t), generalparity G5 2 G2, G4 or G6 requirementsin one strand,mirrored G6 1 n is even; m is odd; sinesonly G7 1 n is even; m is odd; bothsines and cosinesappear 38 MATHEMATICS MAGAZINE Frieze ropes: furtherdirections It maybe unsatisfyingto some readersthat the instructionsabove onlyshow how to constructcurves with desired frieze symmetries. Surelythe factthat these curvesare realizedphysically as woven ropes is the most attractivething about our illustrations.For empiricallyminded readers, we offera way to experimentwith these formulas in searchof theirown patterns to weave intoropes. For those interestedin theorywe expand a bit on the questionswe have not yet answered. To experimentmaking your own designs,you could use any computeralgebra system,such as Matheratica or DERIVE, but we found this a bit cumbersome. Farrishas writtena JAVAapplet allowingyou to tryout the formulas.An additional strandmirroring the firstis availableat a clickof the mouse.To use thisapplet, direct yourJava-enabled web browserto http: //math. edu/fffarris/ frieze.html or http://www.maa.org/pubs/mathmag.html. Probablythe mostinteresting theoretical question involves the differencebetween curvesand ropes. In all our exampleswe have simplyassumed that the rope crosses itselfin an alternatefashion, going over and thenunder every time. We call thisthe simplecoding pattern. It is certainlypossible to use morecomplex coding patterns, for instance,two over, one under, and so on. Such differentcoding patternscan be implementedvery nicely in the simplestpatterns, however for the friezesshown here it is difficultto attainhandsome results. On the otherhand, almost every pattern we have encounteredcan be wovenwith the simplecoding pattern. Experience teaches one to recognizepatterns for which the simple coding patternwill not work. Perhaps a theoreticalresult is possible: the classificationof curvesfor which the simplecoding pattern is suitable. There are furtherquestions about the relationshipbetween coding patternsand symmetry.In several examples,a half-turnhas the effectof negatingthe coding patternof the symmetry.A finerclassification of friezeropes mightuse the two-color friezegroups, outlined, for instance, in Grunbaumand Shephard[3]. Finally,the techniqueswe use here are certainlyadaptable for the constructionof curveswith wallpaper symmetry, at leastif infinitely many strands are used. We would be delightedto see examplesof wallpapersdiscovered using Fourier analysisand wovenfrom rope. How we came to write this note Our articlehad an interestinggenesis. Rossing, a Norwegianradio engineer,responded to Farris's MatheematicsMagazine article[2], sayingthat he had alreadywoven rotationally symmetric patterns like thosedescribed in the article.He had combinedFourier analysis with the old sailors'craft of weaving rosettesfrom rope, and foundmathematical expressions for manyexisting rosettes. The tools Rossingdeveloped had enabled him to producenew rosettesthat he then wove out of rope [4]. Farrisproposed that they carry out the analysisto findthe mostgeneral formula for patternsof each type of friezesymmetry, and see whetherthis would lead to the discoveryof interesting designs suitable for rope work. Once the formulaswere found, we experimentedand foundmany examples for Rossing to weave withrope. As we have not actuallymet in person,we findthis an Internetsuccess story.

REFERENCES

1. JudithN. Cederberg,A Course in ModernGeometries, Springer-Verlag, New York,NY, 1989. 2. FrankA. Farris,Wheels on wheelson wheels-surprisingsyinmetiy, this MAGAZINE 69 (1996), 185-189. 3. BrankoGrunbaum and G. C. Shephard,Tilings and Patterns,W. H. Freeman,New York,NY, 1986. 4. Nils Kr. Rossing,The old artof rope workand modernsignal processing, Knotting Matters, Netvsletter of the InternationalGuild of KnotTyers, December, 1996.