r-fold symmetry in algebraic curves

8/12/2019 R-fold symmetry in algebraic curves

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56ON r-FOLD SYMMETRY OF PLANE ALGEBRAIC CURVES.*

By R. D. CARMICHAEL, Princeton,N. J.

If a planecurve s revolved bouta point nits ownplanethroughnangleof 360?/rnd ifitthencoincideswith ts former osition,t is said tohave r-fold ymmetry ithrespect othepoint; nd the point s called thecenter f r-foldymmetry.The objectof thispaper s to ascertain hean-alyticalconditionswhichare necessary nd sufficiento the existence ofr-fold ymmetrynd toexamine ntothe geometric ropertiesf thecurvesin certain pecialcases. In a previousnotet have givena classificationfplanealgebraiccurveshavingfour-foldymmetrybout a point, nd thishas beenfollowed:by a paperonthegeometric ropertiesfquartic urvespossessing uchfour-foldymmetry.In thepresentdiscussionwe shallconfine urselves oplainalgebraicloci which re suchthatnolocus s composedntirelyf solated oints rofstraightines; notherwords, very ocusconsidered illbe assumed ohaveat leastone partwhich s continuousnd curved. And thisassumptionsmade throughout ithout urther tatement.Evidently hecircle s a curveof infinite-foldymmetry. t is clearthattheconditionf nfinite-foldymmetry ithrespect o theoriginsthatthepolar equationshall be independent f thevectorial ngle; that s, thelocus nthiscase is a circleora systemof concentric ircleswithcenteratthecenter f nfinite-foldymmetry.Thereforet willbe sufficientnwhatfollows o confine urattentiono the cases in whichr is finite.1. Separation ntotwo lasses. Let n be theorder fa curve fr-foldsymmetrynd let it be referred o rectangular artesian oordinates ithorigin t thecenter fr-fold ymmetry.Take theequation n theform(1) X~taztys - ,where t, is a real constantforevery and s andwheret and s each rangeover the values 0, 1, 2, ..., n subject to the condition

t+s -


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57x-pCOs0, y-psinO,we have v=n t, s-'t(2) Y )v Y ats cost 0 sinS 0-O, (t+s=7).

v=O t, s==OPutting -0 I and 0-a0,t- kwhere

4=3600/r nd a-an integer,we have thefollowing quations:

v==n t,s=V(3) Y-, v ats cost 0, sinS 01=0, (t+Sz=v),1)=0 t, s==Ov==n t, s==V(4) Y )v : atscost (01+a c) sins (9+a )-O (t+s-v).v=O t, s==O

Fromthe existenceof the defined -fold ymmetryt followsthatequations 3) and (4) mustyield by solution hesame valuesof P. There-fore hecoefficientsandiffer nlyby a constant actorma; that s,t; s-v t, s=V(5) - atsCOSt 1 sins9,--in at, cost (0, +a 0)sins 9 --a ), (t+s--Yv).t, =O t, - O

Equation (5) must holdfor each value of v from to n andforeach valueof a from to r, a differentquation beingformed or very ase. Then(5) yieldsr(n+l) equations which must all be satisfiedfor verypossiblevalueof0 . It is clear that the existence f thissystem f equations s bothnecessary nd sufficientothe existence fthedefined ymmetry.We shallnow evaluatethe constantsma. If we takea=1 and mul-tiplyequation (4) by ml (which evidently annotbe zero) it follows hatthe result s identicalwithequation 3). Hence a second ddition fq tothe vectorial angle would necessitate a second multiplication f thecoefficientsy m1;that s, two multiplicationsyml producesmin; r

2m2 ==Ml .

Continuinghe additions f 4 to the vectorial ngle,we have2 3


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58At the rth addition f4 to the vectorial ngle the two members f (5) be-come dentical xcept s to the presence f thefactorm, n the secondmem-ber; and thereforem,.mustequal 1. For,ifnot,we musthave

E atscost01 sins 01=? (t-+Fs-=),for n unlimited umber f values of 0, each less than2r; and this s evi-dently mpossible. Since mr mlr and mr 1, we havem1 1. It is evi-dentfrom 5) thatm1 s a real quantity. Therefore(6) ml=+1, whenr is odd;(7) m ?1, when r is even;(8) ma m1a, in every ase.

In order ofind henecessary nd sufficientelationsmongthe coef-ficients we proceed s follows. Forml- + 1, m] -1, equation 5) takestherespective orms:t, s=) t, =1(9) Y ats cost0i sins8l~ 3 ats cost(ol+a 4) sins(O1+a 4), (t+s-v), [A],t,s=0 t, s=Ot, S=) t, S=1)(10) Y atscostOi sins01=(-1()a Y atscost(0I+a4))sins(0, +a4), (t+s-v),B],t, s=O t, s=O

where v ranges over all the values 0, 1, 2, ..., n, different quations beingformed oreachvalueof v. Equation (9) aloneholdswhenr is odd; whenr is evenbothequations 9) and (10) mayhold. Evidently hese equationsare necessary nd sufficiento theexistence fr-fold ymmetry;hat s, forodd-foldymmetry e mustbe able to satisfy 9); foreven-fold ymmetry,either (9) or (10) or both. In case this condition annot be satisfiedforgivenr andn,we are to conclude hatr-fold ymmetryoes not existforcurvesof such degreen. As a case in point,we have the theorem:Four-foldymmetryoes notexistforcurvesofodd degree.We shallsay thatcurveswhich atisfy quations 9) and (10) are ofclassA andB, respectively. In classA therewillbe found oci of bothodd-and even-fold ymmetry;n class B will be found oci of only even-foldsymmetry.Obviously,ftheequationofa curvereferred orectangularxes hasonly erms fevendegreeor onlyterms f odd degree,the curvehas two-fold ymmetry;or n either ase, if a, I is a pointon the curve, o is -a,

*Annals of Mathematics,Vol. 9, No. 2, p. 55.


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59-A~. Conversely,f the originof rectangular oordinates s taken at thecenter f two-foldymmetry,heequationmust vidently ave one of theforms ndicated. If eachterm s of evendegree, t s obvious hat hecurvebelongs o class A; while f each term s of odddegree, he curvebelongs oclass B.2. Determinationfconstantst,for classA. Equation (9) indicatesthat therealfunction

t, s =-v(11) tF_ ats cost 0, sins 0, (t+s-v),t,8=Ois periodicwiththerealperiod#-2 /r. But everyrealfunction fa singlevariable01withthe real period ,/r anbe expanded na Fourier eries inthegeneralform

co co(12) > cicosir 01 > yj sin r 1.i=O i=lIf cos ir 0 and sin r 0i are expanded in termsof sin 01 and cos 01 the resultsare homogeneous f order r in sin0l, cos61; moreover,hecoefficientsiandyido not belong o terms like n sin01 and cos01, and thereforeannotannul each other. Hence, if the expression n (12) is to be identicalwithF, ir v. If ir=-v2j, thecorrespondingartof (12) whenexpanded ntermsof cos 01, sin 01 is of degree v/ 2j; but it becomes of degree v/hroughmultiplicationy theunitfactor cos20?+sin20' j. Evidentlyr cannotdif-ferfrom byan odd number. Hence,as a resultwe have

t, ==vii(13) > ats cost 01 sin- 0, ci cos ir 01-+ Eyi sinir 0,t, s=O i=O i=1

where +s v and ir is always positive nd has as its values someor all ofthepositivenumbers f theseriesv, v-2, v'-4,Now, if v rangesfrom ton,thepreceding esult nables us todeter-minereadilythe values of all of the coefficientst, in terms f a suitablenumber fthem elected s independentonstants. Substitutingheseval-ues in (1) we obtain he mostgeneralform ftheequation f thenthdegreelocuspossessing -fold ymmetryndbelonging o class A as definedbove.Suchequations, or everalvalues ofr andn, are written ut below ntheirmostgeneralform.*

*Forcurvesof four-foldymmetryee mypreviouspapers alreadyreferred o. Curvesoftwo-fold ymmet-ryare disposedof at the close of section1 of thispaper.


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60SOME CURVES OF CLASS A OF 3-FOLD SYMMETRY.

F2 =-c +c, (X2 +y2 ) 0.F3 -F2 +C3X +c4y3 -3c4x2y-3caxy2 ==0.F4- F3 c5 (X2 +y2) 2 -0.F5 F4 +c6x5 +c7Y5 -c7x4y-3c6xy4-2cx3y2 -2c7xy3 O.F6 F5 +C8 (X6 -y6) +6c9xy(x4+y4) -15c8x2y2 (X2-y2)-20c9, yl +C1 0 (X2 +y2) 3-0.Fq F6 + (x2+y2) (ClIx,5+c,-y,5-3cl2 X4y -2c, jx3y-2c, 2X2y3)-0.

SOME CURVES OF CLASS A OF 5-FOLD SYMMETRY.F2-c1+c2 (x2+-y2) 0.F4=-F2c3 (x2 y2)2 0.F5- F4 +c4x5 +c5y5 +5c5x'y+5c4xy4- 1Oc4x3y2-10c5x2y3-0.F6 -F5 +c (X2 +y2 ) I 0.F7 -F6 + (xI +y2) (C7X5 +C8Y5 +5C8x4y+5C7xy4

-1Oc7x3y2 -1OC8X2y30.SOME CURVES OF CLASS A OF 6-FOLD SYMMETRY.

F=-cj?+c (X2 +y2) -0.F4 -F2 +c3 (X2 +Hy2) 2 0.F F4 +c4 (X2 +y) 3+5 (x6 -y6) +6C6XY(X4+y4) -15cx2 (x2 -ye)

- 20c6xay' O.SOME CURVES OF CLASS A OF 7-FOLD SYMMETRY.

F2Cl +C? (X2 +y2)z_O.F4-F2 -1-c3x2+y2 20.F6 -F4+C4 (XI +y2) 3 =0.F7=-F,+c5x' +c6y7 -7C66xy-7c,Xy6 -21c5x5y2-21c6x2y5

+35C6x4y3 + 35c5x3y4=0.3. Determinationf the onstantst, or classB. For class B wehaveseenthatr is even. It maybe shown hat r-fold ymmetrynclassB is aspecialcase of er-foldymmetryn class A. For if #s theangle through

which her-fold r even) symmetricalurveof classB mustbe turned norder o coincidewith ts originalposition, # s the angle throughwhichthe-r-fold ymmetricalurveofclassA mustbe turned hat tmaycoincidewith tsoriginal osition. But the former tillcoincideswith ts first osi-


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61tion fterbeingturned hroughn angle of20. Hence, sincer mustbe ev-en forcurvesof classB, r-fold ymmetryf class B is a specialcase of -r-fold ymmetryf classA.For Ir-fold ymmetryquation 13) becomes

t,s=i iri(14) E at,cost61 sins -0 cos 61+ yisin r01,t, =o i= 2 i=1 2wheret+s=v and ir/2 is always positive nd has as its values someor allthe positivenumbers f theseriesv, -2, 1v 4, This s a necessary on-dition orr-fold ymmetryn class B. From(10) we maywrite

t, s=v t, s=v(15) 1 at. cost6, sin-, , -~ at. cost ( A'1+4)sine (6,+(), (t+s-v).t, =O t, =OSincetheexistenceof equation (10) is a necessary nd sufficientonditionforr-fold ymmetryf curvesof classB, it is readily eenfrom he discus-sion in the preceding paragraphthat the existence at the same timeofequations 14) and (15) is also a necessary nd sufficientonditionorr-fold ymmetryf curvesof class B. This resultenables one to determinetheconstants t8nterms f a suitable number f them hosen s indepen-dentconstants.But if r is twice an odd number, he constantsmaybe morereadilydeterminedn thefollowingmanner: In theequationforthe curveofclassA of er-foldymmetry,nsert he condition ortwo-foldymmetryf classB; that is, let the equationconsistonlyof termsof odd degree. In thiswaywere found heequationsfor6- and 10-foldymmetryivenbelow.

SOME CURVES OF CLASS B OF 6-FOLD SYMMETRY.F3=C1X3 +C2,y-3c2x2y-3c1xy2 O.F15 F3?c3x5+c4Y5 3c4x4y-3c3xy42c3x3y2 -2C4 X2y3O.F, -F5,+ (x2 +-y9) (C5X5 -+-C0y5 3c X4y--3C5xy4 -2C5X3y2 -2c6x2y3)

+ (X2+y2) 2 (C7X3 +C8y3-3C8X2y-3c7xy2) O.SOME CURVES OF CLASS B OF 10-FOLD SYMMETRY.

F5-C_ +2 +5c, 8+5c,X 1O1 -0 X2y -O.F7 -F5 + (x2 +y2) (C3X5+c4y5 +h5C4X4y+5c Xy4-10c3x3y2

-1Oc4x2y3) =0.F, =-F, (X2 +y2 ) 2 (C5X5 ?C6Y5+5C6X4Y+5c5cXy4 -1j05X3y2-10c6x2y3) --O.


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62A different ethod,however, s necessary orcurves f 8-fold ym-metry. In this case we must employ quations (14) and (15); or, whatis the same thing, corollary rommy firstpaper on four-fold ymmetry(alreadyreferred o) and equation 15): namely, henecessary nd sufficientcondition orfour-foldymmetryf curvesof class A is that every erm nthe equation hall be of even degreeand that

ast (-1)tats.(This result s not explicitly tated there, but is easily deduced s a corol-lary from he argument.) This enables us in thepresent ase to write 15)in differentorm. We replace4 by ts value 45?.

t, =v(16) $ [ats(cost01 sins 01+(-1)t coss 01 sint01)]t, s==Ot, s=v- , ats[cost(01 45?)sins(01 +45') + (1)t coss 0 +450)sint(01+450)] },t, =O

where t-]--s=v, beingan even number nd t s. It follows hatthe exis-tenceof equation (16) is the necessary nd sufficientondition or 8-foldsymmetricalurvesofclassB. By itsaid one may determineheequationsof8-fold ymmetricaloci.4. A simplificationn constructinghe quationsn general. Ifr p al p2G2 ... pka7,

wherePi, P2, ..., pkare differentrimes,we may evidently roceed s fol-lows toconstructheequations fnthdegree ocipossessing -fold ymmetry:Construct he equationsof nthdegree loci possessing symmetryfclass A and of orders laa, p2aa2, .., pkal, respectively. From heseconstructthemostgeneral equation n whichthe coefficientsbeyall the imitationsimposed nthe severalequations eparately. Theresult s the mostgeneralform ftheequationofclass A. Proceedsimilarlyorclass B.5. An example. As an illustrative xample onsider special case ofthe seventhdegreecurveofclassB of10-fold ymmetryn the table above.Let clC4~0, C2#0, C3#0. Then heequations of theform:(17)a1 5x4y-lOX2y3+y5) +a2 W +y ) (x51lOyl +5xy4) 0 a1#0, 2#0.Transforming o polar coordinates by the substitutionx-p cos 0, y--psin ?,and substitutingos5 0 and sin5 0forcos50 lOcos3I sin2O+5cos 0sin40 nd5cos4O sin 0-lOcos2O sin3O+sin50, respectively, the equation becomes


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63alp5sin 5 0+a2p7cos5 0=0.

Evidently his maybe replacedby the twoequationsp5 - ol(18) p2_=atan5?,where a--a1/a2.

To the formerf thesetwo equations orrespondsnly he origin. But thispoint s on the locus of the other quation. Hence, so far as plotting hecurve s concerned,quation (17) may be replacedbyequation 18). Evi-dently, tconsists f fivebranches, like exceptforposition. Each branchpasses throughheorigin nd has in itselftwo-fold ymmetry ithrespecttothe origin. Moreover,he origin s obviously point f nflectionor achbranch, nd there re thus five but onlyfive)points f inflectiont the or-igin. Now, since the curvepossesses10-fold ymmetry,ingularitiesot attheorigin an enter nlyby tens. Hence the number f points f nflectionis an oddmultiple f 5. It is easy to see that there s no cusp at theorigin.Hencecusps enter nly n tens, f at all.For the further iscussion f singularitieswe require thefollowingPlucker quations,which re writtenn theordinary otation:(19) m=n(n-1)-- (2 8+3 p),(20) nmm(m -1) - (2 -r+3z)(21) z=3n(n-2) - (6 3+8 p),(22) p-r3m(m-2) - (6 r+8 z).

We now have n 7, z-odd multiple f 5, P=multiple f 10,or zero.Then from 21) it may be seen that 6 8+l8pmustbe an evenmultiple f 5;that s, a multiple f10. But p is a multiple f 10, orzero; hence8 is a mul-tipleof 5. It is obviousfrom 20) thatm , 4; hencefrom 19) it followsthateither or p is zero; and therefore z=0, sincethe curveunder onsid-erationhas doublepoints t theorigin. Nowthe ocus s oftheseventhde-greeand cannothave as many s ten coincident oints; hence,since 4 is amultiple f 5, the number f doublepoints t the origin s 5. Therefore,from 19) it follows hat 8_-5or 15, since singularities ot at the originenteronly by tens. We shall now determinewhichof these is the truevalue.Supposethat there s a doublepointnot at the origin; nd let it be ata distanced from he origin. Then there must be ten such doublepointsat a distanced from heorigin. Pass throughhem circle with radiusd


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64and center t theorigin. Since eachdoublepoint ounts s two points thecircle uts the septiccurve n 20 points. But this s impossible, ence hereis no double point except at the origin. Therefore8=5. Then fromPltucker'squations:m=32, 2 75, r380. Hencethecurve

5x4y-JOX2y3+y5 +c(X2+yl) (x5-10x3y+5xy4)=O, Ce0.is of class 32, is non-cuspidal, nd has fivedoublepoints t the origin,75points finflectionfwhichfive re at theorigin, nd 380 doubletangents.It is obvious hat not all the singularities re real.

DEPARTMENTS.SOLUTIONS OF PROBLEMS.

ALGEBRA.330. ProposedbyR. D. CARMICHAEL, Princeton,N. J.

An importantunctionntheTheory f Numbers s one definedhus:f(x)=1 whenx >0, f(x) =O whenx=0, f(x) -1 whenx 1.n_ - ,,o. n-em X +J)n+ (X+1) -nIt is required o findothernon-trigonometricnalyticexpressions or thisfunction. (There are severalrepresentationsf f(x) by meansof trigono-metric unctions.)

No solution fthis problem as been received.331. Proposed byG. B. M. ZERR, A. M., Ph. D., Philadelphia, Pa.Extractthesquareroot of21+61/2+21//'21-61/'3-6V/7-21/6-21j/14and also of4i 24-21 6-9-4./3.Solution by S. G. BARTON, Ph. D., ClarksonSchool of Technology, otsdam,N. Y., and J. SCHEFFER, A.M.,Hagerstown,Md.(a) Assumethe root o be ofthe form

aS/p2+bi,/3+cw/7d-d.Squaring nd comparingoefficients,e have

r-fold symmetry in algebraic curves

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