Perception & Psychophysics1973, Vol. 14, No.3, 553-564

Some variation of the twisted cord illusion and their analyses* *THADDEUS M. COWANt

Kansas State University, Manhattan, Kansas 66506

Two approaches to the analysis of the twisted cord illusion are discussed. One suggests that the illusion is producedby embedded illusions (e.g., Zollner's illusion); the other points to nonillusion features such as the lines of the cordstrands. Anumber of novel variations of the illusion are presented, and four experiments using these variations aredescribed in an attempt to discern which of the two approaches was more viable. The implications of these results forvarious theories. including a multicomponent theory. are discussed.

Ever since Fraser discovered the twisted cord illusionin 1908, it has been regarded as something of a curiosity.Few have attempted any systematic study of the illusionsince Fraser: it has been kept alive mainly by itsinclusion in introductory texts as an enticement for thecuriosities of students. What may have been a nearrediscovery by Dufour (1930) and a paper by Hyde(1929) on Munsterberg's checkerboard illusion, a relatedphenomenon, are exceptions, but these were moredescriptive than analytical. More recently, Imai (1962)and Adam (1964) have reported effects which areessentially rediscoveries of Fraser's linear forms of hisillusion. Also some interesting variations of the twistedcord illusion have been constructed by Parola and someof his students (1969).

A form of the twisted cord illusion, one that closelyresembles Fraser's original figure, is shown in Fig. 1a.The twisted cord looks spiraled to most of thoseobserving it, but in fact it forms a set of concentric rings.(One can easily demonstrate this for himself by placing apencil tip on any ring and tracing it carefully. Eventuallyhe will return to the starting point.) The number ofdifferent spiral components in the figure is enough todiscourage even the most courageous, which is possiblywhy investigators have avoided this problem. Fraser,undaunted by these complexities, drew severalconclusions regarding the illusion, some of which mayeventually prove to be as insightful as they were bold.These will be introduced and discussed here, thenreferred to periodically throughout this paper.

At the close of his paper, Fraser drew the followingconclusions regarding the illusion:

(a) There is no suggestion ofperspective in the twistedcord figure which some writers have used to explain avariety of other illusions. This is a puzzling statement.Moving from outer to inner edge, the diminishingdistances between the rings and the shrinking widths ofthe rings themselves provide strong distance cues. One

*1 am indebted to the referees of this paper for a number ofsuggestions concerning related studies and possible approaches tothe data. They should not be held responsible for errors ofinterpretation, however.

r Requests for reprints should be sent to Thaddeus M, Cowan.Department of Psychology. Kansas State University. Manhattan,Kansas 66506.

553

can easily imagine that he is looking down a tube inFig. 1a. Yet Fraser's implication that distance cuescannot explain the illusion was substantially correct forreasons other than those he suggested. The presence ofdistance cues in Fig. 1a merely transforms the apparentspiral to an apparent helix. but in no way can thedistance cues be used to explain the illusion. In thecourse of constructing a number of these figures, severalwere made in which the twisted cord rings were equallywide and placed apart by equal distances. The illusionwas not destroyed or greatly weakened. One of thesefigures is shown in Fig. 2.1

(b) /n whatever way the illusion is related to Zollner'sillusion, it is different in one important aspect, and thatis, Zollner's illusion is lost with steady fixation, whereasthe twisted cord illusion is modified but never abolished.Fraser was referring to the lay of the "units ofdirection" across the circular lines of the twisted cord.These units of direction are the "strands" of the cord,and one with its points intact is shown in the inset in thelower right-hand corner of Fig. 1a. Zollner's illusion,shown in Fig. 3a, produces an apparent deviation of linesaway from the angle of the inducing lines. In Fraser'sfigure, however, the lines apparently bend in thedirection of the inducing lines-an effect which, as Imai(1962) and Adam (1964) pointed out, is completelycontrary to the Zollner effect. Fraser, in his twisted cordarticle, demonstrated the illusory effects of the linearforms of these slanting lines 'by the LIFE figuresreproduced in Fig.3b. The illusion in Fig.3b will bespoken of in this paper as the reversed Zollner illusion.Although Adam mentions the twisted cord illusion. shedid not explicitly deal with the curved form of it (only alinear form related to those found in the LIFE figure). Itis likely that she (and Imai) would have evoked thereversed Zollner illusion to explain the twisted cordeffect. The exact way in which this illusion mightproduce the twisted cord effect will be described indetail later in this paper.

Getting back, to Fraser's main point, the apparent lackof satiation is not necessarily linked to the fact thatthere may be a Zollner illusion embedded in his figure.Furthermore, there may be satiation effects, but theyrequire extensive fixation times. This possibility is based

554 COWAN

Fig. 1. Variations of the secondaryspirals. (a) Fraser's original figure. (b) Thestripe spirals tum through approximately+90 deg (+ means in the direction of theillusion) and the stepped spirals tumthrough about +270 deg. (c) Stripe spirals:odeg. Stepped spirals: +180 deg. (d) Stripespirals: -90 deg. Stepped spirals: +90 deg.(e) Stripe spirals: -180 deg, Stepped spirals:odeg, (f) Stripe spirals: -270 deg. Steppedspirals: -90 deg.

on my own phenomenological observations, however.After spending several hours constructing a number ofclockwise forms of this illusion, I found that in somecases the illusion all but disappeared (Fig . 1b and Fig. 2in particular). Yet the counterclockwise versions of thesewere quite overwhelming-even to the point of being farstronger than my initial perceptions (as I rememberthem).

(c) Eye movements are eliminated as a factor, sincethe illusion persists at a 1/8- to 1/2 -secflash exposure. Itis not clear whether Fraser took precautions to eliminatean afterimage, but given that he did, 125 msec (a) eventhough it eliminates eye movements, would provide .enough time for an efferent command to be readied(Festinger, Ono , Burnham, & Bamber , 1967), and (b) is

Fig. 2. Variation of the twisted cord illusion where the cordwidths are constant as are the distances between the cords. Theapparent depth is reduced.

long enough to establish a strong iconic image whichmight be scanned internally (Sperling, 1960). Even ifthere is no internal scanning process, exposures shorterthan 125 msec theoretically may cause certain featuresof the display to be lost while retaining others(Rumrnelhart , 1970). Thus , short exposures mayproduce a change in the saliency of the differentcharacteristics of the figure even though eye movementsare eliminated.

(d) There are two integrative processes: one is theconfluent union of units across the stippled areas inFig. La, and the other involves the collective trends ofthe units of direction . Fraser further surmised that thefirst of these integrative processes is at work in thefoveal region and that the second of these prevails at theperiphery of vision. The neural architecture of the retinaargues against the implication that different retinalprocesses are at work, since one would expect a"confluent union" of two adjacent dark areas to occurwhere the resolving power of the retina is weakest , viz,the periphery. However, the lines in the figure itselfbecome fmer in its central parts . Thus , "confluentunion" may take place in the center of the figure asFraser suggested, not so much because of the structureof the retina, but rather because of the characteristics ofthe physical stimulus.

A word is in order about Fraser's use of the term"confluence." Fraser adhered Closely to the originalmeaning of the word proposed by Mueller-Lyer , whosuggested that distances between closely spaced stimuliappear diminished (Hochberg, 1971) . This is similar tothe Gestalt notion of "closure." As for the term"collective trend," Fraser did not explain what thisentailed. This term will appear later in this paper, atwhich time it will be given more explicit meaning.

Two different hypotheses concerning the illusionemerge upon reflecting on Fraser 's four points. The firstis that the twisted cord effect is influenced by thepresence of simpler illusions embedded in it. Thishypothesis will be called the illusion component

VARIA nONS OF THE TWISTED CORD ILLUSION 555

Fig. 3. (a) Z<iUner's illusion. (b) Fraser'sversion of the reversed Zonner illusion. [Theslashes are angled 11 deg (top) and IS deg(bottom) from the line.]

hypothesis. The other hypothesis, the one favored byFraser, stresses the importance of the units of directionas independent components without reference to theirconnection with simpler illusions. The suggestion thatthese units are independent suggests such processes aseye movements or internal scanning. Although Fraserdid not mention internal scanning, his use of words like"direction" and "trend" implies that he might havereadily accepted such a construct had it been suggestedto him. This hypothesis will be called the unit ofdirection hypothesis.

The remainder of this paper will be spent in discussingthree variants of Fraser's original figure and theirconsequences for the several theoretical statementsdiscussed above. There will be no guarantee that theissue regarding the two general hypotheses referred toabove will be resolved. If there is to be a stated purposeto this paper, it is this: A number of variations of thetwisted cord illusion are possible, and in the course ofdealing with these variations, there may occur someinteresting (and important) consequences for theory.

FIRST VARIANT:DISTORTING THE BACKGROUND SPIRALS

The points in Fraser's units of direction, inconjunction with the stippled area, form the basis of twoantagonistic sets of background spirals that may possiblyinfluence the effect of the illusion. These spirals arequite evident in Fig. 1a. In order that these secondaryspirals be brought into clearer focus, the stippled areaswere filled in (the confluent union was provided by E)and one set of spirals eliminated. The resultingconstruction is reproduced in Fig. 1b. Since thesecondary spirals are the basis of the alternating blackand white background stripes, they will be called the"stripe spirals." There is another set of spirals apparentin Fig. 1b. These spirals will be called "stepped spirals"and are found by tracing any black (of white) areaclockwise from the edge to the center. Another

interesting (and aesthetic) feature of these constructionsis that the stepped spirals have perfect figure-groundsymmetry.

The effects of these background spirals were testedempirically by first distorting them so that they ranagainst the apparent spiral of the twisted cord, thenasking Ss if the cord still appeared to spiral. Thesedistortions are given in the constructions shown in Figs.Ib-I f. The point of zero curvature for the stripe spirals isgiven in Fig. l c, and the stepped spirals of zero curvatureare produced in Fig. 1e. In Fig. 1b , both spiral formstwist in the same direction as the illusion spiral, and inFig. If, both types of secondary spirals run counter tothe illusion. The degree of spiral ness of thesebackground components is given in the figure legend.Their spiralness was measured by the number of degreesthey crossed as they turned from the outer edge to theinner edge. Since it will be useful to speak of thesespirals in terms of their direction, any spiral line thatleads inwardly as it is traced clockwise will be called aclockwise spiral. Counterclockwise spirals arecorrespondingly defined,

EXPERIMENT I

Method

Subjects

Ten undergraduate students, both male and female, with atleast 20(30 corrected vision were used. Each S was runindividually.

Stimuli

The five constructions shown in Figs. Ib-l f, their mirrorimages. and a single black eight-armed spiral were drawn onindividual cards. They were approximately 15 ern in diam.

Procedure

Each S was shown the spirals at a normal reading distance.The first. the black eight-armed spiral, was used to explain what

556 COWAN

Fig. 4. Variation of the twisted cord illusion where the stripespirals are linear and tangent to the inner edge.

was meant by "clockwise" and "counterclockwise" spirals. The5 was then told that some figures would be presented whichshowed a black and white stranded twisted cord superimposed­on some background lines that radiated from the center and thathe would be asked to make a judgment concerning the clockwiseor counterclockwise direction of the twi sted cord. The twistedcord s in Figs. l e and If tend to be hidden in the confusion ofthe background lines, and since the cords are difficult to seehere, one of the other three figures (or its mirror image) waspresented first . In spite of this precau tion, some Ss found itdifficult to see the cords in Figs . l e and If, so the followingprocedure was employed for all constructions. First, E made sure5 saw the cord in Fig. Lb, l c, or l d . Then E placed a pencil tipdirectly over the first cord ring in from the ou tside edge andtraced it not more than halfway around the ctrcumference. ThenE told 5, "I -am going to trace the cord around with my penciltip , and I want you to tell me if the cord I'm following isspiraling clockwise or counterclockwise, or if my pencil hasstarted to .spiral inwardly or outwardly." The movements of thepencil were slow and proceeded in a clockwise direction for allFig . I constructions and their mirror images. The five Fig. Iconstructions and their mirror images were presented to 5 inrandom order.

Results

Of the lOSs, all but one produced responsesconsistent with the illusion in all the constructionsshown in Fig. I and their mirror images. The probabilityof these results occurring by chance according to abinomial test is .011 . The one aberrant S producedresponses that can best be described as random. All ofthe figures were presented to her once again, and she wasasked whether she saw circles or spirals in each. Shereported circles in every case . The reasons for thisunusual behavior are not known , and an attempt toexplain it will not be pursued here .

Discussion

Confluent Union

Fraser suggested that at small visual angles, aconfluent union would occur across the points betweenthe units of direction located in successive rings. Thisconfluence, supplied by E in Figs. Ib-I f, created boththe stepped spirals and the 'stripe' spirals seen in theconstructions. It is evident that the direction of thestripe spirals was not solely responsible for the illusion'seffect , since these spirals turn against the twisted cord inFigs. Ib-Id and the illusion was not destroyed. The samecan be said of the stepped spirals; they becomeprogressively steeper from Fig.lb to Fig. l d , and theillusion did not vary extensively. More critically, theillusion was still intact in Figs. I e and I f, once the cordwas teased out perceptually. Here the turn of thestepped spirals reaches zero and even runs counter to theillusion.

Illusion Component Hypothesis

The possibility of the twisted cord effect beingdetermined by simpler illusion components wasdiscussed earlier with particular reference given to thelay of the strands relative to the cord line . Figure Icsuggests another simple illusion effect created by the layof the strands across the radial lines. Since the strandscross the radial lines at larger angles (closer to 90 deg),the situation here is conducive to Zollner's illusion (seeFig. 3a) . If the slashing lines produce an apparentopposite tilting of the radial lines , then the radials wouldappear off-center-perhaps tangent to some smaller circledrawn around the center point. Imagine that the centerpoint strongly fixes the direction of the radial lines, thuscreating a conflict between this fixation and the tiltproduced by the Zollner effect. Might this be resolvedby an apparent spiraling of the cord? Figures Ib-I e argueagainst this suggestion, since the radial lines in theseconstructions were varied past the point of theperpendicular, thereby eliminating the Zollner effect,and yet the twisted cord illusion persisted. Furthermore,a construction was made in which the radial lines werelinear , somewhat perpendicular to the units of direction,and tangent to the inner edge. The illusion was stillintact in this construction (see Fig: 4).

Conclusions

The variations in Fig. 1 suggest that a Zollner-typeillusion imposed on the radial lines of the twisted cordfigure either does not exist or, if it does exist , has littleor no effect on the twisted cord illusion. The process ofconfluent union as manifested in either the stripe orstepped spirals also plays a minor part, if any, in theperception of the illusion . There remains the effect ofthe units themselves. These may produce the illusion

VARIATIONS OF THE TWISTED CORD ILLUSION 557

· Fig. 5. (a) A reconstruction of Fig. Icwith the radial lines perpendicular to thecord lines. (The stepped lines are spiral ; thecord lines are circular.) (b) Half exchange ofthe circle and spiral lines of Fig. Ic, (Boththe stepped lines and cord lines arespiraled.) (c) FuD exchange of the circlesand spiral lines of Fig. Ic , (The stepped Jinesare circular and the cord lines are spiraled.)

either by their direction (" coUective trends," as Fraserput it) or by the simpler illusion component producedby these units and their relationship with the cord rings(the reversed Zollner illusion) . These constructionsprovided no test for either of these hypotheses.

SECOND VARIANT:CIRCLE AND SPIRAL EXCHANGE

In Fig.5a (which is essentially the same as Fig. Ic),the stepped line is spiraled in a clockwise direction andthe twisted cord describes a circular line . Thus, there isat least one circle line and one spiral line . (The terms"circle line" and "spiral line" should not be takenliterally , i.e., indicating a continuous and completed line.Rather, these terms are used here to describe thesuggestions of circles or spirals given by line segments orsequences of arcs.) In Fig. 5c , the situation is reversed ;the twisted cord is spiraled in a clockwise direction andthe stepped line is circular. Again, there is at least onecircle line and one spiral line . In Fig. 5b, the stepped linespirals counterclockwise and the twisted cord spiralsclockwise. There are no circle lines in this figure in thesense that there are circle lines in Figs. 5a and Sc .However, in this construction (5b), the units of directionare congruent to the circle lines rather than cuttingacross them. In Figs. Sa and 5c, the units cut across thecircumferences and follow what might be regarded as aspiral path to the center. These characteristics of thethree figures are summarized in the first three columns

of Table 1. Note that each figure has one circular lineand two spiral lines associated with it.

In an informal query , several colleagues were shownthe three constructions in Fig. 5 and asked to picture theradiating stripes as appearing on the inside of a tube withthe twisted cord placed against the sides like rifling in agun barrel. When asked to judge the shape and directionof the cord in Fig. Sa, there was no disagreement thatthe cord appeared spiraled in a clockwise direction.Similarly, Fig. 5c was unequivocally judged to bespiraled clockwise. The judgment of Fig.5b, on theother hand, was a different matter. The responses weretypically ones of perplexity and confusion. Some saidthe cord appeared circular; others said it spiraled, andwhen asked to trace it , they invariably selected thestepped spiral and traced it counterclockwise. When toldthat they were not tracing the twisted cord , theirbehavior degenerated into absolute frustration . Thosewho reported circles in Fig. 5b , when asked to trace acircle, also became agitated upon fmding no circle totrace, just as they were perplexed when they could fmdno spirals in the cord lines of Fig. Sa. When asked tocompare Fig. Sa and Fig. 5b , all selected Fig. 5b as beingthe more circular of the two .

When Figs. Sb and 5c were reduced in size to less than5 deg of visual angle, another characteristic not presentin the Fig. 1 constructions became evident ; perception atthis visual angle appeared to be determined by thestepped spiral . That is, Fig. 5c appeared to be a set ofnested circles. and Fig. Sb looked spiraled in a

558 COWAN

Table 1Summary of the Line Characteristics for Figures Sa, b, and c and the General Results of Experiment U

Direction of Line Relative to CenterBend of UnitsRelative to:

Experiment IIPerception at:

Cr -cr indicates the presence of a questionable limen. See text for numerical results.

Fig. SaFig.5bFig.5c

CordLines

Stepped UnitLines Lines

Spirals SpiralsSpirals CirclesCircles Spirals

Radials(Zollner)

Cords(ReversedZollner) Limen*

Sp e- CrCr-CrSp --+ Cr

Supra­limen

SpiralsCirclesSpirals

counterclockwise direction. The predominance of thestepped line also manifested itself in the afterimages ofthe figures. Fraser reported that the illusion was presentin the afterimage of his figure, and indeed the afterimageof Fig. Sa was strongly spiraled, However, the afterimageof Fig. Sb was also spiraled, but in a counterclockwisedirection. The afterimage of Fig. 5c was clearly circular.In spite of the evidence from Fig. 1 to the contrary, itlooked as if Fraser's hypothesis concerning confluence,and the role the foveal area plays in its determination,had been temporarily revived. Stimulation of the morecentral portions of the retina seemed at least to force achange in the saliency of different features of the figure.

This change in saliency at different visual angles, andthe ambiguity surrounding the judgment of Fig. Sb,warranted further investigation. It was felt that certainfeatures of the display would be lost, or at least theirsaliency would change, under presentations of shortduration and that this might prove to be informative.For example, suppose Fig. Sc appeared spiraled abovesome duration threshold and circular below it. Thismight suggest that the twisted cord is most salient atlong durations and the stepped line is most salient atshorter ones. It would follow that both Fig. Sa andFig. Sb, when presented at durations below thethreshold of Fig. Sc, should look spiraled, since thestepped line is spiraled in both of these constructions.Furthermore, if the twisted cord is predominant atdurations above the threshold found for Fig. Sc, Fig. Sbshould at no time appear circular, since the cord line isalso spiraled in this figure.

EXPERIMENT II

Method

Subjects

Twelve male and female undergraduate students with 20/30corrected vision and no evidence of lateral phoria were used.Each S was run separately.

Apparatus and Stimuli

A two-channel tachistoscope was used. The "home" channelcontained a display of black and white stripes radiating from thecenter of the field. These stripes were identical to those in Figs.Sa, b, and c, except the twisted cord was missing. Each of thethree constructions was positioned so that, during exposure, the

radiating stripes exactly matched those seen in the homechannel. The perceptual effect here was one of the continualpresence of the radiating stripes with the twisted cord figurebriefly superimposed on them during exposure. The figuressubtended a visual angle well above 5 deg.

Procedure

A multiple staircase procedure was used with S judging allthree figures. The presentations were limited to the followingdurations (in milliseconds): 1,000, 500, 250, 100, 90, 80,70,60, 50, 40, 30, and 20. Since each figure was started at bothends of this range, six staircase functions were plotted for each5, i.e., one ascending and one descending for each of the threeconstructions. The six staircase conditions were given in blocksof six trials, bu t they were random within that block. Each 5 wasasked to report either circles or spirals. Their instructions read asfollows: "If you look into the eyepiece you will see a set of linesconverging on a point in the center. This is almost like lookingdown a tube with black and white stripes painted on the inside. Iwill superimpose one of three sets of lines on the ones you seehere. Sometimes these will look like nested circles, sometimeslike spirals. It will be as if there was rifling in the tube or a set ofrings placed at equal distances down the tube. Even the same setof lines may look like circles or spirals on different occasions. Iwant you to tell me whether you see circles or spirals. Some ofthese will be shown very quickly, and if you find it difficult togive an answer, give me your general impression of what yousaw. Also, since the presentations will be brief, I wiII give you awarning signal just before the image is flashed." Half the Ss weregiven longer durations on the next presentation if they reportedseeing circles and shorter durations if they saw spirals. This wasreversed for the other half of the Ss. If the ascending anddescending staircase functions for the same figure diverged fromthe upper and lower limits during the first trials, the responsecriteria were reversed so that the two functions could converge.If the ascending and descending functions converged at the rangelimit, that duration was repeated for five trials and dropped if noreversal of the functions was found.

Results and Discussion

The point at which the two functions for a givenfigure first met was taken as the threshold for thatfigure. For example, for one S, Fig. Sc was seen asspiraled at 1,000 msec and circular at 20 msec. The twofunctions met at 35 msec, and at this exposure duration,S oscillated between a report of spirals and circles. Inaddition to the summary of the various characteristicsof, and relationships between, the cords, units, andstepped spirals, Table 1 includes the results ofExperiment II in nonquantitative terms.

VARIATIONS OF THE TWISTED CORD ILLUSION 559

Unit of Direction Hypothesis

Of first importance is the fact that there was completeagreement among the Ss regarding the suprathresholdperceptions. Comparing Ss' suprathreshold responseswith the other columns in Table 1, it can be seen thatthere was perfect agreement between Ss' perceptions ofthese figures and the lay of the units relative to thefigures' centers. The Ss' suprathreshold perceptions ofthe three figures showed no such agreement with the layof the cord or stepped lines. If Fraser's "unit trend"description can be interpreted in terms of the lay of the .units relative to the center, then the suprathresholdresponses reinforce the importance of these trends.

Column 6 in. Table 1 gives the Ss' change inperception at threshold. At some point above a 20-msecexposure duration, 58% of the Ss changed theirperception of Fig. Sa from spirals to circles and, in thecase of Fig. 5c, s8'7c switched their reports from spiralsto circles. However, only 17% of the Ss saw Fig.5bchange from circles to spirals at durations greater than20 msec. These are significant variations according toCochran's Q test (X2(2) = 6.25).

These data tell us that the constructions in Figs. Saand 5c have thresholds at which certain features becomemore salient, whereas the existence of such a thresholdfor Fig. Sb is perhaps questionable. What is importantfor our purposes here is that the perception does notchange consistently from stepped line to cord line, nordoes it change from cord line to stepped line across allfigures. Therefore, it is unlikely that the stepped line (orthe cord line) plays a similar role across all three figures.

One thing the threshold data do not tell us is exactlywhat is happening to perception at threshold. A hintcomes from the results of the informal survey mentionedearlier, that the perception of Fig. Sb was not all thatclear. One possibility is that the subjective contours ofthe cord are lost or, at best, difficult to discern inFig. 5b. These contours are easier to see in Fig. Sa (and5c); indeed, it is precisely these contours that give rise tothe perception of the cord (but not their direction).Perhaps what is being measured in Experiment II is thepresence and absence of subjective contours. This mighthelp explain the fact that little evidence was found for athreshold in Fig. 5b, since there are no subjectivecontours there to be lost.

We are further cautioned by the fact that the steppedline (confluence between units) very obviouslydetermines the perception of these figures at small visualangles. The arguments above apply only to largerprojections of these figures on the retina, and it isapparent that two very different and independentprocesses are at work at large and small visual angles.

Illusion Component Hypothesis

In Figs. Sa and 5c, the units or strands of the cordtracing them clock wise. J: c slanted inwardly relative to

the radial lines, but they are perpendicular to the radialsin Fig. 5b. This was also correlated with Ss' perceptions.Recall that the relationship of the units to the radiallines tends to create a Zollner effect. At the same time,the lay of the units relative to the radials defines the layof these units relative to the center, and this fits thedefinition of "collective trend" used here (see Columns3 and 4 of Table 1). Since Fig. 1 and Fig. 2 providedevidence against the influence of the Zollner effect here,the importance of the Zollner effect as an explanatoryprinciple will be rejected in deference to the unit trends.

In the reversed Zollner illusion, the line of theinducing slashes leans in the direction of the slashes (seeFig. 3b). In Fig. Sa, the strands of the cord, tracing themclockwise, slant inwardly relative to the cord lines. Thus,an apparent inward bending of the line should beproduced, and since the line is circular, this apparentbending perhaps produces the spiral effect. But whatabout Figs. Sb and 5c? Here the units of direction pointoutwardly (again tracing them clockwise) from the spirallines of the cord, thereby bending the spirals outward.perhaps into circles. If the reversed Zollner illusion isresponsible for this effect, would not the cord lines inboth Figs. Sc and Sb appear circular? Perhaps not. Thecord lines in Fig. Sc spiral more steeply (lead to thecenter at a faster rate) than those in Fig. 5b. The cordlines in Fig. 5c may be too steep, whereas theshallowness of the cord lines in Fig. Sb may be preciselyright for the illusion to counteract it and produce theapparent circles-an effect which, if true, would besurprisingly coincidental.

Conclusions

Again some question has been raised regarding therole of confluence as seen in the stepped spirals inproducing the perception of this illusion. Yet thevariations in Figs. Sa, b, and c suggest that the steppedlines control perception under conditions of small visualangle (foveal projection) and afterimages. On the otherhand. the trends of the units of direction were consistentwith the perception of larger representations of thesefigures. implying that this form of the unit of directionhypothesis is still viable. Furthermore, these figurespoint to an interesting interpretation of the term "unittrends." viz, that the trends of the units define stillanother set of spiral lines in the twisted cord figure.

Figures Sb and Sc, while raising some questions aboutthe efficacy of the reversed Zollner illusion, did notprovi de a crucial test of it. Thus, we are again left withtw o possible critical characteristics of the twisted cardfigure. the reversed Zollner illusion component and thetrends of the units of direction.

THIRD VARIANT:SEPARATION OF ALL UNITS OF DIRECTION

One way in which the effects of the units bythen does might be examined would be to remove the

560 COWAN

Fig. 6. The demonstration that theillusion spiral is the same as the units ofdirection spiral lines. (a) A section of thetwisted cord figure with only the black andwhite units of direction. The lined areaswere transparent. (b) A completed versionof the 6a transparency superimposed on theradial stripes producing the twisted cordeffect. (c and d) Clockwise rotation of thetransparency in Fig. 6b through the distanceof a single stripe width.

background lines (the stripe spirals) completely, leavingonly the individual units displayed . The resulting figurespirals weakly , which poses some problems since it hasbeen suggested throughout this paper that thebackground spirals do not have a profound effect on theperception of the illusion. The observation made earliertha t the constructions in Fig. 1 have perfectfigure-ground symmetry provides some help here.Because figure-ground symmetry exists , there are whiteas well as black units of direction, and the removal ofthe black stripe spirals effectively eliminates these whiteunits of direction. Therefore, in order to studyappropriately the effects of the units of direction, bothwhite and black units must be kept intact.

Figure 6 demonstrates the way in which these unitsand their trends were studied by parceling the black andwhite units intact and then gradually phasing out thebackground lines . The type of spiral shown in Fig. 1cwas used because the linear background lines simplifiedthe construction. The black and white units of directionwere painted on a transparency , leaving clear placesrepresented by the lined areas in Fig. 6a. Thistransparency was laid over a set of black and white radialstripes whose widths matched the widths of theoverlapping segments of the units . The rotation of thetransparency through the width of a single radiatingstripe produced a continuous change in pattern goingfrom Fig. 6b through 6c to 6d . The effect was one of

doubling the number of spirals seen, but the degree ofspiral turn seemed to remain constant.

This suggested a rather unusual interpretation of thetwisted cord effect. First, Fraser's trends of the unitsmay indeed be critical, since the slant of the units seemto lie along spiral lines. More importantly, these spirallines may be the very ones seen in the illusion. We tendto say that the twisted cord in Fig. 6b appears spiraled,but that the spirals really do not exist. Thus, we call thespirals "illusions." In point of fact , the spirals are realand, given the presence of subjective contours, the cordis real. The illusion is seen as the real cords lying alongreal spiral lines instead of lying along nonexistent spirallines or their more veridical circular lines.

The following experiment was conducted to see if anysubstantial difference in the degree of spiralness existedbetween the spirals in Figs. 6b and 6d. One of thedifficulties encountered here involved the possiblechange in the spiralness of Fig. 6d due to the increasednumber of spirals rather than to the steepness of thespiral arms. Therefore , Fig.6d was not used in thisexperiment.

EXPERIMENT III

Method

Subjects

Ten male and female undergraduate students having at least

VARIATIONS OF THE TWISTED CORD ILLUSION 56 \

Fig. 7. (a) Control figure to replaceFig.6d. (b) 40S-deg spiral. (c) 4SQ-degspiral. [The differences between (b) and (c)are perceptually greater than the judgeddifferences between Figs. 6b and 7a.]

20/30 correc ted vision were run individually.

Stimuli

The construc tion in Fig. 6b (without the extended radialstripes) was used as the illusion spiral, and the eight-armed spiralshown in Fig. 7a was used as a control spiral. The degree ofspiralness of Fig. 7a was exactly the same as that of Fig. 6d . Nineeight-armed black spirals on a white background served ascomparison spirals. The two extremes of the nine eight-armedspirals turned thr ough 225 and 585 deg ; the remaining sevenwere equally spaced between these extremes followingincrements of 45 deg. The mirror images of these spirals werealso used . All spiral s were about 15 em in diam and were viewedat nonnal reading distance.

Procedure

The 225-deg spiral was called "steep" in this experiment, andthe 585 -deg spiral was referred to as " shallow." After assurancesthat S knew what was meant by steep and shallow spirals , he wasgiven the following instructions. "Y ou will be shown pairs ofspirals like th is [S was shown the 225-deg comparison spiral witheither the spiral shown in Fig. 6b or Fig. 7aJ . I want you to tellme if this spiral [E pointed to the comparison) is steeper orshallower than the other one. Don't be too analytical in makingyour judgment. Just give me your impression of the relativesteepness and shallowness of the two figures." The E then pairedthe 225- and 58S-deg comparison spirals with the two standardspirals (making four comparisons) and asked for S's judgments tomake sure S understood the task. A random multiple staircaseprocedure was used, and S made judgments of both standardspirals. The standard and comparison spirals were shown side byside, but their positions were changed randomly from trial totrial. Clockwise and counterclockwise forms of the stimuli wereshown on alternating trials in an attempt to hinder the satiationof the illusion.

Results and Discussion

The staircase functions of one S showed no sign ofmerging after 10 trials, and her data were discarded .After the staircase functions given by the other Ss met,they were continued for six trials to check stability . Ifthe two functions remained together and started to drift ,the trials continued until they reached what seemed tobe an asymptotic level. The means of the staircase valuesafter the point of first contact were taken as the PSE.

The mean degree of spiralness the Ss selected asmatching Fig. 6b was 454 deg. The degree of spiralnessequated with Fig. 7a was 435 deg. The discrepancy of19 deg was not significant (t(8) :: 1.046]. Even if itwere, however, a judgment of the illusion spiral beingshifted slightly toward shallowness would not be

unexpected, since the possibility of satiating on theillusion was still present in spite of the precautions takento eliminate this problem .

The purpose of this experiment was to show that thediscrepancy between the perceptions of the spirals inFigs. 6b and 7a was not outlandish. The two comparisonspirals (405 and 450 deg) which most closely matchedthe mean judged spirals (435 and 454 deg) are given inFigs. 7b and 7c. Their differences are hardly excessive .

A final task remains , and this is to show which of thetwo factors , the reversed Zollner or the direction of theunits , is most responsible for the twisted cord effect.Adam demonstrated that the reversed Zollner illusion inone of its linear forms weakened as the angle of theinducing slashes increased. Close comparison of Fig. lcand Fig. 6b reveals that the units of direction angles areslightly steeper in Fig.6b than in Fig. l c. In Fig.6c,each unit is a continuation of a unit one step away onthe next ring. In Fig. Ic, the continuation is two or threeunits away. In Fig. 6b, the units cross the cord line atabout a 15-deg angle; in Fig. Ic , this angle is somewhatless (approximately 10 or 11 deg) . The twoconstructions, therefore, might provide a critical testbetween the reversed Zollner illusion componenthypothesis and Fraser's hypothesis of unit trend asinterpreted here .

The unit of direction spiral is steeper in Fig. 6b thanin Fig. lc. If the unit of direction trends form the basisof spirals that govern the perception of the cord, thenthe cord in Fig. 6b should be judged to be a steeperspiral than the cord spiral in Fig. lc. On the other hand ,the reversed Zollner illusion component hypothesiswould predict the contrary, since according to Adam'sfindings the more the, units are angled away from theunit line, the less the line leans in the direction of theunits. Thus, Fig. 6b should appear more shallow thanFig. Ic. This pair of contradictory predictions promptedthe following experiment.

EXPERIMENT IV

Method

Subjects

Twenty undergraduate students, male and female , with 20 /30corrected vision were run . Each S viewed the displaysindividually.

562 COWAN

Table 2Frequency of Responses for Each Figure in Experiment IV

Figure and Judgment

Conven-Angle of LIFE Twisted tionalInducing (More Cord Spiral

Lines Crooked) (Steeper) (Steeper)

II Deg 11 4 315 Deg 9 16 17

Stimuli

Four pairs of displays were used. The critical pair includedtwisted cord constructions similar to those shown in Figs. Ic and6b; the angles with which the units crossed the circular cord lineswere 15 and 11 deg. Two units of direction control spirals wereshown. These were two black spirals on a white background thatmatched the spiralness of the unit of direction spirals in Figs. Icand 6b. Two reversed Zollner illusion control figures wereshown. These were the two LIFE displays shown in Fig. 3b. Theangles of the inducing slashes in the top LIFE figure are 11 deg,and the slashes were made to tilt 15 deg away from the line inthe lower figure. This, of course. matched the unit of directionangles in the two twisted cord constructions used in theexperiment. Two black spirals on a white background whichturned through 585 and 225 deg served as practice spirals.

Procedure

The Ss were told that they would be shown a pair of spiralsand that they were to judge which spiral was steep and whichwas shallow. The E then drew a steep and shallow spiral to makesure S understood what was meant by these descriptions. The Swas then shown the practice spirals to further assure E that heknew what was meant by steep and shallow spirals. The S judgedthe unit of direction control spirals, then the twisted cordillusion spirals. Following this, E showed S the two LIFE figuresand asked S to tell him which of the two LIFE figures containedletters which were the most crooked. The position of eachdisplay within each pair varied between Ss.

Results and Discussion

The results appear in frequency form in Table 2. Only3 of the 20 Ss misjudged the unit of direction controlspirals. The 15-deg unit of direction control spiral wasclearly seen as the steeper of the two. The Ss werealmost evenly split in their judgments of the illusioncomponent control figures, although there was a slighttendency to judge the ll-deg LIFE figure as morecrooked than the 15-deg figure. This is somewhat inaccord with Adam's findings. Thus, according to theillusion component hypothesis, the Ss should be at leastevenly divided in their judgments of the two twistedcord figures, if not slightly favoring the II-deg spiral assteeper. The unit trend hypothesis suggests that the Ssshould favor the 15-deg unit of direction illusion(Fig. 6b) as being steeper. As shown in Table 2, the unittrend hypothesis received more support from these datathan did the illusion component hypothesis. Accordingto a McNemar test for significance of changes, thedifferences between the judgments of the LIFE and

twisted cord figures were significant [X2 (I) = 5.1],whereas the judgment differences between the twistedcord and conventional spirals were not [X2 (I) < I] .

An objection might be raised here concerning thedifferences between the two figures as seen in the widthsand distances between the cord lines. The greaterdistances between the cord lines in Fig. la increase theeffect of depth in this figure. While depth cues cannot beheld responsible for the twisted cord effect (see Fig. 2),they might enhance it slightly, and this would workagainst the unit trend hypothesis in this experiment. Thewidths in Fig. Ic should make the cord in this figureappear more spiraled rather than less, which is possiblywhy the number of Ss judging Fig. Ic as "steeper" was4/20 rather than 3/20.

THEORIES OF ILLUSIONS

To say that the units of direction account for most ofthe illusion's effect only pushes the question back onestep further. We have merely isolated the responsiblecomponent, but we have not described how thiscomponent produces the illusion. The best we can do inanswering this question is to look at some theories ofillusions and how they relate to the twisted cord effect.

Confusion Theory

According to confusion theory, componentsextraneous to a judged component can be seen as part ofor an extension of that judged component. Thus, thefins of the Mueller-Lyer illusion create a total impressionof added length, and the observer cannot eliminate thisimpression from his judgment. There are any number ofspiral components in the twisted cord figure which couldinfluence the judged direction of the cord line. Theexperiments described herein point to the spiralproduced by the units of direction as being dominant inthis respect.

Given that the cord strand spirals are prepotent, is itpossible that they might interact with the perception ofthe cord in a way suggested by confusion theory?Experiments III and IV demonstrated that the spiralnessof the circular cord matched the strand spiral lines.Thus, there is evidence that the judged cues areinfluenced by impressions created by extraneous cuelines, and this satisfies the conditions of confusiontheory.

Confusion theory is more of a descriptive theory thana process theory, however. It describes what the effectsof the added components are, but it does not describehow these effects are produced.

Lateral Inhibition

Ganz (1966) has developed a lateral inhibition theoryfor figural aftereffects. This is a process theory whichpresumably could apply to figures such as the twisted

VARIAnONS OF THE TWISTED CORD ILLUSION 563

cord illusion. The optical aberration theory of Chiang(1968) would also belong in this class. In its simplestterms, Ganz's theory states that lateral inhibition isresponsible for apparent contour displacements whenthose contours are separated by small distances.Furthermore, this inhibition diminishes as the distancebetween the contours decreases. The intersections of thestriped lines and the units of direction are the mostlikely loci for the lateral inhibition effect in the twistedcord figure. Note that the points of intersection vary inangle size from Fig. 1b to I f. One would expect asystematic change in the illusion across these figures,since the change in angle represents variations in thedistance between the critical contours. Perhaps thebiggest problem for the theory of lateral inhibition,though, is that it does not explain why the apparentspiral of the cord coincides with the spiral of the units ofdirection as suggested by Experiments III and IV.

Eye-Movement Theory

Since there are numerous spiral components in thetwisted cord figure, it would seem natural to assume thatthese components would be followed by the eye, andthis in turn would give rise to the apparent spiraling ofthe cord. This is particularly appealing since the eyemight trace the unit of direction spiral line, therebyexplaining the results of Experiments III and IV. Yet theillusion persisted at durations shorter than 100 msec.wh.ich indicates that eye movements are not necessaryfor this effect.

However, the results of Festinger , White. and Allyn(1968) suggest that it is the efferent commands that areresponsible for the perception of some illusions and notthe eye movements themselves. They noted that saccadicmovements across two lines of equal length weredifferent for each line when the two lines wereembedded in an illusion such as the Mueller-Lyer figure.As time passed, the saccadic movements equalized andthe illusion was lost. Yet the illusion did not disappear,nor did it fade when their Ss fixated at a point central tothe figure. In both instances, efferent commands weremade. In the first instance, these commands were being"recalibrated" by the saccadic movements, as Festingeret al put it, and the illusion was lost. In the secondinstance, there was no "recalibration" and the illusionremained intact.

Scanning the twisted cord is quite different fromscanning the Mueller-Lyer illusion, where the saccadicmovements are repeated over the same line. While theefferent commands accompanying the perception of thetwisted cord may always be commands to trace spiralsequal to the unit of direction lines, they probably takedifferent directions. As the figure is scanned, the eyesare likely to move from the outer to the inner edge.They may also tend to skip clockwise, sampling differentspiral arms as they go. Thus, the efferent command has afixed component (outer to inner edge) and a variable

one resulting from the clockwise skipping (left to right.up to down, right to left, down to up, etc.). If there is acomponent of the command that changes continuously.there may be resistance to command "recalibration,"and this is why this illusion does not satiate as readily assome of the simpler illusions.

Figure 5b still presents a problem. One sees circles inthis figure which may well be produced by an efferentcommand to follow circular units of direction. However,the perception of this figure is probably one of generalcircularity and not one of concentric rings of twistedcords. It was suggested earlier that th.is is perhaps due toa loss of subjective contrast in this figure. But thequestion remains whether or not the loss of subjectivecontrast is in any way related to the change in efferentcommands from spirals to circles. It is quite possible thatsubjective contrast is related to the circularity of thecord and not to the direction of the units. If so, thensubjective contrast would be present only in Fig. Sa andmissing in Figs. 5b and 5c. Note that this contradicts theassumptions concerning the relationship betweensubjective contours and perception at threshold made inthe discussion of Experiment II.

A Multicomponent Theory

With a figure as complex as the twisted cord illusion,it might seem likely that there are numerous factorswhich contribute to the effect. Taking Fig. Ic as anexample, the Zollner illusion imposed on the radial lines,the reversed Zollner illusion seen in the circular lines, thespiralness of the units of direction, the spiralness of thestepped lines, and the apparent depth created by thewidth of the cord lines and the distances between themall might contribute in some small but additive way tothe apparent spiralness of the cord. The experimentsdescribed here required, for the most part, one bitdecisions (clockwise-counterclockwise, circles-spirals,etc.); elimination of cues only established the necessity(or lack of it) of that cue for the creation of the illusion.Hence, we can say that the stripe spiral was not anecessary cue, since it was eliminated in Fig. l c and theillusion remained. The question of the degree to whichthe stripe spirals contributed to the illusion in theremaining constructions in Fig. I was not answered here.One would have to extract judgments concerning themagnitude of the effect across all constructions in Fig. Iin order to answer this. The isolation of the change ineffect produced by changes in a single cue may beformidable, since often one cue cannot be variedindependently of another, as in the case of the stripe andstepped spirals.

REFERENCES

Adam, J. A note on visual illusions of direction. AustralianJournal of Psychology. 1964. 16.53-56.

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Chiang. C. A new theory to explain geometric illusions producedby crossing lines. Perception & Psychophysics, 1968, 3,174-176.

Dufour, M. Illusion d'optique relative an micrometreocculaire deKrauss. Comptes Rendus Hebdomadaires de Seances deI'Academie de Sciences (Societe de Biologic), 1930, 105.600-601.

Festinger. L.. Ono, H.. Burnham, c., & Bamber, D. Efferenceand the conscious experience of perception. Journal ofExperimental Psychology Monograph, 1967, 74(4,. WholeNo. 637).

Festinger, L., White, C. W., & Allyn, M. R. Eye movements anddecrement in the Muller-Lyer illusion. Perception &Psychophysics, 1968, 3, 376-382.

Fraser, J. A new visual illusion of direction. British Journal ofPsychology, 1908, 2, 307-320.

Ganz, L. Mechanisms of the figural aftereffects. PsychologicalReview, 1966,73,128-150.

Hochberg, J. Perception I: Color and shape. In J. W. King and L.Riggs (Eds.), Woodworth and ..Schlossberg's experimentalpsychology. (3rd ed.) New York: Holt, Rinehart & Winston,1971. Pp. 395-474.

Hyde, W. F. A variant of the chessboard illusion. American

Journal of Psychology, 1929,41,296-297.Imai, S. Experiments on negative illusions in Zollner's Figure.

Jimbun Gakuho (Journal of Social Sciences & Humanities),Toyko Metropolitan University, 1962, No. 30, 31-44.

Parol a, R. Optical art: Theory and practice. New York: VanNostrand Reinhold, 1969.

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NOTE

1. A variation of the twisted cord illusion drawn by BettyPorter, a student of Rene Parola, was pointed out to me whichbears a striking resemblance to the construction in Fig. 2 (seeParola, 1969, p. 50).

(Received for publication January 22, 1973;revision received July 9, 1973.)