Perception & Psychophysics1976, Vol. 20(2), 129·137

An empirical taxonomy of visual illusions

STANLEY CORENUniversity ojBritish Columbia

Vancouver, British Columbia, Canada V6T 1W5

JOAN S. GIRGUSThe City College oj the City University ojNew York, New York 10010

HOWARD ERLICHMANThe Graduate Centre ojthe City University ojNew York, New York, New York 10036

and

A. RALPH HAKSTIANUniversity ojBritish Columbia, Vancouver, British Columbia, Canada V6T 1W5

A classification system for visual-geometric illusions, based upon the interrelationships betweenbehavioral responses to various distortions was created. Forty-five illusion configurations werepresented to 221 observers. Factor analysis revealed that there are five classes of illusions. A second­order analysis revealed that visual distortions are ultimately reducible to two global types of distor­tions: illiusions of extent and illusions of shape or direction.

In the one hundred and twenty years since Oppel(1855) published the first systematic studies on visualgeometric illusions, nearly a thousand papers haveappeared that deal with distortions evoked by simpletwo-dimensional patterns of lines. Despite thismassive research effort, it is interesting to note thatwe have still not managed to establish a classificationscheme or taxonomy that would allow sensiblegroupings of the many known distortions.

It is true that classification cannot, in and ofitself, provide explanations; however, a meaningfulclassification system has often served as a catalystfor theoretical advances in other areas of scientificendeavor. The classification systems of Mendelyev,in chemistry, and Linnaeus, in biology, immediatelycome to mind in this regard. Often, as in the abovecases, successful taxonomies are based on descriptivecategories, and the appropriate underlying mechanismsare often not isolated until many years after theclassification scheme has been generally acceptedas being useful. A classic example of this occurredshortly after 1816, when Thompson classified archeo­logical finds according to the materials that werepredominant in sets of artifacts, thus defining threeseparate eras that he designated as the stone, bronze,and iron ages. Only much later was this classification

We would like to acknowledge the assistance of RosalindWu, Jeanette Spero, Lucille Spivak, Louis Citron, KathleenMoody, Cynthia Weinman, Gerald Marks. and Eva Kerzner inthe collection and analysis of data. This research was supportedin part by grants from the National Research Council of Canada(A9783) and the National Science Foundation (74-18599).

129

scheme confirmed by stratographic evidence as in­dicating a real temporal sequence of cultures.

Attempts to classify visual geometric illusions dateback to Wundt (1898), who attempted to divideillusions into groups on the basis of the nature ofthe illusory effects themselves. He suggested twoclasses of illusions: illusions of extent and illusionsof direction. Unfortunately, Wundt himself wasunhappy with this division, noting that there wereseveral aberrant illusions that did not clearly fit intoeither class of distortion.

Most recent attempts at establishing typologiesof illusions have followed Wundt's lead in relyingupon the nature of the perceptual error to classifythe configuration. For example, Oyama (1960) hasproposed three major classifications. The firstincludes illusions of angle, direction, straightness,and curvature, a group roughly equivalent to Wundt'sillusions of direction. The remaining two classesare produced by subdividing Wundt's illusions ofextent into illusions of length and distance andillusions of size and area. Oyama was apparentlydissatisfied with these classes, since he then dividedeach general group into subclasses which are so finethat virtually every major illusion pattern forms aseparate entry.

Robinson (1972) follows a similar pattern. His firstgeneral grouping includes illusions of extent andarea, with separate subheads for virtually all themajor illusion figures. His second major groupingcontains illusions in which the apparent direction oflines or size of angles are distorted. He then providesa third grouping, simply labeled "other illusions,"

130 COREN, GIRGUS, ERLICHMAN, AND HAKSTIAN

in which we find the horizontal-vertical illusion anda variety of shape distortions.

Luckiesh (1922) used several criteria to divideillusions into five categories. The categories included(l) illusions of interrupted extent, which could becharacterized by configurations such as the Oppel­Kundt illusion (Figure lJ); (2) the effect of locationin the visual field, which could be characterized bythe horizontal-vertical illusion (Figure 1I); (3) illu­sions of contour, which include a number of Mueller­Lyer variants (Figure 2); (4) illusions of contrast,which include the Ebbinghaus and Oelboeuf illusions(Figures IE and 10, respectively); and (5) illusionsof perspective, which include variations of theSander parallelogram (Figure 1L) and several Orbisonconfigurations. Unfortunately, the basis for classifi­cation is not consistent. The first two categoriesrefer to characteristics of the stimulus configurationitself, while the next two describe the direction ofthe distortion. The fifth category is completelydifferent, referring to a particular theory of illusions.As Robinson (1972) has pointed out, such hetero­geneity does not seem likely to lead to the extrac­tion of underlying principles.

There have been two attempts to classify illusionsaccording to a hypothesized underlying mechanism.Tausch (1954) suggests that all illusions are causedby a depth-processing mechanism that can lead tothe inappropriate application of constancy scalingin a two-dimensional display. Since Tausch reducesall illusory distortions to this single mechanism,the problem of classification solves itself. An alternateclassification scheme has been offered by Piaget(1969). He classifies illusions into two groups, basedupon his theory of centrations, which he uses toexplain changes in illusion magnitude with chrono­logical age. His first category of illusions comprisesthose that decrease with age, while the secondcategory comprises those that increase with age.Unfortunately, only the Ponzo (Figure IF) andEbbinghaus (1E) illusions seem to show consistentincreases with chronological age (Leibowitz &Judisch, 1967; Wapner & Werner, 1957), whilealmost all other illusions seem to diminish in magni­tude with age (Pick & Pick, 1970). There are evensome configurations that show an initial de­crease followed by an increase, which also wouldprovide some problems for this typology.

The problem of classification has resulted in muchexpenditure of experimental effort in attemptingto assess relationships among configurations. Thuswe find Fisher (968) attempting to demonstrateexperimentally that the Ponzo and Mueller-Lyerillusions fall within the same class, Girgus, Coren,and Agdern (1972) attempting to show that theEbbinghaus and Delboeuf illusions are variants of

one another, and, at the other extreme, experimenterssuch as Erlebacher and Sekuler (1969) attemptingto show that the over- and underestimation portionsof the Mueller-Lyer are separate illusory distortions,Quina and Pollack (1972) attempting to differ­entiate the over- and underestimation portions ofthe Ponzo illusion into separate categories, andGirgus and Coren (Note 1) attempting to separateexperimentally all over- and underestimation con­figurations into separate illusory categories. Theseattempts to establish relationships among thevarious illusory configurations experimentally havenot led us very far, since they deal with only limitedsets of configurations, and the overall degree ofcorrelation among the various illusory patterns is notyet well enough established to provide any meaning­ful base line against which to assess actual degreesof relationship. There are procedures, however, thatmay allow us to reasonably assess the interrelationsamong illusions, and thus empirically to establisha useful classification scheme.

Let us consider a hypothetical state of events thatmight lead us toward a behaviorally meaningfultypology. Suppose that there are two underlyingillusion mechanisms. For convenience, we willsimply call them A and B. It would seem reasonableto classify illusions that are predominantly causedby mechanism A into one group and those that arepredominantly caused by mechanism B into another.Of course, if both mechanisms are operating in anyone configuration, there will be some interactionbetween the classes; however, segregation of illusionconfigurations into groups according to their dominantmechanisms still provides a behaviorally significantdivision. The pragmatic difficulty that arises increating such a typology lies in the fact that we mustfirst be able to define the underlying mechanisms.If the specific mechanisms are not correctly identi­fied, the classification scheme is doomed to failure,and unfortunately we do not as yet know all theappropriate mechanisms involved in the formationof visual illusions. Suppose, however, that a particularindividual is highly responsive to mechanism A andonly weakly responsive to mechanism B. In testingsuch an individual, we would expect him to showsizable illusory distortions for configurations that arepredominantly dependent upon mechanism A andonly small illusory distortions for configurationspredominantly dependent upon mechanism B. Anindividual who is more responsive to mechanism Bthan mechanism A should produce the opposite setof results. Notice that we are merely offering thesuggestion that illusory magnitude covaries as a func­tion of the strength of the underlying mechanism;yet this line of analysis suggests a means of pro­ducing a classification system. If there are large

ILLUSION TAXONOMY 131

Fillun~ I. Variancs of c1l15Sical illusions in the stimulus set:the POllllendorff (Al; Wundt (B); Zoellner (C); Delboeuf (D);Ebbinghaus (E); Ponzo (F); Jastrow (G); Baldwin (H); horizontal­vertical (I); Oppel-Kundt (J); divided line (K), and Sanderparallelollram (Ll.

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underlying mechanism or whether each variant represented adifferent illusion which ought to be classified under a separateheading.

Since we are interested in the covariation of illusion measuresacross a number of different configurations, it is important tomake the required judgments as similar as possible so that anydifferences in responses will not artificially appear as a functionof the measurement procedure. It was decided, therefore, toreduce all responses to the estimation of a horizontal linearextent of some part of the figure. For the Mueller-Lyer, Baldwin,and Oppel-Kundt configurations, the relevant extents are obvious.For the Ebbinghaus and Oelboeuf illusions, observers wererequired to judge the diameter of the circles. Judgments ofillusion magnitude for the Poggendorff, Wundt, Zoellner, andJastrow illusions may also be reduced to judgments of linearextent by asking observers to judge the extents indicated as X or Yin Figure 3 for each configuration. Since all of the illusions used(except the Poggendorff) contain an over- and an underestimateddimension, each of these measures was taken separately. The4S resulting illusion stimuli were each drawn on a separate21 x 27.5 em page. In the lower left-hand quadrant of the page,there was a horizontal line on which the observer was asked tomark the apparent linear extent of some part of the figure. Corenand Girgus (1972) have shown that this technique providesresults that are comparable in validity and reliability to thoseobtained by the method of adjustment. A small block ofinstructions and an illustrative diagram in the upper right-handcorner of each page indicated which extent was to be judged.

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1111 " -----J K

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individual differences in the magnitude of visualgeometric illusions, and if these differences dependupon variations in an observer's sensitivity to a givenmechanism or his propensity toward the use of agiven perceptual strategy, then we can group illusionstogether on the basis of those distortions that tendto covary in magnitude, even though we have notspecifically identified the underlying mechanisms orstrategies. The experiment reported below describesan attempt to create an empirical taxonomy ofillusions along these lines.

Figure 2. ,. ariants of the Mueller-L~'er illusions employedin this experiment: standard form (A); exploded (B); Piagetform (D); curved (E); box (F); circle (G); lozenge (HI; Sanfordform (I): minimal (J); Coren form (K).

METHOD

StimuliTwelve of the most common classical illusion configurations

were selected for inclusion. These included the Poggendorff(Figure IA), Wundt (I B), Zoellner (IC), Oelboeuf (10),Ebbinghaus (IE), Ponzo (IF), Jastrow (IG), Baldwin (lH),horizontal-vertical (I I), Oppel-Kundt (I J), divided line (I K),and Sander parallelogram (IL). In addition, the II variants ofthe Mueller-Lyer illusion shown in Figure 2 were included. Thepurpose behind using a reasonably large set of classical illusionfigures was to allow enough representative items so that meaning­ful groupings might emerge. The reason a large number ofvariants of a single illusion were included was to ascertain whetheror not altered forms of a classical illusion, such as are common­ly used in many experiments, actually represented the same

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132 COREN, GIRGUS, ERLICHMAN, AND HAKSTIAN

several mechanisms operating in the same directionto produce the final percept (Coren & Girgus, 1973,1974; Girgus & Coren, 1973). Such multiple causa­tion might very well lead to the positive manifolddisplayed in the correlation matrix. The high degreeof interrelation among the illusion measures alsoseems to suggest that any factors that may be ex­tracted will also be interrelated; hence, an obliquerather than an orthogonal solution seems to beindicated.

An oblique factor analytic solution was obtainedby first performing an unweighted least-squarescommon-factor analysis on the 45 by 45 correlationmatrix. Two criteria were used to determinethe number of factors to be extracted. The firstinvolved inspection of the eigen values of thecorrelation matrix, and the second involved theextraction of factors until the resultant pattern offactor loadings fulfilled the requirements of simplicityof structure (Guilford, 1954; Thurstone, 1947). Bothprocedures suggested the presence of five factors.The extracted factors were transformed by means of thegeneralized Harris-Kaiser (1964) procedure (see alsoHakstian, 1970).

Table I shows the primary-factor matrix for the 45illusion measures. The factor loadings above .40have been underlined. For convenience, those seg­ments that are usually overestimated in the variousillusion configurations are indicated by a + andthose that are usually underestimated are indicatedby a - next to the illusion name. Table 2 showsthe intercorrelations among the extracted factors.It is interesting to note that the intercorrelationsamong the five factors- shown in this table were allpositive, thus again indicating the high degree ofinterrelatedness among illusion judgments obtainedin the study.

Fortunately, the extracted factors seem to makesufficient sense in terms of what is known aboutillusions to provide a meaningful classificationsystem. On the first factor, we find high loadingsfor the Poggendorff illusion and for both judgmentsof the Wundt and Zoellner illusions. In addition, theapparently longer segment of the Sander parallelogramand the apparently longer segment of the dividedline illusion load on this factor. Ignoring the lattertwo for the moment, it is interesting to note thatthe Poggendorff', Wundt, and Zoellner illusionsall contain intersecting line elements. These illusionsare basically directional in nature, and can easily beexplained by such relatively peripheral structuralmechanisms as the blurring of the image in its passagethrough the crystalline lens and optic media (Chiang,1968; Coren, 1969; Ward & Coren, 1976) or lateralinhibitory interactions that displace contour loci(Bekesy, 1967; Coren, 1970; Ganz, 1966) which serve

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SubjectsThe sample comprised 221 undergraduate volunteers from the

City College of the City University of New York.

ProcedureThe illusion stimuli were assembled into booklets. Subjects

received two such booklets, each of which contained the 45 illusionconfigurations in mixed order; thus, each subject made twoindependent judgments on each illusion extent. These two judg­ments were averaged for the purposes of statistical analysis. I

Group administrations were utilized, and response rate was self­paced. although subjects were urged to indicate their firstimpressions.

Figure 3. The horizontal linear extents that subjects were askedto estimate are indicated b~ X or Y for the Poggendorff (A).Jastrow 181. Wundt 10. and Zoellner (0) illusions.

\-.

A factor analytic technique was used to generatea classification scheme. First, the 45 variables wereintercorrelated. In the resulting correlation matrix,virtually all the values were positive, a phenomenonwhich is frequently observed when dealing withmeasures of abilities or intelligence. In this study,only 31 of the 990 correlation coefficients werenegative; the largest of these was - .14, which, givena sample size of 221, is not significantly differentfrom 0 at the .01 level. Thus our analysis begins witha correlation matrix in which only 3010 of the valuesare negative.and these are not significantly differentfrom O. Such a positive manifold obviously indicatesthat, in general, illusion judgments are highly inter­related. It has been previously suggested that mostvisual geometric illusions are probably caused by

RESULTS AND DISCUSSION

ILLUSION TAXONOMY 133

to displace the vertex of angles so as to render themapparently more obtuse. Why one segment of theSander parallelogram and one segment of the dividedline illusion should also load on this factor is notimmediately clear. However, the other five highfactor pattern loadings do suggest that Factor Iinvolved structural, contour interactive mechanisms.

On the second factor we find high factor patterncoefficients for both halves of the Delboeuf andEbbinghaus illusions, the apparently longer segmentof the Ponzo illusion, and the apparently largercomponent of the Jastrow illusion. The apparentlysmaller portion of the Jastrow illusion just missesthe criterion cutoff with a factor loading of 0.38.

Table IFactor Pattern for 45 Illusion Measures

This factor appears to be quite homogeneous, andreadily suggests a theoretical interpretation. Allthe illusion forms with high loadings on this factorseem to involve some sort of cognitive contrast effectin which the apparent difference in the size of thecentral test element and adjacent or surroundingelements is accentuated. Such accentuation of clearlyperceived differences has been recognized since thetime of Helmholtz (1856, 1860, 1866/1962), whoreferred to it as "size contrast." That these illusoryeffects are clearly dependent upon cognitive in­formation processing strategies has been demonstrateda number of times (Coren, 1971; Coren & Miller,1974) and several quantitative models based uponsuch cognitive processes have been offered (Helson,1964; Massaro & Anderson, 1971; Restle, 1971).

Factors 111 and IV are strongly interrelated(r = 0.73). Given the fact that most of the highloadings that appear on these two factors representvariants of the Mueller-Lyer illusion, this is perhapsnot surprising.

Factor 111 shows high factor coefficients for all11 variants of the apparently longer segment of theMueller-Lyer illusion. The only apparently shortersegment that intrudes on this factor is the dumbbellversion. In addition, both segments of the Baldwinillusion, which may easily be interpreted as a variantof the apparently longer segment of the Mueller-Lyerillusion with boxed ends, appear on this factor. It isimportant to note that this is not purely a Mueller­Lyer factor. The apparently longer segment of thehorizontal-vertical illusion and the apparently longersegment of the Oppel-Kundt illusion also manifesthigh factor loadings. In sum, this factor contains,almost exclusively, illusory distortions involvingoverestimation of linear extent. The specific mech­anism underlying this grouping is not immediatelyclear, since quite different mechanisms have usuallybeen proposed to explain the Mueller-Lyer, horizontal­vertical, and Oppel-Kundt illusions.

It is interesting to contrast Factor III with Fac­tor IV. On Factor IV, we find 9 of the 11 apparentlyshorter variants of the Mueller-Lyer illusion loadingsignificantly, with the remaining two apparentlyshorter variants (the variant with boxed ends andthe dumbbell form) just missing the criterion withweightings of 0.39 and 0.31, respectively. In addition,the apparently shorter segment of the Oppel-Kundt

Table 2Correlation Matrix of Factors (pm)

V

.26

.38

.18

.31

IV

.24

.48

.73

.31

III

.12

.32

.73

.18

II

.45

.32

.48

.38

.45

.12

.24

.26

IIIIIIIVV

h2

.65

.50

.55

.59

.63

.59

.23

.64

.51

.77

.61

.48

.54

.27

.68

.55

.64

.58

.06

.52

.68

.64

.55

.65

.67

.67

.54

.66

.66

.65

.66

.57

.54

.62

.59

.52

.66

.72

.66

.56

.50

.52

.61

.59

.48

V

.05

.09-.05-.04

.00

.13-.10

.07

.04

.04

.66

.08

.15

.13-.02-.10-.08

.12-.05-.16

.00

.15

.07

.00

.06

.05

.20

.19

.30-.02-.04

.19-.01

.12

.01-.02

.27

.25

.21

.04-.03

.30

.69-:TO

.16

1.84

IV

.03

.03-.20

.16-.04-.19

.29

.01

.07

.08-.03

.27

.29

.30

.04

.00

.38

.15

.68]8.49:J6.62:T4.50.06.68.24.59:T7.39.16.31

-.04.96

-.03.62.28.63.20.56.05.15.27

-.08

7.43

III

.01-.05

.05-.28

.01

.16-.04

.01-.25

.09

.13

.15

.07

.58

.M

.6736.46TI.69.23.51IS.74:T4.70:IT.62:IT.59.34.64.49.70

-33.81.05.49:or.85]9.13

-.06.28.00

9.04

II

.07

.13

.39

.02

.11

.81

.62

.57

.69

.62

.20

.49

.38'-.22

.21-.06

.12

.07-.05-.03

.16-.12

.13-.07

.17-.03-.10

.04

.03

.11

.19-.07

.06-.07

.00

.13

.02

.12

.22

.02

.00

.09

.16-.09-.06

3.99

.42~.51.84.74

-.08-.10

.23

.15

.07

.05

.08

.01

.02

.02

.24

.05

.07

.04

.04-.12-.05

.12-.07

.02

.24

.05-.11-.12-.10-.19-.08-.33-.15

.02

.07

.05

.03-.07-.03

.22

.45

.02

.56]8

3.57

PoggendorffWundt +Wundt ­Zoellner +Zoellner ­Delboeuf ­Delboeuf+Ebbinghaus ­Ebbinghaus +Ponzo +Ponzo ­Jastrow +Jastrow ­Baldwin +Baldwin ­Horiz-Vert +Horiz-Vert ­Oppel-Kundt +Oppel-Kundt ­Standard ML +Standard ML ­Exploded ML +Exploded ML ­No Shaft ML +No Shaft ML­Piaget ML +Piaget ML­Curved ML +Curved ML­Box ML +Box ML­Circle ML +Circle ML­Lozenge ML +Lozenge ML­Sanford ML +Sanford ML­Minimal ML +Minimal ML­Coren ML +Coren ML­Divided Line +Divided Line -Sander Parallelogram +Sander Parallelogram -

Factor Variance

134 COREN, GIRGUS, ERLICHMAN, AND HAKSTIAN

illusion appears on this factor, while the apparentlyshorter segment of the horizontal-vertical illusionjust misses the cutoff with a factor loading of 0.38.Thus, this factor seems to represent the set of dis­tortions associated with underestimation of linearextents. It is not clear why Factors III and IVsegregate themselves in this fashion. To be sure,the high correlation between them indicates similar­ities between the two classes. Since the two factorscontain separate halves of the same distortions, theywould be expected to manifest some relationshipon purely logical grounds. To find that the apparentlylonger and the apparently shorter sections of theMueller-Lyer load on separate factors is quiteinteresting, since it has been suggested previously,based upon the fact that the two halves of theMueller-Lyer do not covary in the same fashion inresponse to certain parametric manipulations, thatthey may in fact be separate distortions (Erlebacher& Sekuler, 1969). To find that the horizontal-verticaland Oppel-Kundt illusions separate in a similarfashion, along with the Mueller-Lyer illusion, issurprising, since no similar suggestions have beenmade to segregate these configurations on the basisof their over-and underestimation segments.

It is also interesting to note that Factors III andIV are not more highly correlated with Factor II.On the whole, Factor II also involves over- andunderestimations, but the distortions in Factor IIapparently result from a different mechanism thando the distortions in Factors III and IV. It may beimportant to note that all the illusions that load onFactor II, except for the Ponzo illusion, involve theestimation of area, whereas the distortions inFactors III and IV involve linear extents. This cannot,however, provide the only explanation for a differ­ence between these factors, since there is no hint of aseparation between the under- and overestimatedsegments of the illusions in Factor II. It seems likelythat quite different mechanisms underlie the illusionsrepresented in Factors II, Ill, and IV.

The most puzzling factor is Factor V, which showsonly two significant factor pattern coefficients: theapparently shorter segment of the Ponzo illusion andthe apparently shorter segment of the divided-lineillusion. It is possible to view both of these figuresas enclosed segments that occupy only a smallportion of the total bounded space, which wouldsuggest that we may be dealing with the comparisonof an element to its global frame of reference.However, since so few configurations load here,interpretation is difficult. We did not use a largeenough sample of frame-of-reference illusions inthis study to assess this interpretation properly. It issomewhat comforting that this factor accounts forsuch a small portion of the variance, approxi-

mately 112 that of Factors I and II and 1/5 that ofFactors III and IV.

Finally, it should be pointed out that only one ofthe 45 illusion forms used in this study failed to loadsignificantly on any of the five factors describedabove. This is the apparently shorter segment of theSander parallelogram, whose factor pattern coef­ficients ranged from 0.00 to 0.18.

On the basis of these results, we may now tenta­tively offer a classification scheme that can be usedas a taxonomy for illusory distortions. The labelsand generalizations attached to each class must beconsidered as somewhat speculative; however, thegroupings are based on actual behavioral data andtherefore may well reflect common underlyingcausal mechanisms.

(1) Shape and direction illusions. This class ofillusions is characterized by Factor I. This groupingpredominantly includes distortions in apparentshape, parallelism, and colinearity, which seem toarise in patterns with numerous intersecting lineelements. The underlying mechanism is presumablystructural in nature, probably involving opticalaberrations and lateral inhibitory influences. ThePoggendorff, Wundt, and Zoellner illusions arecharacteristic of this class, and presumably othershape and directional distortions, such as the Orbisonor the Jastrow-Lipps illusions, would also fall intothe same calssification.

(2) Size contrast illusions. This classification,characterized by Factor II, represents those illusorydistortions in which the apparent size of an elementappears to be affected by the size of other elementsthat surround it, or fonn its context. Thus, the casualmechanism may involve the use of sharpening orleveling strategies, such as those that have beenquantitatively described by Helson (1964) Massaroand Anderson (1971) and Restle (1971). These effectstypically manifest themselves as cognitive contrast,where a target appears larger when surrounded bysmall elements or smaller when surrounded by largeelements. The Delboeuf, Ebbinghaus, Jastrow, andPonzo illusions are characteristic of the illusionson this factor. Illusions involving contrast betweenthe size of angles would presumably also fall intothis classification.

(3) Overestimation illusions. This classificationis best characterized by Factor III. The illusions thatshow the highest loadings on this factor include allthe apparently longer versions of the Mueller-Lyerillusion, both parts of the Baldwin illusion, theapparently longer segment of the horizontal-verticalillusion, and the apparently longer segment of theOppel-Kundt illusion. It is hard to ascertain whatunderlying causal mechanism may be associatedwith all these figures. For many of them, a depth

processing interpretation, such as that popularizedby Gregory (1968), seems plausible; yet, for others,such as Coren's dot form or the minimal formof the Mueller-Lyer, suchan interpretation seemsdifficult. Simple spatial averaging may playa partin the Baldwin and the Mueller-Lyer figures;however, the Oppel-Kundt and horizontal-verticaldo not seem susceptible to this form of explanation.Thus, we are left with a grouping on the basis ofthe behavioral manifestation that does not im­mediately suggest any underlying mechanism butsuggests a pattern of interrelationship.

(4) Underestimation illusions. This group of illusionsis represented by Factor IV. Since it includes mostof the apparently shorter segments of the Mueller­Lyer illusion, the apparently shorter segment of theOppel-Kundt, and the horizontal-vertical illusions,it seems to be a factor that is the complement toFactor III, representing predominantly under­estimations of linear extent. Again, no mechanismimmediately suggests itself for this class of distortion.It should be kept in mind that the linear illusions ofover- and underestimations seem to be highly correlatedand only an oblique factor analytic solution was ableto separate these dimensions; thus, it is likely thatthe underlying mechanisms for these two factorsmay be somewhat related.

(5) Frame of reference illusions. This classifi­cation, characterized by Factor V, is the mosttentative. Both of the high factor loadings are charac­terized by line segments that are underestimated inlength; however, this pair of illusory distortionsdoes not apparently covary with those of Factor IV,as might be expected if they were simply illusionsof linear underestimation. In some respects, it ispossible to see a configurational identity betweenthese patterns and the variants of the Mueller-Lyerin Factor III, since the inducing elements are ap­pended to 'the ends of the element to be judged.However, the Factor III illusions are all illusionsof overestimation and the illusions in this factorare all of underestimation. On purely logical grounds,one might expect that the Ponzo illusion and thedivided-line illusions would fall into Factor II, sinceboth seem to involve some sort of cognitive contrasteffect. In fact, it is interesting to note that the highestintercorrelation between Factor V and the other fourfactors is with Factor II, the size contrast factor.We have tentatively identified this as a frame-of­reference factor. If this interpretation is correct,illusions like the rod-and-frame ought to fall intothis classification. Further experimental investigationis clearly necessary to specify this grouping moreclearly.

We have, thus, produced a classification schemethat contains five bins into which illusory distortions

ILLUSION TAXONOMY 135

may be sorted. Two of these bins, that containingillusions of direction and shape and that involvingsize contrast, seem to suggest underlying mechanisms,Of the remaining three, two are highly related (illusionsof over- and underestimation) and seem logical, al­though no underlying mechanism immediately suggestsitself. The fifth and final bin must be seen as merelysuggestive at this point.

Higher Order ClassificationDespite the fact that the taxonomy we have derived

seems to be meaningful, it is interesting to consider afurther form of analysis that may allow a more globalpattern of groupings to emerge. This may be done viaa second-order factor solution in which the inter­correlations between the factors shown in Table 2 serveas the data. In order to find higher order, or meta­groupings, the five-by-five matrix was factored by theunweighted least squares method. Two factors wereadequate to explain the higher covariation. These twofactors were transformed to an oblique simple struc­ture using the Harris-Kaiser (1964) procedure. Finally,the primary factor pattern loadings of measured vari­ables projected on the two second-order factorswere computed using the Cattell-White (Cattell, 1965)method. Table 3 shows the resultant matrix, whichcontains the factor loadings for the 45 illusion variantson the two second-order factors. These factors arecorrelated, with r = 0.49.

It is clear that Factor A includes both the over- andunderestimation segments of almost all the variantsof the Mueller-Lyer illusion, both segments of thehorizontal-vertical illusion, both segments of the Oppel­Kundt illusion, and one segment of the Baldwin andSander illusions. This factor seems to involve nearlyall the illusions of linear extent in the test set. Factor Bin the second-order solution contains the Delboeuf,Ebbinghaus, Ponzo, divided line, Wundt, and Zoellnerillusions, with the Jastrow illusion just missing thecriterion value. All these distortions may be seen asinvolving some sort of cognitive contrast. We havealready referred to the Delboeuf, Ebbinghaus, dividedline, Ponzo, and Jastrow illusions as examples of sizecontrast in our discussion of Factor II. It is interestingto note, in this regard, that the various over- and under­estimation portions of these configurations tend to loadon Factor B with opposite signs. Such a result wouldbe consistent with a contrast explanation. The Wundtand Zoellner illusions may load here as examples ofdirection contrast, as originally suggested by Helmholtz(1866/1962).

It is somewhat suggestive to consider the fact thatFactor B deals predominantly with distortions in­volving area, shape, and direction, while Factor Acontains mostly distortions of linear extent. It may wellbe the case that the earlier investigators who grouped

136 COREN, GIRGUS, ERLICHMAN, ANDHAKSTIAN

illusions into two categories, one involving illusions ofextent and the other illusions of shape, may well havebeen on the right track, at least in terms of the twometafactors obtained in this study. Such a globalanalysis is suggestive; although it does not permit thefiner classification that the basic solution allows.

In conclusion, we may describe our data as demon­strating that a taxonomy of visual illusions may becreated based upon behavioral data. The super­ordinate division of illusory distortions seems to bein two general classes. The first contains illusionsinvolving distortions of linear extent, while thesecond contains illusions of area, shape, anddirection. However, complete classification of theillusions in this study requires a subdivision of the

A

Table 3High-Qrder Factor Pattern Loadings for 45 Illusion Measures

PoggendorffWundt +Wundt ­Zoellner +Zoellner ­Delboeuf ­Delboeuf+Ebbinghaus ­Ebbinghaus +Ponzo +Ponzo ­Jastrow +Jastrow ­Baldwin +Baldwin­Horiz-Vert +Horiz-Vert ­Oppel-Kundt +Oppel-Kundt ­Standard ML +Standard ML ­Exploded ML +Exploded ML ­No Shaft ML +No Shaft ML­Piaget ML +Piaget ML­Curved ML +Curved ML­Box ML +Box ML­Ctrcle ML +Circle ML­Lozenge ML +Lozenge ML ­Sanfcrd ML +Sanford ML ­Minimal ML +Minimal ML­Coren ML +Curen ML­Divided Line +Divided Line -Sander Parallelogram +Sander Parallelogram -

.00

.07

.20

.00

.10-.11

.26

.04

.18

.05-.19

.09

.36

.18-.50

.52

.62

.52

.78:Pi.62.08.44.483f.61:IT.74.65.65.65.68.68.57.67:6T

-.50.64.5832.63

-13.24

-.41-:to

B

.34

.46:6T:sT.54

--.74:sT.60

-.60.50

-.46.34.37

-.28-.11

.10

.10

.13

.09

.04-.16-.25-.06-.16-.06

.15

.26-.04

.14-.01

.02-.04

.01-.04

.11-.11

.09-.01

.09-.03

.17-.48

.45-.34-.01

set of stimuli into at least five classes. The first ofthese classes seems predominantly structural innature and involves both shape and directionalillusions. The second class seems to involve a sizecontrast mechanism. The third grouping containsa large number of illusions of linear overestimation,while the fourth class contains a large number ofillusions involving linear underestimation. The fifthclass may involve frame-of-reference illusions.

It might be noted that it has been frequentlysuggested that most visual illusions are caused bymultiplicity of mechanisms acting in concert (Coren& Girgus, i 973, i 974; Girgus & Coren, Note I) .Thus, while the classification scheme resulting fromthis study may suggest primary mechanisms for someof the test stimuli, we cannot begin to specify allthe secondary mechanisms that may contribute tothe various illusions. It seems likely that, for anygiven illusion, several mechanisms, in addition tothat which might define the primary factor loadings,may contribute to the final percept.

It seems somewhat paradoxical that after 120 yearsof research we are only now offering a tentativeclassification scheme for visual illusions. This classi­fication differs from others in that we have allowedthe behavioral data to provide us with a taxonomyrather than forcing the groupings according totheoretical predispositions. It seems possible toextrapolate from these results to predict otherillusions that should fall into particular classifi­cations. Such class membership should be reflectedby high intercorrelation with other members of thatgroup.

Of course the underlying principles that producethe particular natural arrangement of the perceptualphenomena we have uncovered are not yet fullyunderstood. There is some comfort to be derivedfrom the fact that there is enough of an underlyingorder to allow us to organize illusion configurationsin some manner that makes conceptual sense.Ultimately, one may hope that such a taxonomywill be used to reinterpret existing data and relation­ships among configurations, as well as serving as aguide to further research.

REFERENCE NOTE

I. Girgus, J. 5., & Coren, S. Contrast and assimilationillusions: Differences in plasticity. Paper presented at themeeting of the Psychonomic Society, Denver. November. 1975.

REFERENCES

BiKisy, G. VON. Sensory inhibition. Princeton. N.J: PrincetonUniversity Press. 1967.

CAtTELL, R. B. Higher order factor structures and retic­ular-vs-hierarchical formulae for their interpretation. In C.Banks and P. L. Broadhurst (Eds.), Studies in psychology.London: University of London Press. 1965. Pp. 223-266.

obliq ue factorHarris-Kaiser

Measurement.

CHIANG. C. A new theory to explain geometrical illusionsproduced by crossing lines. Perception & Psychophysics,1968. 3. 174-176.

COREN. S. The influence of optical aberrations on themagnitude of the Poggendorff illusion. Perception &Psychophysics. 1969. 6, 185-186.

COREN. S. Lateral inhibition and geometric illusions. Quarterly, Journal ofExperimental Psychology, 1970, 22, 274-278.

COREN. S. A size contrast illusion without physical sizedifferences. American Journal ofPsychology. 1971, 84, 565-566.

COREN. S.• & GIRGUS, J. S. A comparison of five methods ofillusion measurement. Behaviour Research Methods &Instrumentation. 1972. 4. 240-244.

COREN. S.. & GIRGUS. J. S. Visual spatial illusions: Manyexplanations. Science, 1973, 179.5034.

COREN. S., & GIRGUS, J. S. Transfer of illusion decrement asa function of perceived similarity. Journal of ExperimentalPsychology. 1974. 102. 881-887.

COREN. S.. & MILLER, J. Size contrast as a function of figuralsimilarity. Perception & Psychophysics. 1974. 16.355-357.

ERLEBACHER. A.. & SEKULER. R. Explanation of theMueller-Lyer illusion: Confusion theory reexamined.Journal ofExperimental Psychology. 1969. 80.462-467.

FISHER. G. H. Gradients of distortion seen in the context ofthe Ponzo illusion and other contours. Quarterly Journal ofExperimental Psychology. 1968. 20.212-217.

GANZ. L. Mechanism of the figural aftereffects. PsychologicalReview. 1966. 73. 128-150.

GIRGUS. J. S.. & COREN. S. Peripheral and central componentsin formation of visual illusions. American Journal ofOptometry and the Archives of the American OptometricSociety. 1973. 50,533-580.

GIRGUS. J. S.. COREN. S.. & AGDERN. M. The inter-relation­ship between the Ebbinghaus and the Delboeuf illusions.Journal 0.( Experimental Psychology, 1972. 95.453-455.

GREGORY. R. L. Perceptual illusions and brain models.Proceedings of the Royal Society. 1968. Section B, 171.279-296.

GUILFORD. J. Psychometric methods. New York: McGraw-Hili,1954.

HAKSTIAN. A. R. A computer programme fortransformation using the generalizedprocedure. Educational and Psychological1970. 30. 703-705.

HARRIs. C. W .. & KAISER. H. F. Oblique factor analyticsolutions by orthogonal transformations. Psychometrika,1964. 29.347-362.

HELMHOLTZ. H. VON. Treatise on physiological optics. Leipzig:Voss. Part I (1856), Part II (1860). Part III (1866).Republished: New York, Dover. 1962.

HELsON. H. Adaptation-level theory: An experimental andsystematic approach to behaviour. New York: Harper. 1964.

ILLUSION TAXONOMY 137

LEIBOWITZ. H. W., & JUDISCH. J. M. The relationshipbetween age and the magnitude of the Ponzo illusion.American Journal of Psychology. 1967. 80. 105-109.

LUCKIESH, M. Visual illusions. Princeton. N.J: Van Nostrand.1922.

MASSARO. D. W., & ANDERSON. N. H. Judgmental model ofthe Ebbinghaus illusion. Journal of Experimental Psychology,1971. 81. 147-151.

QpPEL. J. J. Uber Geometrisch-optische Tauschungen.Jahresbericht der Frankfurt (Vereins), 1855. 55. 37-47.

OrAMA, T. Japanese studies of the so-called geometrical-opticalillusions. Psychologia, 1960. 3. 7-20.

PIAGET. J. The mechanisms and perception (Trans. G. N.Seagrim). New York: Basic Books. 1969.

PICK. H. L.. & PICK. A. B. Sensory and perceptual development.In B. Mussen (Ed.), Carmichael's manual of childpsychology. New York: Wiley. 1970.

QUINA. K.. & POLLACK. R. H. Effects of test line position andage on the magnitude of the Ponzo illusion. Perception &Psychophysics. 1972. 12. 253-256.

RESTLE. F'. Visual illusion. In M. H. Apley (Ed.), Adaptation­level theory. New York: Academic Press, 1971.

ROBINSON. J. O. The psychology of visual illusion. London:Hutchinson. 1972.

TAUSCH. R. Optische Tauschungen als artisizielle Effekteder Gestaltungsprozesse von Grossen- und Formenkonstanzin der Naturlichen Raumwahmehmung. PsychologischeForschung, 1954. 24. 299-348.

THURSTONE, L. L. Multiple-factor analysis. Chicago: Universityof Chicago Press. 1947.

WAPNER. S.. & WERNER, H. Perceptual development: Aninvestigation within the framework of sensory-tonic fieldtheory. Worcester: Clark University Press, 1957.

WARD. L. M .• & COREN. S. The effect of optically-inducedblur on the magnitude of the Mueller-Lyer illusion.Bulletin of the Psychonomic Society. 1976. 7, 483-484.

WUNDT. W. Die geometrisch-optrischin Tauschungen.Abhandlungen der mathematisch physischen Classe derKoeniglischen Sacohsischen Gesellschaft der Wissenschaften.1891. 24. 53-178.

NOTE

I. The mean intercorrelation between the two measures madeon each configuration was 0.55. with a standard deviation 0.18.This corresponds to a mean reliability of 0.71 for this form ofillusion measurement.

(Received for publication January 26,1976:revision accepted April 28. 1976.)