attneave (1956) the quantitative study of shape and

PSYCHOLOGICAL BULLETINVol. 53, No. 6, 1956

THE QUANTITATIVE STUDY OF SHAPE ANDPATTERN PERCEPTION1

FRED ATTNEAVE AND MALCOLM D. ARNOULTSkill Components Research Laboratory, Air Force Personnel and Training Research Center

The pre-eminent importance offormal or relational factors in per-ception has been abundantly demon-strated during some forty years ofgestalt psychology. It seems extra-ordinary, therefore, that so littleprogress has been made (and, indeed,that so little effort has been ex-pended) toward the systematizingand quantifying of such factors. Ourmost precise knowledge of perceptionis in those areas which have yieldedto psychophysical analysis (e.g., theperception of size, color, and pitch),but there is virtually no psycho-physics of shape or pattern.

Several difficulties may be pointedout at once: (a) Shape is a multidi-mensional variable, though it is oftencarelessly referred to as a "dimen-sion," along with brightness, hue,area, and the like, (b) The numberof dimensions necessary to describea shape is not fixed or constant, butincreases with the complexity of theshape, (c) Even if we know howmany dimensions are necessary in agiven case, the choice of particulardescriptive terms (i.e., of reference-axes in the multidimensional spacewith which we are dealing) remainsa problem; presumably some suchterms have more psychological mean-ingfulness than others.

1 This research was carried out at the SkillComponents Research Laboratory, Air ForcePersonnel and Training Research Center,Lackland Air Force Base, San Antonio,Texas, in support of Project 7706, Task 27001.Permission is granted for reproduction, trans-lation, publication, use, and disposal in wholeor in part by or for the United States Govern-ment.

452

The need for an adequate psycho-physical framework is most obviousin those studies (having to do withdiscrimination, for example, or withpositive or negative transfer) inwhich it is necessary to manipulateshape or pattern as an independentvariable. Unless some meaningfulunits of variation are specifiable,functional relationships cannot beobtained. It is somewhat less obvi-ous, but nonetheless true, that a com-parable need exists in experimentswhich seek to determine how formperception is influenced by extrinsicvariables such as size, contrast,method and degree of familiarization,etc. In studies of this sort, the ex-perimenter commonly uses some small,arbitrarily chosen set of stimuli:sometimes simple geometrical forms;sometimes a group of "nonsense"shapes which he draws in a more orless haphazard manner. If the resultsobtained are "significant" in theusual sense, we have some specifiabledegree of confidence that they are gen-eralizable to people other than thoseused as subjects, but the degree towhich they are generalizable to newstimuli remains a matter of conjec-ture. Yet the latter kind of generaliz-ation is no less important than theformer. Only in rare cases of appliedresearch is the investigator reallycontent with results which hold onlyfor the particular stimulus objectsemployed experimentally.

Egon Brunswik (9, 10, 11) is per-haps the only psychologist who hasever given due weight to the im-portance of stimulus-sampling, or of

STUDY OF SHAPE AND PATTERN PERCEPTION 453

situation-sampling in general. Al-though the approach of this paperis somewhat different from Bruns-wik's, for reasons which are devel-oped below, we wish to acknowledgefreely Brunswik's influence upon ourown thinking, and to commend hiswritings on this subject to any readerunacquainted with them. Brunswiktakes the reasonable position that re-sults with "ecological validity" maybe obtained only by the use of experi-mental materials which are drawnfrom, and hence representative of,the real situations to which onewishes to generalize. Thus, in thestudy of shape perception, it wouldbe desirable to experiment with theshapes of natural objects. Suppose,however, that we wish to investigatethe learning and memory of shapeswith which subjects are initially un-familiar: the requirement of unfamili-arity will obviously preclude the ex-perimental use of shapes which arecommonly encountered. Is there anysensible procedure for choosing stim-ulus-materials in this sort of situa-tion?

It is our belief, at this time, thatthe problem of generalizing from ex-perimental stimuli may profitablybe broken into two parts. First,there is the problem of specifying thestimulus-domain, i.e., the problemof drawing a sample of stimuli froma parent population characterized bycertain determinate statistical pa-rameters. The stimulus-domain, orparent population, includes all thosestimuli to which the results may begeneralized, and is defined by the sta-tistical parameters which characterizeit. In the following section we shallindicate a variety of particular meth-ods for drawing "random" patternsand shapes from such clearly definedhypothetical populations, to whichexperimental results may then be gen-

eralized with measurable confidence.The second problem, which is

really a special case of the first, isthat of drawing a sample which has"ecological validity." If our real aimis to generalize to natural forms, orto some subset thereof, it is neces-sary to estimate the psychologicallyimportant statistical parameters ofthese natural forms in order that ex-perimental materials may be con-structed to possess the same parame-ters. Thus, we are brought back tothe acute need for a general psycho-physics of form. In the final sectionwe shall discuss the kinds of physicalanalysis and measurement which ap-pear appropriate to such a psycho-physics.

THE CONSTRUCTION OF STIMULIAll the methods described below

for constructing nonsense shapes andpatterns have in common the factthat the particular characteristics ofeach figure are randomly determined.Each method is, in effect, a set ofrules by which points are plotted andconnected in accordance with valuesobtained from a table of randomnumbers. Each method, or set ofrules, thus determines a domain ofstimuli. The stimuli actually con-structed for use in a given experimentwill, if they are all constructed ac-cording to the same rules, be a ran-dom sample of the stimulus-domaindefined by the set of rules. The ex-perimental results, consequently, maybe generalized both to the entirestimulus-domain and to the appropri-ate subject population.2

1 The kind of double-generalization pro-posed here would require an error term whichincluded the variance due to subjects, thevariance due to stimuli, and the interactionbetween them. In what is perhaps the mostobvious analysis-of-variance design, the sw&-jectsy.stimuUy.treatments mean square wouldbe the appropriate error term to use.

454 FRED ATTNEAVE AND MALCOLM D. ARNOULT

The experimenter who desires touse stimuli constructed in this man-ner must determine what set of ruleswill provide him with a stimuluspopulation having the character-istics he wants. If one desires to gen-eralize experimental results to theworld of real objects (chairs, air-planes, people, etc.), it is necessaryto have a stimulus sample possessingecological validity. To constructnonsense stimuli of this sort onemust know the pertinent parametersof the stimulus-domain of real ob-jects and use these parameters inconstructing the experimental stim-uli. In the next section we shall dis-cuss some of the problems inherentin this methodological requirementand some of the attempts which havebeen made to solve them.

In the present section, some gen-eral methods for constructing stimuliare described in sufficient detail thatthe reader, if he desires, may repeatthe operations in order to developadditional stimuli belonging to thevarious stimulus-domains denned bythe methods. It should be kept inmind, however, that these methodsare described merely as examplesand are not intended to constitute acomprehensive catalog of all possiblemethods. Descriptions will be givenof methods for generating shapeshaving either closed or open contours,

for generating various kinds of pat-terns, and for introducing systematicvariations or transformations ofshapes or patterns.

Closed Contours—Angular Shapes

Method 1. Starting with a sheet ofgraph paper—say 100X100—succes-sive pairs of numbers between 1 and100 are selected from a table of ran-dom numbers. Each pair will deter-mine a point which can be plotted onthe 100X100 matrix. The total num-ber of such points to be plotted canbe determined either randomly orarbitrarily.

When all the points have beenplotted, a straightedge is used toconnect the most peripheral pointsin such a way as to form a polygonhaving only convex angles. Thisoperation will usually leave some un-connected points within the polygon(Fig. la). When a point falls withinsome small, arbitrarily chosen dis-tance of the proper perimeter (e.g.,the point between segments 7 and 8in Fig. la) it is included even thoughit makes a slightly concave angle,since otherwise an indentation prac-tically dividing the shape into twoparts might later occur. The sides ofthe polygon are numbered, and thepoints remaining inside are assignedletters. The table of random numbersis then used to determine which of the

20 40 6O 8O IOO SO 4O 60 80 100 £0 40 60 80 100

FIG. 1. SUCCESSIVE STAGES IN THE CONSTRUCTION OF A "RANDOM" FIGUREACCORDING TO METHOD 1 (SEE TEXT)


central points is connected to whichside. In the example given, Point Cwas connected to Side 2, forming inthe process Side 10 (Fig. IV). At thisstage in the construction, the possi-bilities of connecting points havebeen changed. Point A may now betaken into Sides 3, 4, 5, 6, 7, 8, or 10,but not into Sides 1, 2, or 9. Point Bmay be connected only to Side 2 orSide 10. If Point A is connected toSide 5, forming new Side 11, thereremains only the possibility of con-necting Point B to Side 2 or Side 10(see Fig. li). Connecting Point Bto Side 10 completes the shape, whichfinally appears as shown in Fig. Ic.

It will be noted that every stepin the procedure is determined eitherrandomly or by the elimination of allother possibilities. Furthermore,every step is completely determinateand can be duplicated by anyone us-ing the same rules and the same selec-tions from the table of random num-bers.

Method 2. This method of con-structing random shapes is alsostarted by plotting successive pairsof random numbers as coordinates ongraph paper. As each point is plotted

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(a)

it is given a number so that even-tually all are numbered serially.These points are then connected inthe order in which their serial num-bers first appear in a table of randomnumbers, except that numbers whichviolate certain rules of constructionare rejected. The incomplete con-struction shown in Fig. 2a will pro-vide examples of permitted and non-permitted connections. The rules forconnecting points are as follows:

a. No line may be drawn twice.Assume, in Fig. 2a, that the last linedrawn was from Point 2 to Point 5.If the next number in the table were2, it would be rejected since that con-nection has already been made.

b. No line may be drawn whichcompletely encloses a point withinthe perimeter of the figure. FromPoint 5 it would not be permissible todraw a line to Point 6 or to Point 4,since either action would completelyenclose Points 3 and 8.

c. No two points may be directlyconnected if they are already con-nected by a path which follows per-imeter lines without passing throughany other plotted points. For ex-ample, Point 5 may not be connected

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(b)

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FIG. 2. EXAMPLE OF NONSENSE SHAPE CONSTRUCTED BY THE RULES OF METHOD 2: a.INCOMPLETE CONSTRUCTION DEMONSTRATING PERMISSIBLE AND NONPERMISSIBLF.

CONNECTIONS; J. THE COMPLETED SHAPE


to Points 3 or 7, Point 3 may not beconnected to Points 5 or 6, and Point2 may not be connected to Point 4.

d. The figure is complete wheneach point has been connected to atleast two other points. It sometimeshappens that the table of randomnumbers leads one to a point whichalready has all the other connectionsallowed it. In this case one of theother points is chosen randomly as anew origin and the regular process iscontinued. The incomplete shape ofFig. la is shown in a completed formin Fig. 2Z>.

As is the case with all the methodsdescribed in this paper, this methodis completely objective. The result-ing figure could be reproduced, ifnecessary, from a set of coded instruc-tions consisting only of the numbersoriginally selected from the table.

Unlike Method 1, Method 2 usu-ally generates shapes containing someangles in addition to all those atoriginally plotted points. This dif-ference is emphasized by Rule c ofMethod 2.

In Method 1 there are no restric-tions on the ways in which the plottedpoints may be connected except that(a) the figure must be closed, and (&)connecting lines may not cross, i.e.,the completed figure may have anglesonly at the original points.

In Method 2, on the other hand,there may be "emergent" angles atplaces other than originally plottedpoints, and the figures produced tendto be characterized by "good continu-ation." Again, it is Rule c of Method2 which causes many of the perimeterlines of the final figure to be continua-tions of other perimeter lines.

Comparing the two methods interms of the informational contentof the shapes produced shows thatin Method 1 information (in additionto that required to locate the original

points) is used only in connecting theinterior points to the sides of theoriginal perimeter, whereas in Method2 information is used in making allconnections between plotted points.For this reason a Method 2 shapecomposed of n original points andcontaining n-\-k angles (k represent-ing the number of "emergent" points)will contain more information than aMethod 1 shape composed of n origi-nal (and final) points. Because of thegood continuation introduced intothe figure, however, the Method 2shape having n+k points will containless information than would a Method1 shape having n + k original points.

Method 3. Fitts, Weinstein, Rap-paport, Anderson, and Leonard (15)have developed a technique for con-structing "metric" figures, the in-formational content of which may beeasily and accurately determined.Starting with a somewhat smallermatrix—say, 8X8—the number ofcells to be filled (from the bottom up)in each column of the matrix is ran-domly determined. This method pro-duces shapes which belong to a rela-tively small stimulus-domain andwhich are equal in informational con-tent. A variation of this method in-volves allowing each possible column-height to appear only once in eachshape, with the order of appearancedetermined randomly. This secondstimulus-domain contains memberswhich are equal in area and, conse-quently, contain less informationthan the shapes first described. Stillanother variation may be introducedby reflecting each shape on one of itsaxes to produce a symmetrical shapecontaining no more information thanits nonsymmetrical predecessor. Ex-amples of these various classes ofmetric figures may be found in Refer-ence 15.


H

FIG. 3. METHOD FOR INTRODUCING "RANDOM" CURVES INTO AN ANGULAR NONSENSESHAPE. THE ORIGINAL SHAPE is THE SAME ONE WHICH APPEARED IN FIG. Ic

Closed Contours—Curved Shapes

Method 4. This method describesa procedure for making wholly orpartially curved shapes from theangular shapes constructed byMethod 1 or 2. This procedure mayappear to be somewhat involved, butactually it requires more time to de-scribe than to perform. Essentially,it consists merely of replacing angleswith inscribed arcs, of curvaturechosen randomly within limits im-posed by the figure.

For purposes of demonstrating the

method, let us start with the shapedescribed and constructed underMethod 1 (Figs, la-lc). It is de-cided (arbitrarily or randomly) thatfour of the twelve angles are to becurved. Let us suppose that AnglesC, F, J, and K (Fig. 3) are chosen.(For convenience of exposition theangles have been assigned the lettersA through L.) The first step in theprocess consists in constructing lineCp, which is the bisector of /.BCD.Then, the shorter of the two arms ofthe angle (in this case, line BC} is


divided into equal units. These unitsmay be chosen for convenience. Forexample, Fig. 3 was constructed on a100X100 matrix having matrix unitsequal to 0.20 in., and Line BC wasarbitrarily divided into segments of0,25 in. each. It should be noted thatthe divisions of the line are num-bered in sequence, starting alwaysfrom the apex of the angle.

One of these numbered points online BC is now chosen at random anda perpendicular from Line Cp to itis constructed (Line 5-q). This line(5-q) now becomes the radius of anarc which is inscribed within /.BCD.The arc is tangent to Line BC andLine CD at points equidistant fromC. Thus, /.BCD has now been re-placed by a curve (actually, two lin-ear segments and an arc) going fromB to D.

Point F has been curved by thesame process. Angle EFG is bisectedby line Fr, and Line FG is dividedinto equal segments. Division 8 hav-ing been chosen at random, line 8-5is constructed and used as a radiusfor inscribing a curve within Z EFG.

The next two constructions dem-onstrate the complex curvature whichmay result when successive pointsare chosen to be curved. Point /is curved by the process describedabove, with Line 13-w being used asthe radius of an arc inscribed withinZ UK. However, in curving PointK it is necessary to inscribe an arcwithin ZJ'KL, not within /.JKL.Point J' is the point at which the arcconstructed with radius 13-w be-comes tangent to line JK.

If it is so desired, all the points ofan angular figure may be curved. Itshould be noted, however, that theshorter arm of every angle is dividedinto segments, and that its divisionsare numbered beginning with zero. Ifthe zero is the random choice, the re-

sulting curve will have zero radius,i.e., that angle remains as originallydrawn.

Method 5. Angular shapes can bechanged into curved shapes by a pro-cess of photographic blurring. Thefigure is first photographed and then,with the help of an enlarger, isprinted out-of-focus on high contrastpaper. The resulting image has a con-tour which is curved, but which isalso graded in density. A repetitionof the process of photographing andprinting, however, will eliminate thedensity gradient, producing a shapewith contours which are rounded andwell-defined. The amount of blurmay, of course, be carefully con-trolled, and a graded series of curvedshapes may be made from a singleprototype shape.

Open Contours.

Method 6. There are many ways inwhich open-contour nonsense shapesmay be constructed from a table ofrandom numbers, but all that wehave used have been variations onone basis method. Starting from theapproximate center of a matrix ofconvenient size, a line is drawn to oneof the eight intersections nearest thestarting point. These eight intersec-tions (or, more generally, directions)have been assigned numbers as shownin Fig. 4a. The intersection on thegraph paper at which the first lineterminates becomes the origin forthe second line to be drawn, and soon. A difficulty with this method isthat there is no intrinsic criterion forcompleteness in such a figure. Oneobjective rule is to determine, beforebeginning the construction, the totalnumber of digits to be selected fromthe table and to consider the figurecomplete when that number of lineshas been drawn.

Many variations on this basic tech-


(a) (b )

FIG. 4. CONSTRUCTION OF OPEN-CONTOUR"RANDOM" SHAPE: a NUMBERING OF POSSIBLEINTERSECTIONS; b. TYPICAL NONSENSE SHAPE

nique may be introduced. For ex-ample, for some purposes it may bedesired to allow only four directionsin which the contour may vary; also,the length of each line may be deter-mined randomly as well as the direc-tion. Partially or wholly curved con-tours may be produced by thismethod as follows: the radius of cur-vature of the arc drawn to connectsuccessive intersections along thehorizontal and vertical axes of thematrix is set as one-half the lengthof a matrix unit. To connect twointersections diagonally separated,the arc would have a radius equal toone matrix unit. Thus, for example,one might determine randomly foreach line constructed: (a) which twointersections will be connected, (J)whether the connection is to be linearor curved, and (c) the direction ofcurvature. Figure 46 was drawn bythis technique. Additional variationson these methods may be providedby using semi-log, log-log, or polarcoordinate matrices on which to con-struct the nonsense contours.

Patterns

Method 7. Although the more obvi-ous ways of generating random pat-terns have been used by a number ofinvestigators, the possibilities of this

approach to the construction of com-plex visual displays have never beenadequately explored. In general thepractice has been to construct amatrix of some given size and then todetermine randomly which cells areto be filled. Patterns of dots wereconstructed in this fashion by Kauf-mann, et al., (19), French (16), andKlemmer and Frick (20), for exam-ple. Attneave used the same ap-proach, including the introduction ofa symmetry factor, in a study of theeffect of redundancy on memory forpatterns (4). In another slight varia-tion Arnoult used random shapes aselements in constructing random pat-terns for use in a learning experiment(2). Patterns generated in this fash-ion are very attractive as stimuli be-cause it is usually possible to com-pute fairly precisely the informa-tional content of the display.

Systematic VariationsFrequently it is desired to con-

struct "families" of shapes havingknown physical relationships amongthe individual members. Again, thereare many possible techniques for ac-complishing this end. The followingtwo methods represent two kinds ofsystematic variations which have re-cently been used.

Method 8. A prototype shape isconstructed by any of the methodsso far described. Then, each pointis moved to a new location and theconnecting lines redrawn as before.In moving the points, any of the fol-lowing parameters may either beheld constant or varied randomly:(a) the number of points moved, (b)the particular points moved in mak-ing successive variations on the sameprototype, (c) the distance throughwhich a point is moved, and (d) thedirection of movement. A number ofvariations made from a given proto-


type will form a distribution of shapeswhich "vary about" the prototype.Stimuli of this sort were used re-cently by Attneave in testing thehypothesis that knowledge of theprototype shape, or "schema," wouldfacilitate discrimination of the varia-tions in paired-associate learning (6),and by Arnoult in a study of the ef-fect of predifferentiation trainingon recognition (1). A typical proto-type shape and its variations areshown in Fig. 5.

FIG. 5. A PROTOTYPE SHAPE AND "FAMILY"OF RANDOM VARIATIONS

Method P. A somewhat differenttechnique for creating "families" ofshapes has been developed at Stan-ford by LaBerge and Lawrence (23).Initially, a random shape is con-structed by a method essentiallythe same as those described inMethod 1 and Method 2 (actually,LaBerge and Lawrence simply con-nected randomly chosen points intothe polygon of minimal perimeter).Then, each point on the contour isassigned randomly chosen "x" and"y" increments to its coordinates,and these new coordinates are plottedand connected on a fresh matrix.These same increments are then

added to the new coordinates and athird figure is constructed. This pro-cess may be continued until one hasconstructed a row of, say, six figures,each differing from its immediateneighbors by a constant amount ofdistortion as measured by the dis-tance through which the points move.The next step is to label the former"x" increments as "ys" and theformer "y" increments as "xs."These new increments are added tothe coordinates of the points of all sixof the figures already constructed,and the process of constructing suc-cessive shapes is repeated until thereis a column of six shapes for each ofthe original six shapes. The final re-sult is a matrix of 36 shapes in whichany two adjacent shapes in a row orcolumn are equally spaced in termsof the average distance the pointshave moved. Matrices of stimuliof this sort are currently being usedby LaBerge and Lawrence in studiesof transfer.

As has been emphasized a numberof times in the preceding discussion,these methods for constructing "ran-dom" shapes are only a few whichhave been selected to show some ofthe classes of shapes which can beconstructed. The number of differ-ent sets of rules which can be de-veloped for plotting and connectingpoints taken from a table of randomnumbers is limited only by the fer-tility of the individual experimenter'simagination. It should be reiterated,however, that using stimuli con-structed by these "random" methodsdoes not insure that the generaliza-tions resulting from the research willbe pertinent to all other kinds ofvisual stimuli. It guarantees onlythat the results will be generaliz-able within a particular stimulus-domain, i.e., to any other stimuli con-structed by the same rules.


ANALYSIS OF NATURAL FORMSLet us now return to a problem

which the methods discussed in theprevious section by no means obvi-ate. We still need a technique, or aset of techniques, by means of whichphysical measurements of a psycho-logically relevant sort may be ob-tained for forms which we have notconstructed ourselves. Any methodof "random" construction must em-ploy some set of rules, either arbi-trary or otherwise, and these ruleswill strictly determine the class-char-acteristics, or statistical parameters,of the shapes constructed. Weshould like to be able to devise rulessuch that our synthetic shapes mightpossess the statistical characteristics(but not the familiarity) of naturalshapes to which we wish to gen-eralize. At present, we lack not onlya factual knowledge of the values ofthese statistical parameters, but alsoa methodology to guide us in theirdetermination. Likewise, when someexperimental variation of form isfound to produce a certain effect inthe laboratory, it is necessary thatthe variable in question be identifia-ble and measurable outside the labo-ratory if the results are to be gen-eralized. Unfortunately, however, itis much harder to measure form thanto manipulate it.

Relatively few scientists have seri-ously applied themselves to the prob-lems of analyzing and describingform; these problems seem to havefallen into the cracks between sci-ences, and no general quantitativemorphonomy has ever developed.D'Arcy Thompson's Growth and Form(27) is virtually the only major workin the field: it is a fascinating andimpressive book, but its contributionto the identification of psychophysi-cal variables is limited. Rashevsky,whose work in mathematical bio-

physics is in some respects a continu-ation of Thompson's, has been moredirectly concerned with psychologi-cally relevant measures of form.

A bstraction of contour. Consideringthat the first step in the analysis of ashape is the abstraction of its con-tour, Rashevsky (25, p. 449) deviseda simple hypothetical nerve-net withthis function. Suppose that the stim-ulation of the retina is projected tosome central area as an activity ofsharply localized excitatory fibersand of inhibitory fibers slightly morediffuse in their projection. If certainconstants of the system have propervalues, excitation from any area ofuniform brightness will be sup-pressed, except at a contour wheresuch an area is bounded by a darkerone which provides less inhibition.

This nerve-net has a fairly closeanalogue in the following photo-graphic process. A negative and apositive transparency, separated bya thin plastic sheet, are preciselysuperimposed so that they "cancel"each other when viewed from a rightangle. A print is made by transmit-ting light from a diffuse source (e.g.,the ground glass of a contact printer)through the superimposed positiveand negative to a high-contrast paperplaced in contact with the negative.In the case of a black object on awhite ground, or vice versa, lightcan angle through both positive andnegative only at the contour, andthe resulting print is indistinguish-able from an outline drawing of theobject. In the case of more complexpictures, the abstraction of sharpbrightness-gradients preserves tex-ture, as well as contour: this is illus-trated clearly in Fig. 6. A picture ob-tained in this way may be thought ofas a differential (with respect tobrightness) of the original, involvinga "delta" of finite magnitude. If


VH/ W • - ' • ' • : • • " ..•; •V/ ' . . . . •>.•• . ' • ' • • . - . - •? '

FIG. 6. A DIFFERENTIAL PICTUREThe photograph of D'Arcy Thompson from which this was derived is by Bjflrn Soldan; it

appeared originally in Isis and was reproduced in the August, 1952, Scientific American. In theoriginal, the lightest portions are Thompson's forehead and beard, and the darkest portion isthe back of his coat. These have approximately equal brightness in the differential picture.


a smaller "delta" had been taken inthe derivation of Fig. 6 (by reducingthe space between the superimposedpositive and negative), the iris andpupil of Thompson's eye, for ex-ample, would appear in outline in-stead of as a black dot.

In 1948 one of the authors (Att-neave), in collaboration with JohnM. Stroud, attempted to developthis photographic technique to a de-gree of precision such that the totalreflectance of the differential picturemight serve as an index of the com-plexity of the original. That attemptwas unsuccessful for several reasons,having to do chiefly with the unrelia-bility of photographic operations:e.g., the initial step of making a posi-tive and a negative which would ade-quately cancel always required con-siderable cut-and-try. It may beadded that the process is a close rela-tive of one which has long been usedto produce a "bas-relief" effect, andthat the Eastman Laboratories haverecently employed a similar tech-nique with color film to obtain photo-graphs which look remarkably likepaintings.

An electronic device lately de-veloped by Kovasznay and Josephat the National Bureau of Standardsappears to accomplish much the sameresult as the photographic processdescribed above, but in a manner sub-ject to more precise control. Thebeam of a cathode ray tube, movingin a complex scan which covers thefield in two orthogonal dimensions,transmits light through a photo-graphic transparency to a photo-electric cell. The electrical signal thusgenerated isdifferentiated and squaredelectronically, and then fed into a re-ceiving scope where it modulates abeam synchronized with the trans-mitting beam. Illustrations of theresults, which are presented in the

descriptive note of Kovasznay andJoseph (21), could be mistaken forthe efforts of a somewhat naive artist.

A group of engineers in the Lin-coln Laboratory of M.I.T., includingOliver G. Selfridge, Gerald P. Din-neen, and Marshall Freimer, are cur-rently experimenting with the use ofdigital computers to perform opera-tions relevant to object identifica-tion. They have been successful inprogramming a contour-abstractingoperation; this is preceded by an av-eraging operation, which rids the fig-ure of irrelevant detail, and followedby an operation which abstractsangles, or regions of high curvature,from the contour (26).

The mere abstraction of contour,whether by an objective process orwith the aid of the experimenter'sown perceptual machinery, does notin itself constitute quantification. Itdoes, however, contribute to the iso-lation of that which is to be quanti-fied: i.e., form. Whenever we speakof form, we are referring to a some-what vague set of properties whichare invariant under transformationsof color and brightness, size, place,and orientation; our definition mayor may not be extended to specifyinvariance under projective (or per-spective) transformations. Contouris characterized by invariance undercolor and brightness transformations.Attneave (3) has previously pointedout the related (though not equiva-lent) fact that contours are regionsof relatively high informational con-tent.

A nalysis of contour. There are vari-ous practical reasons for wishing tobe able to describe a contour in termswhich are independent of its size,place, and orientation. For example,subjects are often required to drawfigures from memory: such drawingscannot be fairly evaluated by any


simple method of superimposing adrawing upon the original and meas-uring deviations, because of differ-ences in scale, etc. If both the origi-nal and the reproduction could berepresented in terms descriptive ofform alone, they could then be com-pared objectively.

Such a representation may take theform of a single function. If the re-ciprocal of the radius of curvature ofa closed contour is plotted againstdistance along the contour, a peri-odic function results. This functionmay be normalized (i.e., rendered in-dependent of the scale of the originalfigure) by assigning a value of unityto the perimeter of the figure and ex-pressing radius of curvature in com-parable terms, or by setting equal tounity the area under one period ofthe function. An angle is representedby a vertical line which rises (or falls,in the case of a concave angle) to in-finity; a spike of this sort, of infiniteheight, infinitesimal width, and de-terminate area, is the so-called 5-function of Dirac, and is amenable tomathematical treatment.3

If one feels more comfortable deal-ing with finite ordinates, the follow-ing system may be used. Imagine aminiature tricycle, guided over acourse such that a point midway be-tween the rear wheels precisely fol-lows the contour. The angle 9 bywhich the front wheel deviates froma forward position may be plottedagainst distance travelled by thefront wheel to give a periodic func-tion descriptive of the contour. Thefront wheel will move in an arc con-centric with the segment of the con-tour being followed. Wherever anangle occurs in the contour, the angle

* This system of representation has been de-veloped in considerable detail by OliverStrauss (personal communication).

0 of the front wheel will be 90°; thusthe function will always have somevalue between plus and minus 90°.Radius of curvature, r, is related to0 by the equation r = L cot 0, inwhich L is the distance between thefront and rear wheels. Normalizingmay be accomplished by giving theperimeter of the figure unit value, andsetting L at some standard fractionalvalue. If L is made to equal l/2ir,regular polygons will be representedby square waves regularly alternat-ing between 0 and 90°, a circle willbecome a horizontal line with anordinate of 45°, and certain otherregularities will be uniquely repre-sented; this value of L is somewhatlarge for convenient use with morecomplex shapes, however. The in-terested reader will have little diffi-culty in working out further detailsof the system. It has the advantageof specifying an actual measuring de-vice which is practical and simple toconstruct. Automatic recording ofthe function could be arranged withtwo pairs of selsyns: one translatingthe rotation of the front wheel into amovement of the recording paper;the other coupling the angular posi-tion of the front wheel with the posi-tion of a recording pen.

Both of the functions just de-scribed have a serious disadvantage.Suppose we wish to compare twoshapes which have a part-to-part orpart-to-whole similarity—say, theoutline of a cow's head with the out-line of a whole cow. The normalizingfactors which will be employed on abasis of perimeter or area will obvi-ously not be such as to give compara-ble representation to the similar por-tions of the outlines.

The method next to be describedavoids this difficulty, though it is notwithout limitations of its own. In-stead of describing the contour by


means of a continuous function, wemay attempt to analyze it into partswhich are individually homogeneous,and hence amenable to approximatedescription in terms of a few stand-ardized dimensions. It is usuallypossible to construct a polygon abouta figure made up of complex linesand curves, as in Fig. 7, by drawingtangents (a) at points of zero curva-ture (e.g., CD, IJ, etc.: whenever acurve changes from concave to con-vex, it must have an intermediatepoint of zero curvature), (b) at pointsof minimal curvature, where a de-crease in curvature is followed byan increase (e.g., FG), and (c) at dis-continuities of slope, or angles (e.g.,AB, GH, etc.). The series of linesthus formed may be described simplyby stating the slope and length ofeach line in succession, but this de-scription is peculiar to a given orien-tation and size of the figure. It maybe rendered orientation-free and scale-free by specifying instead, for eachpair of adjacent segments, (a) thechange in direction (in degrees], and(b) the change in the logarithm oflength, as the contour is followed in aclockwise direction.4 Curves aretreated as "rounded-off" angles: i.e.,a curve is approximated by an arclocated tangent to two successivelines of the polygon we have beendiscussing. In most cases, the size ofthe arc will be limited by the length

4 Several other possible pairs of coordinatesconvey the same information. What is re-quired, essentially, is to describe the shapesof successive segments of the polygon, takenin pairs. Measures of any two angles of sucha triangle, or any two ratios of sides or differ-ences between logarithms of sides, or anycombination of an angle and a comparison ofsides, is adequate to specify the shape of thetriangle. The combination above is chosenfor its intuitive appeal; also because errors ofmeasurement have a more uniform effect onthese coordinates than on certain others.

of the shorter of the two segments.Hence curvature is conveniently ex-pressed by a third coordinate speci-fying (c) the proportion of the distancebetween the apex of the angle and theend of the shorter segment at which thearc best approximating the curve istangent. This coordinate will usuallyhave some value between 0 and 1.0,with 0 indicating an abrupt angle(radius of curvature equal to zero)and 1.0 indicating an arc which istangent to the shorter segmentat its end. In the case of Fig. 7,for example, (c) would have avalue of 0 at A and M, a value of1.0 at G, and a value of about .8 at C

N

B

FIG. 7. ILLUSTRATION OF METHOD FORQUANTIZING IRREGULAR CONTOUR

(note that the arc best approximat-ing a curve will not necessarily havethe same point of tangency as theoriginal curve). When the arc ap-proximating a curve turns throughmore than 180°, as in the case of thebulbous projection in the JKL re-gion, the value of (c) will not remainbetween 0 and 1, since some of thepoints of tangency are on extensions


of segments of the polygon, ratherthan on the segments themselves.The values of (c) associated with /,K, and L would be about 3.8, 3.6,and .5, respectively.6

The reader will recognize this sys-tem of analysis as essentially the re-versal of a method for constructing"random" shapes which was de-

6 The two sets of numbers below, which arepresented as a demonstration of the practica-bility of the system and as an amusement forthe reader, describe recognizable profiles ofthe two authors. Successive tri-coordinatesare given, thus: 01, 61, a; at, bi, £2; etc. In theactual reconstruction of a contour from suchcoordinates, a line of any desired length andslope is drawn to start. The first triad ofcoordinates gives the relationship of thesecond line of the contour to the arbitrarilydrawn starting line, and so on. Cumulativeerror will be avoided if the values of a arecumulatively added to the slope, in degrees, ofthe starting line (with due regard for the circu-larity of the scale) to obtain the slope of eachsegment; likewise if the values of b are cumu-latively added to the logio length of the start-ing line to obtain the logio length of each seg-ment. Positive values of a denote clockwiseturns; negative values counterclockwise (con-sistent reversal results in a mirror-image). Inspecifying values of (c), the symbol "<" isused to mean "less than .1," i.e., that theangle is rounded to a slight, practically tin-measurable, degree.

+61, -.56, .3; -84, +.08, < ; +27, -.04,.1; +35, +.66, .2; +1SS, -.06, 1.0; -145,-.39, 0; +32, +.13, 1.0; +27, -.43, <;-56, +.61, .1; +107, -.38, .4; -69, -.12,.1; +37, -.62, <; -88, +.38, 0; +105,+ .11, 1.0; -79, +.33, .1; +86, +.34, .5;-20, -.14, <; -25, -.76, .4; -66, +.23,0; -55, -.15, 1.0; +30, +.09, 0; +113,-.02,0; -56, +.15, .1.

+43, +.14, .4; +26, -.22, .4; -31, +.02,.3; +37, +.21, .4; +36, -.21, .6; +41, -.41,.3; -50, +.45, 0; -6, -.18, < ; +29, +.03,.8; -43, +.26, <; +12, -.18, <; +88,-.11, .5; +23, -.27, .4; -124, +.31, .5;+90, -.12, .8; -29, -.28, <; +70, -.23,1.0; -114, +.39, 0; +66, +.16, .3; -65,-.04, .4; +44, -.17, <; -43, +.77, .7;+ 121, -.28, .9; -26, +.04, <; +105, -.08,.7; -62, -.49, .6; +26, +.09, <; -113,+.13, .3; -25, +.25, .5; +44, -.26, .8; -83,-.24, 0; +19, +.42, 1.0.

scribed in the previous section. Thesystem has several advantages: (a)It yields a description which doesnot vary with size and orientation.(b) Since the use of a general normal-izing factor is avoided, part-similari-ties between the contours of two ob-jects are reflected in their numericaldescriptions. Likewise, repetitioussequences of elements in the samecontour (but not parallel lines) arereflected, and could be quantified byan autocorrelational technique. Iftwo similar shapes (e.g., an originaland a subject's reproduction frommemory) were compared by crosscorrelation of their numerical de-scriptions, it would be desirable tocalculate values for several "displace-ments" of one set of coordinates uponthe other (as in autocorrelation), inorder to allow for qualitative omis-sions or additions of elements, (c)There is reason to believe that thenumber of tricoordinates required todescribe a shape constitutes a firstorder approximation of its psycho-logical complexity (i.e., the numberof psychologically discrete parts whichit contains). Fehrer (14) used a simi-lar measure (number of internallyhomogeneous lines) on her figures,and found that complexity, so meas-ured, was closely related to difficultyin a reproduction-learning situation.Attneave (5) recently confirmed thatthe number of sides in a polygon isthe primary determinant of its judgedcomplexity. A better approximationwould require some adjustment forrepetitious sequences of elements,mentioned above (see Rashevsky,25, p. 486 ff.; also Attneave, 3, 4).

The major disadvantage of the sys-tem is that some figures (spirals arean obvious case) do not yield uniquedescriptions. This limitation arisesinevitably from the approximationof all curves with straight lines and


arcs, and the ignoring of higher-orderinvariances. It is interesting to specu-late that the system might be to somedegree psychomimetic even in thislimitation, and that objects for whichit does not yield unique descriptionsare less likely to evoke reliable per-ceptual responses, with the result thatthey may be perceived as "amor-phous" or "unstable," and be diffi-cult to remember.

Measuring operations like the fore-going, which involve following abouta contour, are laborious to accom-plish manually, It appears, however,that they are quite amenable toautomation by electronic and me-chanical means. For example, anelectronic contour-follower, describedby Beurle (7), has already been con-structed. A point of light is movedrapidly through a very small circle;when its path crosses the contour, asignal is obtained. The phase rela-tionship of this signal to the circularmovement is used to guide the circlealong the contour, i.e., to move thepoint of light about the contour in acycloidal path. A record of the move-ment of the circle, taken from theservo control loop, constitutes a de-scription of the shape which mayfurther be transformed and analyzedby computer-type circuits.

Measurement of gestalt-variables.We have been considering analyticalsystems by means of which the for-mal properties of contours may bedescribed in detail. Also of interestis another set of variables which donot provide a description from whichthe shape can be reconstructed, butwhich do abstract important prop-erties of the shape as a whole. Weshall refer to these as "gestalt-varia-bles," or "gestalt-measures," evenwhen they serve to summarize somequantizing or analytical process: e.g.,the number of sides in a polygon is

such a variable; so is the number oftri-coordinates necessary to describea shape by the system discussedabove. Likewise, the mean value ofthe c-coordinate in that system mightbe taken as a crude measure of over-all curvedness-vs.-angularity. Itshould be clear that the "statisticalparameters" of populations of shapes,referred to earlier, necessarily pertainto distributions of certain gestalt-measures.

The more restricted notion of agestalt as a system in which everypart is affected by every other parthas been incorporated by Rashevsky(25, p. 451 ff.) into a hypotheticalnerve-net. Suppose that the contourof an object is projected to some sheetof neurons in the cortex as an iso-morphic excitation (Rashevsky'smechanism for contour-abstractionhas already been described). Sup-pose further a distribution of inhibi-tory fibers such that, in the nexthigher projection area, every pointon the contour (i.e., every excitedneuron) receives inhibition from ev-ery other point in an amount whichvaries as a function (presumably de-creasing) of the distance between thepoints. At this level, the variousneurons to which the contour is pro-jected will retain more or less residualexcitation, depending upon the de-gree to which each is isolated fromthe others. A given contour will becharacterized (though not uniquely)by some distribution of residual exci-tations which will be invariant withrespect to its place and orientationin the field (but not with respect toits size). The integral, or mean, ofthis distribution would constitute ameasure of the "simplicity" or com-pactness of the figure (provided sizewere held constant, or corrected for);e.g., a circle would have the highestpossible value, since its points are as


far from one another as is possible ina closed contour, and jagged or sinu-ous shapes would have low values(see Householder, 18). The neuro-logical terms in which this model ispresented need not be taken too seri-ously; Rashevsky's basic idea mightequally well be applied to the pro-gramming of a man-made computer,or to a series of photographic opera-tions.

Deutsch (12) has recently sug-gested a model for shape perceptionwhich is somewhat akin to Rashev-sky's. Since it may be described verysimply in terms of geometrical con-cepts, we shall ignore the neuralmechanisms which Deutsch proposesas its basis. Suppose that a perpen-dicular is drawn to a closed contourat every point along its length. Eachsuch perpendicular will contain asegment which lies inside, and isbounded by, the contour. Thelengths of these segments will havesome distribution which will dependupon the shape of the contour; thisdistribution may be rendered size-invariant by expressing the lengthof each segment as a proportion ofthe length of the contour. In the caseof a circle, a square, or any otherregular polygon with an even numberof sides, the distribution will consistof a single spike, since all the seg-ments will be of equal length.Deutsch suggests that the most prim-itive mechanism of form-discrimina-tion may abstract a distribution ofthis sort; at the human or primatelevel it would obviously need supple-menting with some finer mechanism,perhaps one involving contour-fol-lowing. He points out that rats havemore difficulty discriminating asquare from a circle than from a tri-angle, and predicts further that regu-lar polygons with even numbers ofsides should be more difficult to dis-

criminate from one another thanfrom odd-sided polygons.

Merely to order shapes along acompactness-dispersion continuumrequires nothing so elaborate as theRashevsky model outlined above.The relationship of the perimeter of ashape to its area provides an attrac-tively simple means of measuringthis characteristic. The quotientP/A, which has been employed bysome investigators (8, 17), is unsatis-factory from our standpoint becauseit varies with size as well as withshape, but either P*/A or P/\/A issize-invariant. These ratios may betransformed in various ways to suitthe user's convenience; e.g., the meas-ure

Z>=1-

expresses dispersion as some numberbetween zero and one, assigning zerovalue to the most compact figure pos-sible, the circle. Dispersion (as meas-ured by any such relationship ofperimeter to area) is not the same ascomplexity (in the sense of number ofparts). Although a deeply convolutedor jagged figure will indeed tend tohave a high dispersion value, so willa very thin rectangle or ellipse.

Bitterman, Krauskopf, and Hoch-berg (8, 22) have found that underconditions of low illumination orshort exposure, shapes are perceivedin much the same way as if they werephysically diffused, or blurred. Theseexperimenters created a physical dif-fusion model by cutting filter paperinto various shapes and impregnat-ing it with an inhibitor of bacterialgrowth. This inhibitor was then al-lowed to diffuse from the paper intobacterial cultures. The shapes whichmost resembled each other after dif-fusion were those most often confused


under adverse viewing conditions.Likewise, identification of impover-ished stimuli was most impaired inthe case of shapes characterized byrelatively small detail, which wouldbe averaged out in a diffusion process.

These findings are interesting andimportant, but the clumsy and some-what bizarre bacterial model doesnot lend itself to quantitative predic-tion. There is no apparent reasonwhy it might not be replaced with amodel employing optical blur, inwhich case diffusion would be meas-ured by the radius of the blur circle.An image may readily be blurred toa measurable degree in an ordinaryphotographic enlarger, and then re-sharpened by means of high-contrastpaper or film (cf. Method 5 under"The Construction of Stimuli''). Thisresharpening process introduces an-other parameter, that of the black-white threshold to be used in print-ing. It is easiest photographicallysimply to employ long exposure anddevelopment, with the result that awhite-on-black figure will diffuseoutward into the field to the full ex-tent of the radius of the blur circle.If it is desired that concavities andconvexities be affected symmetri-cally, however (note that a psycho-logical question requiring an empiri-cal answer is thus raised), it is neces-sary to resharpen the image intoblack and white about some inter-mediate gray such that a linear con-tour between black and white fieldswill be restored to its original posi-tion.6 This may be accomplishedwith the aid of a suitable test-figure.

6 Dinneen (13) has succeeded in program-ming a digital computer to perform averaging-and-resharpening operations of almost ex-actly this sort. His paper, which containscopious illustrations of the effect of varyingresharpening threshold, is recommended tothe reader who finds the above discussion in-sufficiently informative.

Over a wide range of values on theresharpening threshold parameter,the process of blurring and resharpen-ing will decrease the dispersion(P*/A, or D) of any shape except acircle, which is already the most com-pact shape possible. For any suchvalue, dispersion will tend to decreaseas amount of blur increases, but theform of this function—which we shallcall a blur-response function—willvary with the shape involved andwill describe certain important char-acteristics of the shape. Since thedecrease in the function is associatedwith the "washing out" of progres-sively larger detail as the blur circleincreases in size, any sharp drop indi-cates that the shape contains con-siderable detail of a magnitude indi-cated by the blur circle at that point.The blur-response function (or, per-haps better, its derivative) is thus apotential aid in the statistical evalua-tion of "magnitude of critical detail,"which Bitterman, et al., found to beof primary importance in determin-ing the identifiability of an impover-ished shape (8). A full exploration ofthe properties of such functions (par-ticularly in the case of shapes char-acterized by certain types of regu-larity, or redundancy) is beyond thescope of this paper; our purpose hereis merely to suggest their feasibilityand possible usefulness. One furtherpoint should be made, however;neither the blur-response functionnor any other gestalt measure canpossibly predict the relative identifi-ability of shapes except in a limited,statistical way. The kinds and de-grees of similarity which an impov-erished shape bears to all the othershapes with which it might be con-fused will clearly affect the difficultywith which it is identified (quiteapart from any intrinsic propertiesit may have), and these similarities


may be evaluated, if at all, only byrecourse to analytical measures. Aparticular detail in a shape may ormay not be critical to identification,depending upon the specific discrimi-nations which identification requires.

Gestalt measures, as defined earlier,all involve a reduction in the dimen-sionality of figures (sometimes, thoughnot necessarily, to a single dimension)with a concomitant discarding of in-formation. The number of operationsby means of which a shape may be"collapsed" to lower dimensionalityis indefinitely large, as Selfridge (26)has recently pointed out. At thesimplest level, for example, we mayliterally collapse a shape upon anyspatial axis by plotting, as a functionof distance along that axis, the thick-ness of the shape in the orthogonaldimension (26, Fig. 3). The axis in-volved need not even be linear; e.g.,it might be a circle about the centerof gravity of the shape (cf. Pitts and

McCulloch, 24).Of all the conceivable physical

measures of shape, analytical as wellas gestalt, there are undoubtedlymany that have little or no valuefrom a psychophysical point of view.On the other hand, it appears un-likely that any single system of physi-cal measurement can be optimal forall psychophysical situations: in otherwords, we are suggesting that formperception involves a number of dif-ferent psychological mechanismswhich function in a complementary,and to some degree overlapping, man-ner. Unfortunately, there is no quickand easy way to determine whichphysical measurements have greatestpsychological relevance; only experi-mentation can answer this question.The preceding discussion and reviewmay at least serve, however, to allevi-ate somewhat the paucity of hypoth-eses which in the past has charac-terized this research area.

REFERENCES

1. ARNOULT, M. D. Recognition of shapesfollowing paired-associate pretraining.Paper read by title at USAF—NRCSci. Sympos., Washington, D. C., No-vember, 1955.

2. ARNOULT. M. D. A comparison of train-ing methods in the recognition of spatialpatterns. USA F Personnel Train. Res.Center, Res. Rep., No. AFPTRC-TN-56-27.

3. ATTNEAVE, F. Some informational as-pects of visual perception. Psychol.Rev., 1954, 61, 183-193.

4. ATTNEAVE, F. Symmetry, information,and memory for patterns. Amer. J,Psychol., 1955, 68, 209-222.

5. ATTNEAVE, F. Physical determinants ofthe judged complexity of shapes. /.exp, Psychol. (in press).

6. ATTNEAVE, F. Effect of familiarizationwith a class-prototype on identification-learning of shapes. Amer. Psychologist,1955, 10, 400-401. (Abstract)

7. BEURLE, R. L. Discussion in W. Jackson(Ed.), Communication theory. New

York: Academic Press, 1953. Pp. 323-326.

8. BlTTERMAN, M. E., KRAUSKOPF, J., &HOCHBERG, J. E. Threshold for visualform: a diffusion model. Amer. J.Psychol., 1954, 67, 205-219.

9. BRUNSWIK, E. Systematic and representa-tive design of psychological experiments:with results in physical and social per-ception. Berkeley: Univer. of Cali-fornia Press, 1947.

10. BRUNSWIK, E. The conceptual frameworkof psychology. Chicago: Univer. of Chi-cago Press, 1952.

11. BRUNSWIK, E., & KAMIYA, J. Ecologicalcue-validity of "proximity" and ofother Gestalt factors. Amer. J. Psychol.,1953, 66, 20-32.

12. DEUTSCH, J. A. A theory of shape recog-nition. Brit. J. Psychol, 1955, 46, 30-37.

13. DINNEEN, G. P. Programming patternrecognition. Cambridge: Massachu-setts Inst. of Technology, Lincoln Lab.,1955.

14. FEHRER, E. V. An investigation of the


learning of visually perceived forms.Amer. J. Psychol., 1935, 47, 187-221.

15. FITTS, P. M., WEINSTEIN, M., RAPPA-PORT, M., ANDERSON, N., & LEONARD,A. J. Stimulus correlates of visual pat-tern recognition. /. exp. Psychol., 1956,51, 1-11.

16. FRENCH, R. S. The discrimination of dotpatterns as a function of number andaverage separation of dots. J. exp.Psychol., 1953, 46, 1-9.

17. HOCHBERG, J. E., GLEITMAN, H., & MAC-BRIDE, P. D. Visual thresholds as afunction of simplicity of form. Amer.Psychol., 1948, 3, 341-342. (Abstract)

18. HOUSEHOLDER, A. S. Concerning Ra-shevsky's theory of the "Gestalt."Butt. Math. Biophys., 1939, 1, 63-73.

19. KAUFMANN, E. L., LORD, M. W., REESE,T. W., & VOLKMANN, J. The discrimi-nation of visual number. Amer. J.Psychol, 1949, 62, 498-525.

20. KLEMMER, E. T., & FRICK, F. C. Assimi-lation of information from dot andmatrix patterns. J. exp. Psychol., 1953,45, 15-19.

21. KOVASZNAY, L. S. G., & JOSEPH, H. M.Processing of two-dimensional patternsby scanning techniques. Science, 1953,118, 475-477.

22. KRAUSKOPF, J., DURYEA, R. A., &BITTERMAN, M. E. Threshold for vis-ual form: further experiments. Amer. J.Psychol., 1954, 67, 427-440.

23. LABERGE, D. L., & LAWRENCE, D. H. Amethod of generating visual forms ofgraded similarity. Amer. Psychologist,1955, 10, 401. (Abstract)

24. PITTS, W., & McCuixocH, W. S. Howwe know universals. The perception ofauditory and visual forms. Bull. Math.Biophys., 1947, 9,127-147.

25. RASHEVSKY, N. Mathematical biophysics.Chicago: Univer. of Chicago Press,1948.

26. SELFRIDGE, O. G. Pattern recognition andlearning. Cambridge: MassachusettsInst. of Technology, Lincoln Lab., 1955.

27. THOMPSON, D. W. Growth and form. NewYork: MacMillan, 1942.

Received March. IP, 1956.

attneave (1956) the quantitative study of shape and

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