Diagram showing the bas icconcept for using the computerto produce typewritten o rvisual output.

X (1)= 100

X (2) = 100

X (3) = -100

X (4) = -100

X (5) = 100

Y-ARRAY

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& [X(5), Y(5)]

X-AXIS

Y(1).= 100 -100 100

Y (2) = -100

Y (3) = -100

Y (4) = 100 -100IX(3), Y(3)] [X(2), Y(2)]Y (5) =. 100

X-ARRAY

Example of drawing a square withthe microfilm plotter: Four pointsare located, each with a valueof 100, i n a quadrant o f the X-Ycoordinate system. An instructionto the microf i lm p lo t ter t oconnect these points would thusproduce a square.

INPUT

(e.g., PUNCHED -CARD READER)

Y-AXIS

MEMORY,

STOREDPROGRAM

DATASTORAGE

ARITHMETICPROCESSOR

OUTPUT

H O ) ( e . g . .TYPEWRITER.CATHODE RAY TUBE)

exposed frame. The next instruction would tellthe microfilm plotter to draw a line between thefive points whose coordinates are contained inthe X and Y arrays. Thus, the point (X(1),Y(1))would be connected to (X(2),Y(2)), (X(2),Y(2))would be connected to (X(3),Y(3)), and so on.The last instruction would once again advancethe film.

The arrays are blocks of sequential locations inthe computer's memory that have been specifiedby the programmer as being associated withsome name. For instance, in the above case theX and Y arrays each contain five storage loca-tions where X(1) refers to the first location whichcontains the number 100.

In the preceding example of a square, the co-ordinates are all known and are therefore putinto the X and Y arrays as constants. For morecomplicated pictures, the program might in -clude instructions for mathematically calculat-ing these coordinates. The essence of all this isthat a picture can be specified numerically as aset of points which can be obtained as resultsof numeric calculations in a computer program.For complicated pictures, the only additionalrequirements would be more points and smallerline segments.

RANDOMNESS

A sequence of numbers would be described asrandom if an observer were unable to determinea formula for exactly predicting each number inthe sequence. These numbers might representthe coordinates of points in a computer-gener-ated picture. However, a practical picture can-not be infinite in size, and therefore some limitor range must be placed upon the set of per-missible random numbers. For example, onlyrandom numbers between —67 and +240 mightbe allowed. Within a permissible set, certainnumbers might occur more often than others, oralternatively the occurrence of all the numbersmight be equally probable.If all the random numbers in a sequence fall be-tween the limits a and b, where b is greater thana, and if the occurrence of all these numbers isequally probable, then the numbers are said tohave a uniform probability density. Such a se-quence is specified by only the limits a and b.The occurrence of any number within these lim-its is just as probable as the occurrence of anyother number.

Aside from the uniform probability density thereis another density of considerable importance.It is called the normal or Gaussian density (afterthe mathematician Gauss who first formulatedit mathematically). For a Gaussian sequence ofrandom numbers, the numbers tend to clusterabout an average. The larger the number com-pared with this average, the less probable is itsoccurrence. Sometimes the Gaussian sequenceis "truncated" so that numbers much larger thanthe average are not allowed. The Gaussian den-sity is also characterized by its standard devia-tion which is a measure of the spread of therandom numbers about the average. For a verylong sequence of Gaussian random numbers,68.3 percent of the numbers fall within plus orminus one standard deviation of the average.95.5 percent within two standard deviations, and99.7 percent within three standard deviations.

Numerical techniques developed for scientificand engineering use are available for generat-ing a sequence of random numbers with either auniform or Gaussian density. These techniqueshave been incorporated into special computerprograms called subroutines which calculatethe specified random sequence.

TWO-DIMENSIONAL PICTURES

A computer program can be written instructingthe machine to compute coordinates of pointswhich, when connected together with straightlines, produce a picture.

Some interesting pictures can b e generatedwith t h e computer by programming differentrandom elements into the picture. In the firstattempts along these lines, the random numbersubroutines for uniform and Gaussian densities

Picture with points determinedby a random process and thenconnected b y the microfi lmplotter.

"Gaussian-Quadratic,"A. Michael Noll, 1965.

were used to generate arrays of numbers whichbecame the coordinates of points. These pointswere then sequentially connected together withstraight lines. The lines in the Gaussian picturecluster about the center since the average ofthe random coordinates was chosen to be thecenter of the picture. I f desired, the computercould generate series of pictures in which thenumber of lines or the standard deviation wasvaried. The best picture could then be chosenfrom the series.

In general, completely random two-dimensionalpictures are not very interesting. However, thecomputer is also able to mix together random-ness and order in mathematically specified pro-portions to achieve a desired effect. The initialattempts at such mixing used Gaussian random-ness for the X-axis coordinates but introduced aspecified and non-random mathematical func-tion fo r generating the Y-axis coordinates. Aparticularly good example o f this mixing ap-proach i s shown i n t h e picture, "GaussianQuadratic". Ninety-nine lines join together 100points whose horizontal positions are Gaussian.The vertical positions increase quadratically,i.e., the first point has a vertical position fromthe bottom of the picture given by 12 + 5x1, thesecond point 22 -+- 5x2, the third point 32 + 5x3,etc. The maximum picture'size is limited to 1024units wide by 1024 units high, and thus the 30thpoint would be off the top of the picture (302 +5x30 = 1050). To prevent this from happening,the vertical positions at the top are reflected tothe bottom of the picture and then continue torise. The result is a line that starts at the bottomof the picture and randomly zigzags to the topin continually increasing steps; a t the top theline is "translated" to the bottom to continue itsrise. The standard deviation o f the Gaussiandensity is 150.The exact proportions of "Gaussian Quadratic"were arrived at in a process of trial and error.The computer very rapidly produced series ofpictures in which the different factors were uni-formly changed. In this manner it became pos-sible to bring an intuitive feeling for the pictorialeffects of these factors into play.

"Vertical-Horizontal" pictures were generatedby a scheme in which only one of the two co-ordinates was changed (alternatingly) from onepoint to the next. The coordinates were other-wise random with a uniform probability density."Vertical-Horizontal No. 1" consists of 50 lineswith equal ranges in both directions; the num-ber of lines in "Vertical-Horizontal No. 2" wasincreased t o 300. "Vertical-Horizontal No. 3 "consists of 100 lines with a range of —200 to+200 along the X-axis and a range of —500 to+500 along the Y-axis.

The preceding examples indicate that the com-puter in association with some method for ob-taining visual output can be used to generate

Computer-produced pictures byA. Michael Noll.

Right: "Vertical-Horizontal No. 1"

Bottom right:"Vertical-HorizontalNo. 2"

Far right: "Vertical-HorizontalNo. 3" A . Michael Noll, 1965: Inthese pictures only one of thetwo coordinate values (X or Y) waschanged from one point to thenext along a continuous line.The change alternated betweenthe X and the Y value. Otherwisethe positions of the points werechosen at random with auniform probabi l i ty density.

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random and quasi-random abstract images. Theartistic merit of the image thus generated is amatter o f personal opinion, bu t i t should betaken into account that the medium was in thiscase strongly limited by the exclusive applica-tion of black straight lines.Many "op art" paintings are very regular andmathematical in design. The computer is ex-tremely adept a t constructing purely mathe-matical pictures and hence should be of con-siderable value to "op" artists. The drudgery ofdrawing or painting complex designs such asthose in moire patterns can be easily done bythe machine. A s an example, Bridget Riley'spainting "Current" is a series of parallel linesthat mathematically can be specified as sinewaves with linearly increasing period. Such aformulation o f her painting enables the com-puter to calculate an array of points based upona simple mathematical formula. The microfilmplotter then connects the points to produce thefinished result.

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The fact that an "op art" picture can be ade-quately produced b y a computer should no tdetract from the artistic merit of either the artistor the picture resulting from the artist-computercollaboration. The creative process takes placein the mind o f the artist; the final painting i sonly the artist's rendition of his mental image.This i s particularly true o f " op ar t " althoughother types of art do involve the artist's inter-action with his materials as part of the creativeprocess.

Since most "op art" is definitely mathematicalit is not at all surprising that the computer canduplicate "op" paintings. However, what couldthe computer do with different forms o f ab-stract paintings?

This problem was approached by trying a quasi-random duplication wi th the computer o f anabstract painting. The painting chosen for thisinvestigation was Piet Mondrian's "Composi-tion With Lines" (1917). This black and white

"Ninety computer-generated sinusoids w i th l inear lyincreasing per iod." The t o p l ine o f th is picture wasmathematically expressed as a sinusoid curve. Thecomputer was then instructed to repeat the l ine 90 times.The result approximates closely Bridget Riley'spainting "Current . "

Bottom left: "Composition withLines," (1917) by Piet Mondrian,@ ilijksmuseum KroIler-Muller.

Bottom right: "ComputerComposition With Lines," byA. Michael Noll in associationwith an IBM 7094 digital computerand General Dynamics SC-4020microfilm plotter, @, A. MichaelNoll, 1965. This compositionapproximates Piet Mondrian's"Composition with Lines" (1917).

painting is from Mondrian's earlier period whenhe was experimenting with representations ofthe vertical and horizontal mot i fs o f nature."Composition With Lines" was chosen becauseit was composed entirely o f solid black barswhich could be drawn very easily by the micro-film plotter.A cursory examination o f Mondrian's paintingreveals that, first, the outline of the painting isnearly circular except for cropping of the sides,top and bottom; second, the bars falling withina region in the upper half of the painting areshortened in length: third, the length and widthof t he bars otherwise appear random; a n dfourth, the placement of the bars is not randombut seems to follow some scheme so that theentire space is almost uniformly covered.

A computer program was written utilizing thefirst three of the observations about Mondrian'spainting. The placement o f the bars was ran-dom with a uniform density. The vertical andhorizontal bars were approximated as a seriesof closely spaced, and therefore overlapping,parallel line segments. These bars were placedrandomly within a circle o f radius 450 units.The choice of vertical o r horizontal bars wasequally likely. The width of the bars was equal-ly probable between 7 and 10 units while thelength o f the bars was equally probable be-tween 10 and 60 units. I f the center of the barfell inside a certain parabolic region with i tsorigin at the center of the circle, the length ofthe bar was reduced by a factor proportional tothe distance o f the bar from the edge o f theparabola. A trial-and-error approach ensuredthat the final picture was reasonably similar tothe Mondrian painting.

In a psychological experiment, a sample of onehundred artistically naive subjects were givenreproductions of the Mondrian and of the com-puter picture.' These people were instructed toindicate which o f the pictures they preferredand to identify the computer picture. In this par-ticular group, 59')/o of the subjects preferred thecomputer picture and 28% identified correctlythe computer pictures, i.e. 72°4, thought theMondrian picture was done by computer. Thesample group was composed of technical andclerical personnel who probably had no biasagainst computers. As indicated by questionarycomments of these subjects, there was a trendamong the non-technical people to associaterandomness with creativity. Therefore, the re-sults of this experiment neither discredit Mon-drian nor imply that the computer is a greaterartist than Mondrian, but raise the question towhat extent randomness has aesthetical andemotional appeal. To answer th i s question,more experiments wi th controlled groups o fsubjects would be required.

The human-or-machine experiment, testing thepreference of the two pictures, one computer-generated, the other by Mondrian, could raisea few more questions. For instance, althoughonly one particular sequence of random num-bers was used by the computer in producingthis picture, i t is not known i f other picturesgenerated b y statistically ident ical randomsequences would be preferred over the Mon-drian. The computer picture would clearly bemore random than the Mondrian painting. How-ever, more elaborate schemes could be usedto produce pictures that even more closely re-semble the Mondrian.

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As a follow-up, another series of Mondrian-likecomputer pictures was generated. The schemeused to produce these pictures utilized randombar lengths and random b a r widths wi th inspecified ranges. The bars were shortened i fthey fell within a parabolic region in the upperhalf of the picture. Only vertical bars were per-mitted along the sides of the picture. The actualpositions of the bars were determined by addinga uniform-density random perturbation t o anotherwise completely uniform grid-like set o fpositions. This random perturbation has a spe-cified range; the range is zero and increasesgeometrically to a range of -± 250.

The conclusion of these investigations of com-puter-generated two-dimensional p ic tures i sthat the exciting potential o f computers in artconsists in their capability o f producing mix-tures of random elements with mathematicallyspecified formulae for order. The experimentsreported here involved only black and whitepictures, but in the very near future color pic-ture tubes will be controlled by computers, andinfinitely variable color mixing will be possible.Presently, any artist desiring to use the com-puter wou ld requ i re a fa i r l y sophisticatedknowledge o f computer programming. How-ever, special "programming languages" tha t

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closely suit the needs of any particular artistcould be developed, and these languages couldbe as natural to use as the conventional brushesand oils. Until recently the time lag between therunning of the computer program and the fin-ished picture has been several hours. Now,however, n e w display devices which imme-diately create an image on a large picture tubeare being made available. Special lightpens arealso available for writing and drawing on theface o f the cathode ray tube and in this wayit is possible to modify the picture. Such de-vices allow the computer-artist to sit at the con-sole o f the machine and to interact with themachine to produce a picture immediately. Inthe future it may even be possible for an artistto rent a console with a display device andwork with a computer over distances. Manypeople would share the same mammoth facili-ties o f a central ut i l i ty computer. A s leisuretime increases and computer costs decrease,the use of time-shared computers for creativeactivities may become quite commonplace.

THREE-DIMENSIONAL PICTURES(COMPUTER SCULPTURE)

That three-dimensional pictures can be gener-ated by the computer and a microfilm plotter by

Perspective projection of a three-dimensional cube onto atwo-dimensional picture plane.

STATIONPOINT

addition of a third dimension, is only a logicalstep further. This has already proved particularlyvaluable in depicting the motions of the basilarmembranes in the ear in response to varioussounds. The third dimension makes many phe-nomena far easier to understand and visualize.

The illusion of depth can be created if minutelydifferent pictures are presented separately tothe left and right eyes. These two pictures areobtained as perspective drawings of some ob-ject seen from two slightly different directions.This technique is formally called stereographicprojection.

In usual practice a perspective drawing is pro-duced by choosing a point (representing the eyeand formally called the station point) from whichthe object is viewed. A picture plane is then in-serted between the object and the station point,and projection lines are drawn from the objectto the station point. The points of intersection ofthese projection lines with the picture plane arejoined together to produce the perspective draw-ing shown below. In this way a three-dimensionalobject is projected onto a two-dimensional plane.

Two perspective drawings obtained from twopicture planes and two station points producewhat i s called a stereographic drawing. Sincethe two station points are located i n slightlydifferent positions, two slightly different per-spectives are obtained. When viewed stereop-tically, these minute differences in the pictures

presented to the left and right eyes are translatedby the brain into a depth effect. The computeris unable to physically draw lines from the objectto the station points, and therefore an analytictreatment of the projection technique is required.The derivation o f these projection formulae isstraightforward.

The equations resulting from the analytic geom-etry approach t o perspective projection havebeen incorporated into a computer program forgenerating perspective pairs which when viewedstereoptically produce a depth i l lusion. T h ethree-dimensional ob jec t i s specified b y therectangular coordinates of its points, i.e., eachpoint in the object is determined by three numer-ical coordinates rather than jus t two as f o rtwo-dimensional objects. The program then cal-culates the projections of these points for boththe left and the right perspectives. The projec-tions are two-dimensional, and therefore onlytwo coordinates are required. The microfi lmplotter draws the left and right perspectives on35-mm microf i lm wh ich can b e viewed i n astandard stereoscope.

In looking at the left and right perspective draw-ings shown adjacent to each other, a depth effectcan be obtained by decoupling one's eyes suffi-ciently to produce double images. This task ismade easier by first gazing beyond the page atsome distant wall or object and then droppingthe eyes without refocusing back to the page.Sometimes a piece of paper placed between andperpendicular to the two perspectives may be

PROJECTIONLINES

CUBE

Three-dimensional picture pair ofa random structure consistingof 50 lines falling at randomwithin a cube. (figure a)

Three-dimensional picture pair of arandom line connecting 50points chosen to fall randomlywithin a cube. (figure b)

Three-dimensional picture pair ofstructure consisting of 30random-length vertical lines and20 random-length horizontallines. (figure c)

Three-dimensional picture pair of85 mutually perpendicularconnected line segments, (figure d)

Far right: Two-dimensionalversions of structures a, b,.c and d.

helpful. The desired result is t o obtain threeseparate images. The center image wi l l looksolid and wil l become clear i f one looks at i tlong enough.

Fourcomputer-generated three-dimensional pic-tures and thei r two-dimensional counterpartswere used as stimuli in a preliminary subjectivetest to determine aesthetic judgments for three-and two-dimensional pictures shown above.The first picture is comprised of 50 lines deter-mined by end points chosen at random to fallwithin a cube so that all locations within thecube are equally probable. The second consistsof 49 line segments connecting sequentially 50random points once again chosen to fall withina cube. Thirty random-length vertical lines andtwenty random-length horizontal lines with endpoints chosen randomly within a cuboid areshown in the third picture. The object shown inthe fourth picture consists of 85 line segmentsconnecting 86 points randomly chosen within acube. The constraint has been added that onlyone coordinate can change from point to point,with the result that the lines are always per-pendicular to each other.

(b)

(c)

(d)

Final subjective ordering of three- and two-dimensionalpatterns. The numerical rating scale is a measure of thepsychological distance between the patterns.

The computer produced three-dimensional pairsfor the four objects in which both perspctiveswere identical: when viewed stereoptically suchpairs look two-dimensional.

Also two-dimensional versions of the four ob-jects were produced by computer. The resultwas that e ight pictures were n o w available:three- and two-dimensional pairs o f four di f-ferent three-dimensional objects. T h e e igh tstimuli were presented t o 52 subjects (9 a r tstudents a n d 4 3 technically-trained people)whose task was to order the stimuli accordingto the i r artistic preferences fo r the pictures.Each subject was tested individually and viewedthe eight stimuli i n a hand-held stereoscope.A simple test was administered to each subjectto be certain that he could fuse the stereo pairsand obtain a satisfactory depth effect.

A statistical analysis o f the data was used toobtain f inal ordering o f the pictures and theassociated rating values as shown at right. Thenumerical rating values are a measure of thepsychological distance between t he different

'stimuli. The results of this experiment indicatethat t h e three-dimensional version o f eachpattern w a s always preferred ove r i t s two-dimensional counterpart as shown r ight A lso ,the lowest rated three-dimensional pattern waspreferred over the highest rated two-dimensionalpattern. In all cases, the disconnected line pat-terns were preferred over the connected l inepatterns. A l l th is would seem to indicate thatthe 52 subjects participating i n the test pre-ferred three-dimensional "disconnected" l i neobjects. Th is i s quite surprising since i t wasexpected tha t most people would prefer themore orderly connected patterns.

This experiment may have important ramifica-tions in the general area o f aesthetics. Onceagain, a s i n t h e Mondrian test , computer-generated patterns were used as stimuli in apsychological test, bu t this t ime some of thestimuli were three dimensional. This means thatit should be possible to give meaningful teststo determine the difference in aesthetic judg-ments between t w o - a n d three-dimensionalpresentations. Computer-generated pictures arecompletely objective and can be constructedfrom mathematically-controlled mix tu res o forder and random elements.

EQUAL PREFERENCE SCALE

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Computer sculpture combiningrandom scattering of linesabout specified, but never drawn,trend lines.

However i t i s quite obvious that considerablymore testing wi l l be required t o answer ade-quately the aesthetic questions raised by theprevious experiment.

As an example of the ease with which order canbe united with randomness, the computer wasprogrammed to generate its version of RichardLippoIdIs sculpture "Orpheus and Apollo" whichhangs in the lobby of New York City's Philhar-monic Hall and is shown below. This work con-sists of long slat plates of brass that have beenhung from the ceiling by thin wires. For all prac-tical purposes, the plates can be represented bysingle straight lines. When Lippold's work is

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so visualized, i t becomes possible to describethe sculpture in terms of imaginary trend linesabout which the bars have been placed. Thecomputer approach was to specify each 'trendline by giving the coordinates of its end points,and the computer then distributed lines ran-domly about this trend line. These lines wererandom distances from the trend line and alsohad random angular positions in space. In theresult, a total o f s ix such trend lines is used(shown above).

The three-dimensional projection program hasthe flexibility of specifying any viewing position.In this way i t is possible to obtain views of acomputer sculpture from any specified positionwithout the necessity o f actually constructingthe sculpture. This facility should be valuablein visualizing complicated sculptures beforethe expense of final construction.

The previous comments about the desirabilityof display consoles for the user to si t at andsee the picture as i t i s being made by thecomputer a re equal ly appl icable t o three-dimensional pictures. Here, the lef t and r ightimages could appear adjacent to each other,and some type o f simple optical device couldbe used t o achieve a depth effect. Facilitieswould a lso b e desirable f o r shaping three-dimensional objects and instantaneously seeingthe results. This might be accomplished withsome form of electronic "gloves" whose posi-tions would be sensed by the computer.

THREE-DIMENSIONAL MOVIES(COMPUTER KINETIC SCULPTURE)

Three-dimensional pictures are definitely static,and some form of dynamic display was thereforeinvestigated next. The result was an extensionof the three-dimensional program to producethree-dimensional movies. A lmost a l l o f t h estatic three-dimensional pictures could haveas easily been made out of wire using conven-tional sculpture techniques. The introductionof randomness and the absence of supportingelements are two advantages of the computersculpture but are really of little importance.

But if the computer can produce a single three-dimensional picture, then i t also can producea series of three-dimensional pictures to make athree-dimensional movie. Now the static charac-ter of the computer sculpture is gone and in itsplace are the almost limitless possibilities o fthree-dimensional movement and shape transi-tions.

The three-dimensional projection programs weremodified so that three-dimensional movies couldbe easily specified. The primary change in theprograms was one which allowed only thoseportions o f the picture that differed from theprevious picture to be projected. ThiS elimina-ted unnecessary calculations by the computerand resulted in faster programs. The microfilmplotter was fitted with a 16-mm camera so thatthe developed f i lm c o u l d b e viewed w i th astandard 16-mm movie projector.

The procedure for generating a three-dimen-sional movie with the computer is to mathema-tically specify the three-dimensional coordinatesof the points in a line representation of the de-sired object. The projection program then com-putes the corresponding points for the left andright perspectives and generates instructions forthe microfilm plotter to draw a single frame ofthe movie. The next position o r shape of theobject is mathematically specified, the perspec-tive points are computed, and instructions aregenerated for producing another frame of themovie. This process is repeated until the moviehas been completed on a frame-by-frame basis.The whole procedure is somewhat similar to theconventional animation process.

Some problems existed at first in obtaining astereoscopic process suitable fo r viewing thefilms. Fortunately, this was solved with a com-mercially available prism device for separatelypolarizing and then superimposing the left andright perspectives when projected on a screen.The viewer wears polarized glasses so that eacheye sees only a single image. The two perspec-tives might also be projected alongside eachother, but the observer would then have to lookcrosseyed a t the two images in order to ob-tain a fused center image.

Selected frames from a computer-generated three-dimensionalmovie of a randomly changingrandom object or a new formof "kinetic sculpture:.

In the f irst attempt a t a computer-generatedthree-dimensional movie the object consistedof 39 line segments sequentially connecting 40points picked a t random (uniform density) tofall within a cube as shown 'at left. At randomlychosen times one of the points is given a newrandom position within the cube, and the twolines attached to i t are instantaneously twistedto new orientations. Th is i s t ru ly a "k inet icsculpture". Finally the computer has producedan art ist ic e ffect unattainable b y any o thermeans. There have been attempts previously byartists to paint by hand three-dimensional films,but attaining the detail o f the computer f i lmwould be extremely time consuming and wouldprobably require a lifetime.

The three-dimensional projections were intro-duced a s a n added improvement o n t w o -dimensional pictures. In a way, a fourth dimen-sion in the form of time was added in the three-dimensional movies. Now a fifth dimension isadded as a computer-generated three-dimen-sional mov ie o f four-spatial-dimensional o b -jects. The world as we know it has three spatialdimensions. However, i t i s qui te possible t ospeak mathematically o f four and even higherspatial dimensions. Indeed, many scientific andtechnical data have four or even more dimen-sions, and cannot be fully plotted or representedin three-dimensional space. However, dimen-sions higher than three can have no physicalsignificance t o u s since we are only able tovisualize three spatial dimensions. In the sameway that a three-dimensional object is specifiedby three coordinates, a four-dimensional ob-ject is specified by four coordinates. In otherwords, i f a l ine representation is made of thefour-dimensional object, then each point is loca-ted in four-dimensional space by four numbers.

Although it is possible mathematically to specifyfour-dimensional objects, it is impossible to seesuch an object. However, i t is possible to pro-duce a three-dimensional projection of a four-dimensional object. Th is i s accomplished b ymathematical formulae similar to those used toproduce a two-dimensional perspective o f athree-dimensional object. The four-dimensionalobject can be rotated in four-dimensional spaceby mathematical operations on the coordinatesrepresenting i ts points. The resulting orienta-tion can then be projected down to three dimen-sions. A l l these operations are mathematicaland hence can be performed by the computer.The result is a three-dimensional movie of thethree-dimension conical projection of a rotatingfour-dimensional object.

The four-dimensional analogue o f t h e cubeis ca l led a four-dimensional hypercube. Acomputer-generated movie of a rotating hyper-cube was made (shown right). The rotationswhich involve the fourth dimension result inthe object appearing t o turn inside out. The

Selected frames from movie ofthe three-dimensionalprojection of a rotating four-dimensional hypercube.

motion itself i s very intriguing and, althoughvery complicated, immediately implies a sophis-ticated generating process.

The programs and mathematical techniques forfour-dimensional projections and rotations arequite general and can easily be extended to evenhigher dimensions.

Three-dimensional movie production would begreatly facilitated by having immediate displayof the movie as i t is being generated by thecomputer. As in the case of three-dimensionalstatic pictures, some form o f optical devicecould be used for viewing the two perspectivesto obtain a depth effect. The user could controldifferent parameters and see their immediateeffects on the movie. As an example, the anglesof rotation of the four-dimensional object mightbe controlled by knobs that the user could turnwhile seeing the effects o f the rotation. Suchdevices wil l very soon be available, and suchexperiments in the presentation of objects anddata will undoubtedly be attempted.

COMPUTER CHOREOGRAPHYStick figure motion on a stage can be shown bya sequence of movie frames. Each stick figureconsists of a single line for the body, a single-line shoulder, and single-line arms. The armpositions are completely variable, and the sizeof each body element can be individually speci-fied. The whole stick figure can be rotated to anyspecified angle and located at any position onthe stage. In one particular example, six figureswere used(illustrated page78). Three move theirarms uniformly up and down. The stage motionis random (any position is equally likely), butonly one coordinate changes each time so thatthe motion i s always parallel t o the edges ofthe stage. The motion from position to positionis at a uniform rate that is individually specifiedfor each figure. At random times all stage motionceases, and the three figures with the movingarms make one complete turn. The effect isreminiscent o f the motion o f atoms in a gas.The stage motion then continues. The three-dimensional movie was generated on the com-puter using the method described in the previoussection.

This example demonstrates the possibilities forintroducing controlled combinations o f orderand randomness. The stick figures used werequite simple bu t did, however, require manydetailed specifications. The extension to moreelaborate stick figures would require new pro-gramming techniques in which movement of thelimbs is combined into basic movements whichmight then be combined on a n even higherlevel. In this way the most complicated dancemotion could be specified as a combination ofrelatively simple movements. Human movementis extremely complex and obtaining the equa-tions for as simple a motion as walking wouldbe formidable. A better attack on this problem

Far right: Selected frames from acomputer-generated three-dimensional movie of stick figurerepresentation of human motionon a stage or a form of"computer choreography."

might b e f o r t h e c o m p u t e r i t se l f t o ana lyzehuman mo t i on us ing devices that have just be-come ava i lab le f o r c o n v e r t i n g p i c t o r i a l d a t ainto mach ine digest ib le data. A l ib rary o f basicmovements c o u l d b e bu i l t u p w i th in t h e c o m -puter, and par t icu lar movements cou ld be thenput together at will.

The d a n c e wor ld i s c u r r e n t l y i n t h e m i d s t o fproblems concerning dance notation. The prob-lem might be likened to a composer who requiresa f u l l s y m p h o n y o rches t ra a t h i s d isposa l i norder to try different orchestrat ions as he thinksof them. When the f inal mus ica l wo rk has beenobtained, each mus ic ian h a s t o remember h isown p a r t s ince the re has been n o m e t h o d o fnotation. In dance, the choreographer requiresthe who le bal let corps to be present for him tocreate t h e bal let , a n d w h e n f in ished n o no ta -tion is available. Some dance notat ions do exist,but t h e y a r e ne i t he r unders tood n o r used b ymost choreographers. A l so , m o s t o f the no ta-tions are so extremely elaborate that t ranscr ip-tion i s very t i m e consuming . Thus , dance n o -tation i s unders tandab ly l i t t l e u s e d a l t h o u g heveryone recogn izes t h e need f o r i t . F i lm , i n -cidentally, i s n o t the answer, s ince th is wou ldonly be a method o f recording one ballet corps'part icular i n te rp re ta t i on o f a w o r k . P e r h a p scomputers m igh t offer a way out o f the presentdilemma.

One solut ion t o the notat ion prob lem wou ld beto use a f i lmed vers ion o f the bal let as input toa compu te r. T h r e e d i f fe ren t camera pos i t ionscould be used simultaneously, and the dancerswould wear l ight sources so that only l ight wouldbe obtained o n the f inal f i lms. T h e f i lms wou ldthen be analyzed by the computer and convertedinto mot ion patterns. These patterns could thenbe t ranslated i n t o a n y des i red d a n c e nota t ionand d rawn o n mic ro f i lm o r o ther sui table hardcopy. H o w e v e r, t h i s a p p r o a c h w o u l d r e q u i r esome sophisticated technology. Instead, changesin t h e present choreograph ic creat ive processmight in the long run be better.

Many choreographers requ i re the actual p res-ence of the bal let corps to immediately performthe dance s teps as the choreographer createsthem. Th is is very wasteful o f the dancers' t imewhich c o u l d b e bet te r spent in rehears ing andfurther t ra in ing. I n a way, i t i s a l so restr ic t ivebecause the resu l t ing ba l le t i s s t rongly relatedto one par t icu lar corps o f dancers. As an al ter-native to this present approach, one can assumethat t he choreographer has a dig i ta l compu te rwith some form o f real-t ime visual display at hisdisposal. Ins tead o f us i ng t h e ba l le t c o r p s ashis c h o r e o g r a p h i c i n s t r u m e n t , t h e c h o r e o g -rapher could in teract wi th the computer dur ingthe creative process. Stick figure representationsof t h e dancers c o u l d appea r i n s o m e fo rm o fthree-dimensional d i s p l a y o n t h e f a c e o f a nelectronic d isp lay tube. The choreographer, bymanipulat ing d i f fe rent bu t tons o n the console,

could c o n t r o l t h e m o v e m e n t a n d p rog ress o fthe ballet. Di fferent dance movements migh t bestored i n t h e compu te r ' s m e m o r y a n d p u t t o -gether a t wil l . Indiv idual movement restr ic t ionsfor each dancer cou ld even be in t roduced in tothe process. A t the comple t ion o f the ballet, al lthe m o v e m e n t s o f a l l t h e d a n c e r s w o u l d b estored within the computer in digital form. Thesemovements cou ld t hen b e automat ica l ly t rans-lated by the compute r in to any desired form o fdance no ta t i on a n d p r o d u c e d a s ou tpu t . T h enotated m o t i o n s w o u l d t h e n b e g i ven t o t h edancers. In th is manner, the choreographer anddancers w o u l d b e m u t u a l l y f r e e d f r o m e a c hother, and each ab le to ind iv idual ly pursue h isown tasks . T h e no ta t i on p r o b l e m s w o u l d b eautomatical ly solved s ince the ba l le t wou ld bebuilt up wi th in the computer and could be easilytranscribed by the computer.

These are qui te radical proposals for the danceworld, a n d m o s t cer ta in ly n o t every cho reog -rapher w o u l d w i s h t o c r e a t e d a n c e w i t h t h einanimate mach ine . H o w e v e r, i f ba l l e t s w e r especified i n compu te r form much research in tothe d a n c e p r o c e s s a n d m o v e m e n t w o u l d b epossible. Some of the quest ions to be answeredby s u c h invest igat ions i nc lude t h e d i f ferencesin s t y l e a n d m o v e m e n t f o r d i f fe rent cho reog -raphers, the relat ion between mus ic and dance,and the mechanics of human movement.

FUTURE POSSIBILITIES

Presently, e q u i p m e n t f o r ob ta in ing v isual d i s -plays w h i l e t h e p r o g r a m i s r u n n i n g o n t h ecomputer is becoming available. With this equip-ment, a n a r t i s t -p rogrammer w i l l b e ab le t o s i tat the console and immediately see the pictor ialresults o f h i s p rogram. I f desired, parameterscan b e changed, a n d t h e n e w p ic tu re w i l l i m -mediately appear on the face of the display tube.In t h i s way, an ar t i s t -p rogrammer can in te rac twith the computer through an intermediary dis-play unit. The next exper iments in computer artwill u n d o u b t e d l y b e conce rned w i t h s u c h i n -teractions.

Suppose some art ist decides that he wou ld l iketo t ry using the computer as an art ist ic medium.Unfortunately, a number of problems would veryrapidly arise. The f i rs t w o u l d be the avai labi l i tyof a compu te r with some form o f visual output .Computers a re present ly s i tuated a t indust r ia lresearch laborator ies o r a t universi t ies and aretherefore somewhat inaccessible to most artistsunless they are associated with a university. Butmost universities do not even have display equip-ment. A l so , compu te r p rog ramming i s a d isc i -pline ent i re ly d i f fe rent f r o m any th ing eve r ap-proached i n m o s t a r t i s t s ' t r a i n i n g . S p e c i a lprogramming languages must be learned beforeone can ever hope t o wr i te a p rog ram f o r t hecomputer. A l l o f th is easily leads to a pessimis-tic a t t i tude toward the fu ture o f compute r art.

Nevertheless, an opt imis t ic approach is reason-able. N e w c o m p u t e r languages w i l l shor t l y b edeveloped t h a t w i l l m a k e i t very easy t o c o m -municate instruct ions to the machine. Computerswith d isp lay equ ipment w i l l a lso become morereadily accessible w i th t he net result that manymore p e o p l e , i n c l u d i n g a r t i s t s , w i l l b e c o m ecomputer o r i en ted . I n t i m e t h i s n e w a r t i s t i cmedium wi l l be exploi ted to p roduce previouslyunknown e f fec ts c o m b i n i n g c o l o r, dep th , m o -tion, and randomness in creative combinat ions.New i n fo rma t i on a b o u t t h e psycho log ica l a s -pects of the aesthetic experience will also result.

The n e w techniques discovered and developedby ar t is ts as they use compute rs cou ld b e fedback to the scient ists who or ig inated the "a r t , "and thus th is cou ld be the beg inn ing o f a newera o f c l o s e r coope ra t i on a n d b e t t e r u n d e r -standing between the arts and science.

REFERENCES

Computer programming:1. Sherman, Philip M. Programming and Coding Digital Com-puters. John Wiley, New York, 1963.2. Bernstein, Jeremy. The Analytical Engine. Random House,New York, 1964.

Computer art:3. Rockman, Arnold, and Mezei, Leslie, ''The Electronic Com-puter as an Artist," Canadian Art, Vol. XXI, No, 6, November/December 1964, pp. 365-367.

Computer music:4. Mathews, M. V., "The Digital Computer as a Musical In-strument," Science, Vol. 142, No. 3592, November 1, 1963,pp. 553-557.5. Pierce, J. R., Mathews, M. V., and Risset, J. C., "FurtherExperiments on the Use of the Computer in Connection withMusic," Gravesano Dialler, No. 27 / 28, November 1965, pp.85-97.

Dance notation:6. Hutchinson, Ann, Labanotation. New Directions Book, NewYork, 1954.

N-dimensional objects:7. Kendall, M. G. A Course in the Geometry of n-Dimensions.Hafner Publishing Company, New York, 1961,8. Coxeter, H. S. M. Regular Polytopes. The MacMillan Com-pany. New York, 1963.

Computer-generated fi lms:9. Zajac, E. E., "Computer-Made Perspective Movies as aScientific a n d Communication Too l , " Communications o fAssociation fo r Computing Machinery, Vol. 7, March 1964.pp. 169-170.10. Knowlton, K. C., "A Computer Technique for ProducingAnimated Movies,- AFIPS Conference Proceedings, Vol. 25.1964, pp. 67-87.11. Nob, A. Michael, "Stereographic Projections by DigitalComputer," Computers and Automation, Vol. 14, No. 5, May1965, pp. 32-34.12. "Computer-Generated Three-Dimensional Movies," Com-puters and Automation, Vol . 14, No, 11, November 1965,pp. 20-23.13. "Human o r Machine: A Subjective Comparison o f PietMondrian's 'Composit ion W i t h L ines ' a n d a Computer-Generated Picture,- The Psychological Record, Vol. 16, No.1, January 1966, pp. 1-10.