Leonardo

Perception and Notation: A Core Curriculum in the ArtsAuthor(s): Edward WestSource: Leonardo, Vol. 23, No. 2/3, New Foundations: Classroom Lessons inArt/Science/Technology for the 1990s (1990), pp. 183-188Published by: The MIT PressStable URL: http://www.jstor.org/stable/1578603 .

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PROGRAMS FOR THE COMING ERA

Perception and Notation:

A Core Curriculum in the Arts

Edward West

I he age of specialization has made an edu- cational virtue of knowing more about less. As in Ray Brad- bury's Fahrenheit 451 [ 1 ], we have committed our one text to memory, but tragically we have lost our link to literature. Undergraduate education in America has become the learn- ing of one text.

When art became academic, it was separated from the complexity of the life that had nurtured it. It became a discipline among disciplines, each contributing to a pre- sumed whole that was, in reality, splintered as each disci- pline became self-justifying and resistant to conceptual chal- lenges. Art, physically separated from other disciplines, was further divided internally into various media.

With few curricular bridges, dialogue, of necessity, di- minishes. We are left with fragments-courses from which the student is to surmise a whole, but which by their discrete- ness only confirm that the film maker and the painter (for instance) are like the biologist and the botanist-both deal with life, but each deals with a separate life.

General education (history, science, language) is viewed by many art majors as a barrier to art study. Their art studios have not clarified and reinforced the role that general study might play in their art. We as educators have been content to assume that we need only address their art-specific skills, leaving their general education to others. We must realize that an approach that ignores synthetic educational ex- periences places our students in an experiential ghetto.

Fig. 1. Nathan Lerner, Eggs and Box, photograph, 111/4 x 131/4 in, 1938. (Courtesy Nathan Lerner) The creation of spatial sensation through the reduction of light is the focus of Lerner's light-box experiments. Lerner describes these experiments in Ref. [13].

? 1990 ISAST Pergamon Press plc. Printed in Great Britain. 0024-094X/90 $3.00+0.00

Synthesis is a goal of art. For artists, borders must never be boundaries. It is therefore in- cumbent on us as art educators to create an interdisciplinary curricular model that reveals the prismatic nature of ex- perience in the arts, a model that will combine not only the media options within the arts but also other disciplines in order to reassert the com- monality of principles in the arts and to reinforce concepts of scholarship for artists.

'Things are not difficult to accomplish", Brancusi advises us. "What is difficult is to pre- pare ourselves to do them" [2]. The majority of art programs offer 'foundation' curricula as

ABSTRACT

The author discusses the creation of a core curriculum of interdisciplinary study for artists implemented at the University of Michigan and adapted by the Uni- versity of Hawaii to replace a tradi- tional art foundation. Addressing the cognitive development of students, the author describes Perception and Notation as a series of three studio seminars- Light, Space and Time-that stress the fundamental relatedness of principles for all visual artists.

the only intersection for all media. It is this arena, then, that provides the primary opportunity to reinforce synthetic education as fundamental preparation for all art majors.

Traditionally, foundation preparation in the arts empha- sizes drawing and design, as if these skills were the sole basis for subsequent study. Such a view ignores both the reality of art making in the late twentieth century and the aspirations of a diverse student population. Furthermore, as a curricu- lar image it is inappropriate. A foundation is a horizontal base, prefigured in time, location, size and shape to accom- modate the building of a future structure. It is a fixed image, constant in form, that predicts that we will all build the same house, or at least that our houses will conform to a single foundational frame. Such a rigid image is, finally, inade- quate to address a future that we cannot predict.

In the debate that preceded the adoption of this cur- riculum at the University of Michigan and the University of Hawaii, we, as faculty, accepted the premise that as artists we share with one another our existence as sensing beings who, in order to transmit our thoughts, visually notate our ex- perience. This idea allowed us to shift our attention away from particular media toward a holistic concern with per- ception and notation as the thematic loci of fundamental art education.

Implicit in such a shift is a redefinition of faculty mem- bers' roles in the education of the apprentice artist. The

Edward West (educator), School of Art, University of Michigan, Ann Arbor, MI 48109, U.S.A.

Received 10 November 1988.

LEONARDO, Vol. 23, No. 2/3, pp. 183-188, 1990 183

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teacher's involvement with the student as a discrete perceptual center, a special 'framing' of both biological and cul- tural components, encourages the stu- dent as an individual with a unique potential for artistic contribution. A shift from media-focused study to per- ceptual study stresses process over pro- duct and allows the student the free- dom to be curious and therefore to experience the joy of discovery.

Growth is the gathering of strength through adaptive change. If education is to be organic, it must discard its dis- crete notions of what constitutes ade- quate information for the art student. It must recognize that art requires the integration of ideas, not just the acqui- sition of skills.

If apprentice artists are to grow, they must each seek a pattern of adaptive change, an organic process that allows them to be both responsive and re- sponsible. To ensure a richness of re- sponse to the particular needs of the emerging artist, the courses in this cur- riculum are, ideally, team taught.

Perception and Notation is meant to serve as a touchstone of interdiscipli- nary thought for artists throughout their education. The individual courses are studio seminars that reinforce writ- ten and oral skills and assist research and concept development in conjunc- tion with studio practice.

The three courses that constitute the core of visual study are (1) Perception and Notation: Light, (2) Perception and Notation: Space and (3) Percep- tion and Notation: Time. There is no set sequence; students choose the course that is appropriate to their re- spective programs of study.

The descriptions that follow are an attempt to outline a progression of thought that can be adapted to the needs of individual art departments. As an open matrix, the curriculum is meant to illustrate the possibilities for a synthetic art education.

PERCEPTION AND

NOTATION: LIGHT

The dominance of painting and draw- ing within the vast majority of univer- sity-level art programs has resulted in courses that teach value as drawing and color as pigment. The inclusiveness of this particular course on light promotes a balanced structure of inquiry for stu- dents interested in different media. The course is designed as an investiga- tion of vision as the dominant vehicle

of perception and of light as the agent of that perception.

Our naive experience of light is of the sky as a luminous dome suspended above us. The sky is a vast theater of changing events, and light is its animate force with many faces (e.g. the rainbow, lightning). The joy and terror that we experience in the sky's light are the sensual beginnings of our inquiry into the nature of light.

Intellectually, we understand that the sun is the major source of the illumination that we experience, and that it in part determines day and night, the seasons and our sensations of hot and cold. But it is through light embodied, on earth, as fire that we intuitively understand, and obtain a measure of control over, incandes- cence and radiant energy.

Fire is the primal force that is utilized in this course, beyond language and culture, to unite the students in won- der. The students are exposed to art- works that seize the emotional force of fire, and they are then challenged to channel this energy into artistic expres- sion. The artist is the primal human, and fire is his or her first medium. Class visits to glass and ceramics studios rein- force energy as the subject of study. Pyrotechnics involve the students in the science of fire. Incandescence is con- trasted with luminescence (phospho- rescence and fluorescence).

When fire was tamed by means of the filament, we achieved control of light and its possible qualification. Artists who utilize the modulation of light as their vehicle of expression address the artistic qualification of light. In this course, stage lighting is examined as an essential component of theater expe- rience. The pioneering work of Nathan Lerner with his light-box experiments (Fig. 1) exemplifies the creation of an individualized theater of light.

The primary sensation of light is de- fined by the polarity of light and dark. From our experience of the levels of luminosity we create achromatic value as our notational response. Value is ex- pressed through photographic mate- rials to demonstrate transparency and opacity and therefore transmission and reflection. Rembrandt's work illu- minates a study of relative brightness and links perception of light to nota- tion. Chiaroscuro is discussed as the artistic articulation of light and shadow. The works of Caravaggio and Titian are studied as exemplars of artistic concern for the perception and notation of light.

Projected light and cast and attached shadows are utilized in the discussion of figure/ground relationships. Cast sha- dows are used in silhouette studies to express shape, while artful manipula- tion of shadows demonstrates trans- formations of figures. Attached-shadow gradients and brightness create vol- ume. Students are instructed to create works that incorporate attached sha- dows as spatial components of three-di- mensional compositions and therefore to consider light and object as a unified experience.

The viewer's position in space is used to study ambient light, light available at a point in space, i.e. the viewer's posi- tion as well as the ocular display af- forded by that position. Students may initially fail to grasp the importance of the study of our ocular capabilities (i.e. vision), but by linking the revelations of science to artistic articulations (e.g. David Hockney's photo collages based on saccadic eye movement), students can appreciate the interaction of art, science and technology. Pinhole pho- tography is useful to demonstrate this interdependence of disciplines.

The physics of light, as discussed in Newton's Opticks [3], is introduced to allow the students to establish expecta- tions that are prerequisite to their ex- pressive control of light. The principles of dispersion, refraction and selective absorption provide a foundation for subsequent study. The artistic explora- tion of these principles is evidenced in the work of the Impressionists and the Pointillists (e.g. Monet and Seurat).

The electromagnetic spectrum is the Rosetta stone of radiant energy, de- lineating the palette of possibilities from radio waves to infrared and visible light. It is hoped that even a rudimen- tary, scientific understanding of how the physical properties of light are mea- sured will inform and alter the stu- dents' aesthetic responses and open new vistas of artistic expression, such as those available in the area of hologra- phy.

Color is considered a specialized as- pect of light and vision, as manifest in wavelength. But again, it is sensation that precedes both measurement and notation. Red is used to introduce the basic qualifications of color, e.g. hue and saturation. Although the ability to see red is not culture-bound, our assign- ment of qualities to red reveals cultural assumptions. Culturally codified color is discussed as a visual language, the syntax of which modifies our responses to light [4]. Impressions of the warmth

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and coldness of colors are pursued in student projects. Color is classified through the Ostwald, Munsell and Goethe systems [5].

The ocular perception of color is linked to our study of tri-color theory. Additive and subtractive aspects of light and pigment are examined in the course. Albers's [6] extensive study of the interaction of colors establishes a vocabulary for the use of color as a visual language. The student is intro- duced to the notation of this language as it is expressed in the various dialects of the arts (photography's color recep- tors, fiber's dyes, paint's pigment, ce- ramists' glazes, printers' ink, sculptors' patinas and oxides).

In this overview of light and color we assume that the art major will en- counter specialized study of these top- ics in media programs, and we acknow- ledge that filling in the landscape is the process of education and the con- sequence of experience. As a begin- ning, the course is an open field. Each student provides a unique closure through his or her educational choices.

PERCEPTION AND NOTATION: SPACE

The point of departure for our study of space is the individual as the locus of perception. Piaget and Inhelder's cognitive developmental study [7] of the progressive acquisition of spatial constructs-from topological, through projective, to Euclidean space-is util- ized in the organization of the course.

Students each select a 'vantage point' in the landscape from which to respond to subsequent information and assign- ments. They are introduced to topo- logic concerns for proximity, separa- tion, order, enclosure and continuity, which are defined through tactile, audi- tory, olfactory and visual cues drawn from our environment. Studies of terri- toriality confirm that personal space is determined by a mixture of percep- tions from all five of our physical senses. The development of personal space re- veals perceived qualitative distinctions (e.g. sacred place, haven) between different locales.

The curriculum restores a balance between vision and the other senses by giving assignments that emphasize the contribution of touch, taste, smell and sound to our experience of space. Through haptic perception (touch without sight) the students tactually encounter shape; they are asked to

Fig. 2. Marshall Island Navigational Stick Chart, ca. 1900. (Museum fur Volkerkunde, Munich, West Germany. Photo courtesy Bishop Museum, Honolulu, HI.) These elegant objects, which served as training maps for young ocean navigators, instructively illustrate for students the notation of spatial planes.

make drawings from their haptic experience. Our auditory experiences contribute to our conception of spatial distance. Vocal exercises and perform- ances help establish the perceptual field of our attention. And insofar as smell is a primary link to memory, stu- dents are asked to exercise their per- sonal memories to reveal the impact of this sense on recall.

Our most intimate physical contact, beyond that with our own bodies, is with the bodies of others. In this type of re- lationship we observe structural simi- larities (symmetry) and see ourselves as one of many. Body language (gestural communication) and territoriality are social expressions of how we are in re- lation to one another. The subtlety of these issues of relation is addressed through discussion of the work of per- formance artists.

The progression from topologic space to projective space is character- ized by the dominance of vision over the other senses. This transition is visu- ally exemplified in the change from Naive, Medieval and Persian art, as ex- amples of topologic concerns, to Ren- aissance art, where projective space emerges as the dominant form. Stu- dents are first asked to make drawings from their chosen vantage points to reveal relational concerns (e.g. in front of, behind) and then to extend their work into perspective studies.

Binocular and monocular optical models illustrate depth cues and pre- cede studies in perspective. Western perspective, as the static expression of projective space, is explored and con- trasted with the Navajos' world of forces and actions to provide an important cultural counterpoint. Further cultural contrast is provided through examina- tion of Eastern notations of space.

The projective 'line' of vision is further extended into the visual 'plane' through scanning, which partially explains our horizontal bias in spatial representation. This construction of a plane from a line is done in the studio through the use of woven structures.

Plaited and knotted forms are later used as expressions of lines amassed as volumes.

Experientially, the students are asked to delimit the planar surfaces (e.g. bed, football field, plaza, neigh- borhood, nation) that they see as the arenas for their actions. Movement across these 'planes' permits inquiry into kinesthesis and perspective trans- formation. Our movement between ob- jects (events) becomes our measure of distance, linking time to space.

The polarities of the cardinal points, east-west and north-south, are ex- perienced in our bodies as left-right and in front of (forward)-behind (back- ward), respectively. As such, the cardi- nal points echo the structural ori- entation of our bodies and are our primary codes for orientation in the landscape, our guides to navigation. The need to venture, to extend our- selves into the landscape and safely re- turn to our respective points of depar- ture, is the beginning of the study of Euclidean space.

Such operations as foraging suggest the utilization of different points in the landscape and the need for under- standing their relative positions within our defined planes. Such orientation requires that we codify what we learn from our passage through space, to document for ourselves and for others spatial relationships observed in the landscape.

Mazes and labyrinths are presented to the students as means of inducing disorientation in space and confound- ing navigation. Students are trained in mapping to give them the rudiments of scale and coordinate systems (geo- metric constructs) as well as the no- tational tools associated with cartog- raphy's rich visual tradition. The students use compasses to orient them- selves to the cardinal points and sex- tants to orient themselves to fixed ob- jects as they map their planes. This mapping task is no less than the con- struction of a world wherein complex spatial relationships must be elegantly

West, Perception and Notation: A Core Curriculum in the Arts 185

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Fig. 3. Mamoru Sato, Fieldscape 1983.3, aluminum and stainless steel, 2Y2 x 10 x 5 ft, 1983; installed Hilo Hospital, Hilo, HI. (Photo: Mamoru Sato) This kinetic sculpture melds engineered precision with aesthetic pleasure. The development of multiple skills is a focus of the course of study.

expressed to be of use. Marshall Island stick charts (Fig. 2) illustrate such ele- gance through an economy of means and a directness of form.

Measurement is fundamental to mapping. In our encounter with the world we impose the idiosyncratic mea- sure of our own bodies (feet long, hands high). Marcel Duchamp's Stand- ard Stoppages is the license for students to develop personal systems of meas- urement that can be used in the di- vision of their planar structures.

This measurement of personal planes in turn is linked to the structural study of natural forms and to the con- cept of the module. Our sense of pro- portion, the elegance of division, is con- firmed in the Fibonacci series and the Golden Section. The dialogue between the three-dimensional experience of space and its two-dimensional no- tational expression is facilitated by com- puter-aided design and drafting sys- tems, which allow students to progress rapidly in their understanding of the interrelationship between projective and Euclidean space.

Architecture is a primary expression of the culturally bound sense of vol- ume. Alexander and others [8] reveal that social and architectonic structures are a consequence of the felt need for and the expression of evolved order. Architecture provides a rich experi- ence of volumes. Students are sensi- tized to our built environment and in- troduced to the concept of site-specific structures. They are then asked to pro- pose the construction of a 'volume' on

their respective planes, expressed first as a two-dimensional notation and sec- ond as a scale model to allow the dimen- sional expression of the proposed space-a fictional architecture, a theater to allow the students to choose vantage points and qualities of re- lational interaction between actors and objects.

Most of the studies and exercises up to this point in the course have been earthbound. We thus conclude the course with studies of the sky and cosmologic space. Historically, human- ity's search for order in the universe has led to the anthropomorphization of the constellations as entities that populate and rule the night sky. We perceive the cosmos as infinity and we reach out with telescopes and radio waves and space- ships in our exploration of the dark expanse. Students in the course are thus asked to extend their earthly con- structs to an astral arena by creating their own constellations, imposing their own imaginations on the sky and cosmos. The works of Otto Peine here suggest the viability of cosmologic space as an arena for artists to unite the two worlds of the earth below and the sky above [9].

If we as artists are to reveal our com- plex relationship to space, we must be explorers, willing to extend ourselves beyond our perceived boundaries in a search for the connection between our individual point and the larger land- scape. Conceptually, students are chal- lenged to broaden their intellectual landscapes to allow confident move-

ments into the world to claim larger arenas of operation for their lives.

PERCEPTION AND NOTATION: TIME

Light and space are more directly per- ceptible than time. Students often believe that time is too ephemeral for substantive study or too esoteric for in- clusion in their art training. This course is designed to introduce temporal study as a fundamental in the training of artists. The aim is to help students rec- ognize, in a process of inner trans- formation, a temporal experience of 'becoming'. As educators we must be willing to forego the immediate gratification of an object-oriented cur- riculum for an investment in the growth pattern of a process-oriented education.

In an attempt to make time palpable, we begin the class with the concepts of time as succession and as duration. Ver- nacular expressions such as 'two shakes of a lamb's tail' and 'a stitch in time' are utilized in visual charades to induce the student to link images to concepts of succession and duration. The student seeks out symbols of temporal states in the popular culture (advertising, traffic lights, etc.) to document the existing visual language for temporal states.

An active world surrounds us in the external rhythms of nature and in the internal pulses of our arteries. The cycles of night and day and of the sea- sons reveal the rhythm of our lives and provide the basic timekeeping divisions that express the successive and du- rational states of our existence. Students examine timekeepers as the visual representations of temporal division.

From cycles of life and death we come to experience our lives as dura- tions divided by past, present and fu- ture. The past is divided, for the pur- pose of study, into the historic past (the territory of the archaeologist and the historian) and the personal past (the territory of memory). The historic past is known from data fragments garnered through research. The function of research is reinforced for the student through the recreation of a chosen his- torical period in drawings of event frag- ments, chronologies of pertinent data, descriptions of physical environments and so on to illustrate the role of the historian as one who gives fo.rm to past occurrences.

The personal past is the rich soil

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worked by Freud in his revolutionary theories of the relationship of our past to our present and future. His revela- tions altered our conception of our- selves and became the source of the Surrealists' involvement with the sub- conscious and dream imagery. Stu- dents are asked to describe fully in writ- ten form incidents from their personal pasts or to illustrate images from their dreams. These accounts are then used in visual and narrative storyboard as- signments. Journals are the written no- tations over time of responses to our experiences. Diaries and artists' jour- nals are explored to suggest the com- plex mix of ideas and sensory input that informs the process of realizing an art- work.

Photography preserves our faces and gives visual access to our ancestors. The phonograph allows us to "speak for- ward in time to the unknown and listen backward to the dead" [10]. We ex- plore mark making as the expression of our desire to be survived by an image that we have created.

Memory is the presence of the past in our 'now'; it allows us to consider the past in relation to our present actions. The present is distinguished by our attention; interactive computer games are used in this course to introduce the computer as a partner in inquiry and to stress the role of focused attention in our movement through time.

The performing arts are distin- guished by the creation of sensations in the present, ephemeral offerings to audiences. This kind of art is consi- dered through the historical contribu- tions of the Dadaists and the Surrealists and through contemporary work rooted in ritual. The students are asked to perform in ways that assert the pre- sent. Humor and surprise are charac- teristics of the present that help the students structure their performances. Shock as a tradition in the arts (e.g. as employed by Duchamp and Man Ray) is examined to reveal new terrain for artistic activity.

Technological advances have altered our present in a complex way and made it a simultaneity of events, suggesting a collage of pieces that compose a larger whole. Television is the perfect illus- tration of multiple realities presented simultaneously. People now expect a plurality of expression as the norm and find it in music (fusion), architecture (postmodernism), painting (cubism) and film (contrast editing).

But the present is most dramatically experienced in moments of terror or

rapture. The student experiences art- works that aspire to the exaltation of the present (e.g. Bernini's Ecstacy of St. Theresa ). The Chinese Book of Changes [11] is introduced to examine the be- lief that past, present and future can all be contained within one moment. And film is examined as a form that asserts the present tense through simulta- neous action, exemplified in another medium in the works of writers such as Alain Robbe-Grillet and Gertrude Stein.

Metamorphosis, the experience of our own becoming, prefigures a pre- sumed future. We anticipate the changes to come and through anticipa- tion form our respective futures. The future is anticipated by activity and by change, therefore motion and speed become central metaphors of the mod- ern. The Futurists, recognizing that motion has a strong visual appeal, turned away from the past in favor of a future of machines (e.g. automobiles and trains) racing into a streamlined tomorrow.

Our romance with speed is ongoing. From the Impressionists' concern with moving subjects to our quest to break speed records, we accelerate toward the biological limits of our capacity to expe- rience speed. Motion perception and kinesthetics provide the informational base for the subsequent notation of motion. In the course, motion per- ception is explored through film, flip books, comic book illustrations, zoe- trope strips and other forms of suc- cessive representation of motion. The study of kinetic sculptures is ap- proached through toys that reveal simple machine motion. The machine aesthetic is discussed, including George Rickey's use of aerial and maritime navigation in his kinetic pieces [12]. In Mamoru Sato's Fieldscape 1983.3 (Fig. 3), the natural manipulation of the wind simulates the experience of the open agricultural fields of Hawaii.

Musical scores are expressions of in- tended action, temporal models for fu- ture activity. The student encounters musical structure, dance notation and other forms of scripts for the future and employs storyboards to plan future visual, auditory and temporal ex- periences.

Overall, the success of the course is measured by the students' utilization of the past as a means to occupy the present fully and to anticipate the fu- ture, which is an open field of possibili- ties.

CONCLUSION

Our early success in implementing this curriculum has been gratifying, but I am still not satisfied. I hope to find other programs and individuals who engage in related strategies for im- proving the learning environment. In- creasingly, art curricula should become reflections of our composite under- standing of the diverse media of expres- sion, not reflections of what is merely convenient for us to teach. If we are to save art education, we must bolster our inquiries with challenges that instruct across media lines.

References

1. Ray Bradbury, Fahrenheit 451 (New York: Simon and Schuster, 1967).

2. Constantin Brancusi, "Aphorisms", This Quarter 1, No. 1 (1925) p. 236.

3. Isaac Newton, Opticks (New York: Dover, 1952).

4. Ikko Tanaka and Kazuko Koike, eds., Japanese Coloring (Tokyo: Libro Port, 1982).

5. Wilhelm Ostwald, Colour Science, Vols. 1-2, J. Scott Taylor, trans. (London: Winsor and Newton, 1931, 1933); A. H. Munsell, Munsell Book of Color, Vols. 1-2 (Baltimore, MD: Munsell Color Com- panyv, 1929,1942) ;J. W. von Goethe Theory of Colors, C. L. Eastlake, trans. (London: John Murray, 1840).

6. Josef Albers, The Interaction of Color (New Haven, CT: Yale Univ. Press, 1971).

7. Jean Piaget and Barbel Inhelder, The Child's Conception of Space (London: Routledge and Kegan Paul, 1956).

8. C. Alexander, et al., Pattern Language (New York: Oxford Univ. Press, 1977).

9. Otto Peine and Elizabeth Goldring, eds., Center- beam (Cambridge, MA: MIT Press, 1980).

10. G. S. Lee, cited in Stephen Kern, The Culture of Time and Space, 1880-1918 (Cambridge, MA: Har- vard Univ. Press, 1983) p. 39.

11. R. Wilheim and C. Baynes, trans., TheIChingor Book of Changes (Princeton, NJ: Princeton Univ. Press, 1967).

12. George Rickey and Nan Rosenthal, George Rickey (New York: Abrams, 1977).

13. Laszlo Moholy-Nagy, ed., Vision in Motion (Chi- cago: P. Theobold, 1947) p. 200.

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GLOSSARY

Fibonacci series-a system of expanding relation- ships connected with the Golden Section, repre- sented in the spiral of the nautilus shell. Leonardo of Pisa, called Fibonacci (1175-1230), discovered that if a ladder of whole numbers is constructed so that each number on the right-hand side is the sum of the number pair on the preceding rung, the mathematical ratio between the two numbers on the same rung rapidly approaches the Golden Sec- tion.

Golden Section-also known as the Golden Mean, a basis of proportion consisting of the division of a straight line into two parts in such a way that the proportion of the smaller to the greater is the same as the greater to the whole. The exact proportion is 0.618:1, roughly 3 to 5 or 5 to 8. The Golden Section was formulated by Vitruvius and appeared again in 1509 when Luca Pacioli published The Divine Proportion in Venice, with illustrations by Leonardo da Vinci.

zoetrope strips-precursor of the motion picture; motion created through animation of static im- ages.

188 West, Perception and Notation: A Core Curriculum in the Arts

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