Feedback would be appreciated - I would like to propose two 3-hour Humanities courses to be offered in the second summer session of 2000 and possibly again in later summers: Hum 309/ MthSc 408: Geometry of Design Hum 309: Changing Perspectives Brief Summary The first of these courses (Design) would deal with the mathematics underlying ornament and perspective drawing and would be taught as a regular lecture/studio course at the Villa. The second course (Perspectives) would use the mathematical base developed in the first course to study changes in artistic style and thinking through out history. The focal point of this course would be in two 3-4 day field trips: the first to Ravenna and Florence and the second to Lausanne, Bern, and Basel in Switzerland. The courses would be taught concurrently throughout the session with most of the time in the first three weeks devoted to DESIGN and the field trips falling in the second three weeks. DESIGN would be a corequisite to PERSPECTIVES. DESIGN could be cross listed as MthSc 408 (Special Topics in Geometry) and could be used to satisfy the math gen ed requirement by substituting for a "terminal" calculus course: 106 for the architects, 207 for English majors. 408 could NOT be used to satisfy a math prereq! Description: Hum 309/ MthSc 408: Geometry of Design There are at least four major topics that such a course could cover: symmetry tiling polyhedra projective geometry (I am attaching a list of resource books dealing with the first three of these topics.) Each of these by itself could be the subject of a complete semester long course, and indeed over the past twelve years, I have at various times chosen each of these as the "special topic" for a MthSc 408 course. Within easy reach of Genoa there is ample material for each of these topics: Symmetry in the mosaics in Ravenna and the ornament of columns, portals and tracery in virtually any cathedral. Tiling patterns in building facades, brick streets, and the floor of San Marco in Venice. Stellated polyhedra depicted in the floor of San Marco, the famous painting of Luca di Pacioli in Naples with polyhedral models in the background, and Etruscan ceremonial dodecahedra. Intuitive perspective techniques in the paintings to Giotto in Assisi and elsewhere and the full emergence of scientific perspective in the works of Renaissance painters. For summer of 2000 I want to focus on symmetry and perspective. The symmetry component would involve learning the classification of linear, planar, and spatial symmetries. Every figure possesses a group of symmetries. The term "group" is a technical term which encapsulates one of the most important concepts in modern mathematics. The abstract study of groups is the main subject matter of MthSc 412. I have always approached groups in 408 in a very concrete, hands-on way to prepare students for the abstraction of 412. Roughly speaking, a group is a "symmetry pattern". In fact, there is a result of group theory called Leonardo's theorem because it was discovered by da Vinci when he was analyzing the possible symmetry patterns that can arise by attaching chapels to the choir of a cathedral. In the DESIGN course, we will learn to recognize the seven frieze groups -- that is, the possible symmetry patterns of linear ornament. Numerous examples abound everywhere: cornice work on buildings, frames around paintings, and ornamented columns are examples. Crowe (a geometer) and Washburn (an archeologist) have used these to classify and date pottery shards from archeological sites. Crowe's HiMAp module (High School Mathematics Applications) is cited in the resources and is a text I often use. Students will also design frieze ornaments of each of the seven types. There are seventeen planar symmetry groups which are usually called "wallpaper" groups because they represent the possible symmetry patterns of wallpaper. These are much more complex than the frieze groups and a complete mathematical treatment is impractical. I will teach the students the crystallographic notation for naming these groups and show them how to use a flow chart (such as the one in Crowe's HiMAp Module) for classifying these patterns. The task of designing patterns of the 17 types will be divided among the class. There are several hundred spatial symmetry patterns, called crystallographic groups because they represent the possible symmetries of crystals. This is way beyond the scope of most undergraduate courses, but I will at least briefly mention some of the main building blocks of these groups -- namely, space filing polyhedra. We all know that we can stack bricks without leaving gaps or holes. But there are many other shapes that can be stacked to exactly fill space. And savvy designers have used these in intriguing ways. For example, there is a "jungle gym" in a playground in Freiburg, Germany, based on the truncated octahedron rather than the cube. The museum in Freiburg also contain celtic jewelry in the shape of a cuboctahedron. My current approach to MthSc408 also involves introducing matrix and complex number representations of symmetries. I will omit these more demanding mathematical topics for Genoa in favor of a "synthetic" approach that is more visual and concrete and leaves time for the second very important topic: perspective. The course will introduce the basic concepts of projective geometry: points at infinity lying on a line at infinity. Drawing exercises will involve taking a geometric configuration and changing which line is considered the line at infinity. The rudiments of perspective drawing will also be taught, but will be limited to simple geometric figures such as a pair of cubes. The humanities aspect of this topic is a main component of the second course. Description: Hum 309: Changing Perspectives We all know that social conditions are reflected in art. The purpose of this course is to further pursue the thesis put forward in my study of the stained glass in Koenigsfelden -- namely, that world view and social conditions are also mirrored in the mathematical aspects of art, specifically, in the use of symmetry and perspective. Rigid societies tend to produce highly symmetrical but flat art. There figures are planar and lack perspective. More flexible societies produce art with greater depth, more perspective, and symmetries that are hidden and more subtle. The mosaics in Ravenna provide an excellent example. Orthodox Byzantium, represented by Justinian, was a rigid, tradition bound society and religiously intolerant. The Arians, however, were relatively recent Christian converts, descended from gothic invaders. They were new to the region and culture, brought fresh impulses, and were religiously tolerant. Under gothic rule, both orthodox and Arian churches were allowed in Ravenna. After Justinian's reconquest, only orthodoxy was permitted. In San Vitale, Justinian and his retinue stare sternly down at the altar from a flat, two-dimensional space. The figures are show full, frontal bilateral symmetry. The mosaic decoration in San Vitale also uses primarily bilateral symmetry in its ornaments. By contrast the Masoleum of Galla Placidia nearby, an Arian structure, is rich and lively, with shading indicating depth in a Greek key border and more subtle symmetries involving half-turns and glide reflections in the ornamentation. This theme can be repeated over and over again. The Romanesque period was a period of turmoil, caused by the lack of central authority and repeated invasions by Huns in the east and Moslems in the south. The oldest stained glass window in Switzerland is a Romanesque madonna, bilaterally symmetric, impersonal, full frontal, without perspective or depth. This contrasts sharply with the late gothic glass at Koenigsfelden where naive perspective is employed, often to amusing effect. Figures are turned in space, but the backgrounds remain stylized with fascinating tilings by geometric shapes. I propose making the first field trip to Florence to visit the birthplace of modern scientific perspective. It is easy to trace its development historically just by walking thru the Uffizi Gallery. The baptistry of the Duomo also has beautiful late Romanesque mosaics. Then on to Ravenna to compare the differences mentioned above in Arian and Orthodox mosaic art. The second field trip would take us up to Swtzerland thru Milan. First stop would be the Romanesque cathedral at Lausanne with its magnificent zodiac window, a perfect illustration of Romanesque stained glass. We might also pause briefly at the Chateau Chillon of Byron's poem to look at the intriguing three-faced representation of the Trinity in the castle chapel. Then on to Zurich. The Flumser Madonna, mentioned above, is inconspicuously housed in the Landesmuseum, just across the street from the main train station. But once in Zurich, a viewing of the Chagall windows in the Paulus Kirche is a must. The Zurich visit could easily be done in three hours, then catch the next train to Basel, stopping in Koenigsfeld (at Brugg-Windisch) on the way. The Basler Munster is a wonderful blend of Gothic towers and Romanesque nave and choir. All the glass is 19th century but there are some nice Gothic frescoes in the crypt. And the Romanesque capitals around the choir are magnificent. We would spent a day or two in Basel, also visiting the international acclaimed Kunstmuseum and possibly the more recently openned Museum fuer Gegenwaertige Kunst (Modern Art). Half a day would be spent in Dornach at the Goetheanum, one of the first modern concrete buildings designed by Rudolf Steiner, artist, educator, and mystic. Steiner stressed the importance of projective geometry in developing flexible thinking and his architecture reminds one of his Spanish contemporary Gaudi. On the return trip we would visit Bern, the home of Paul Klee, whose museum has an extensive collection of Klee's innovative art work. Klee and Einstein both played the violin (Klee well, Einstein poorly), and both lived in Bern at the same time. Einstein developed a physical theory of altered time and space whereas Klee developed a visual expression of altered time and space. Note: Bern-Zurich-Basel form a triangle with trains running every hour between them. They are all roughly one hour apart by train. In summary, I want to take the students through several artistic and cultural revolutions, noting the treatment of perspective and symmetry along the way: Arian to Orthodox in Ravenna. Gothic to Rennaisance in Florence Romanesque to Gothic in Lausanne, Basel, and Koenigsfelden Victorian to "modern" at Dornach and Bern. Dr. Robert E. Jamison O-15 Martin Hall Professor of Mathematical Sciences OFFICE: (864) 656-5219 Clemson University FAX: (864) 656-5230 Clemson, SC 29634-0975 rejam@clemson.edu

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