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Curves in Traditional Architecture in East Asia Hiroshi Yanai

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? I f so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Dirk

Editor

n e of the pleasures for people travelling to East Asia may be to s e e various curves decorating traditional architecture. Curves are observed everywhere: in gables, in eaves, over the entrances, and sometimes in jacket walls of castles. In particular, roofs play the most important role in their appearance, as the faqade does in European architectures. However, if you watch the curves very careftflly, you will notice the difference among the curves in different regions in East Asia. The curves in Chinese architecture have, in general, stronger curvature than those in Japan. Korean curves are somewhere in between. One might say that they reflect the aesthetic senses of the nations; there must be also some philosophical or religious connotations. Some old procedures are known, in Japan, to draw such curves, which can be translated into western mathematics, although many traditional architects copy and modify older curves nowadays. According to many old manuscripts or textbooks, Japanese curves are most f~equently parabolas. In some cases, they are formed by line segments which result from the process of numerical solution of the differential equation y" = 0 with boundary condition at both ends.

O

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Huylebrouck,

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In other cases, they are approximated by an envelope of stretched strings. Some of the procedures and equations are illustrated in the figures. The author knows little about the mathematical aspects of the curves in China, Korea, and in other countries like Indonesia, Kazakhstan, and Thailand where beautiful and characteristic curves are also observed in architecture. Mathematical forms are basic not only to architectural engineering, but also to the cultural coimotation hidden in architecture. Figures are reprinted with permission from Yanai, H., "Curves of Stone Walls-Mathematics in Style," Communications of the Operations Research Society of Japan 33 (1988), No. 6 (in Japanese) Yanai, H., "Curves in Traditional Japanese Architecture," Communications of the Opera-tions Research Society of Japan 36 (1991), No. 3 (in Japanese) Yanai, H., "Curves in TraditionalArchitecture and Civil Engineering," Forma Vol. 14, No. 4 (in English) Prof. Hiroshi Yanai Faculty of Science and Technology KEIO University 3-14-1, Hiyoshi, Kohokuku Yokohama 223-8522, Japan

Wakayama, Japan

Xian, China Pavilions

where L is the length of the beeline, N is the number of the nodes and 0 ks the angle of the inclination. Drawing Procedure of Gable

Drawing Procedure of Stone Wall

Kanazawa, Japan

VOLUME 23, NUMBER 1, 2001

The Shape of Divinity Kim Williams

he mathematical tourist visiting medieval cathedrals in Europe certainly has a wealth of material to sort out. We have already seen tracery [A] and hammerbeam ceilings [H3] discussed in these pages. I would like to add another item to the list of noteworthy architectural features: the symbolism of geometric shapes, and in par-

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ticular, the vesica piscis. The vesica,

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

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which translated literally means "fish bladder", is also known as a mandorla, Italian for almond. The shape is created by the intersection of two circles of equal diameter, the perimeter of each passing through the center point of the other (Fig. 1). Its frequent appearance in medieval ornamentation, particularly in the sculpted reliefs that fill the arches over entrance portals (an element properly known as a t y m p a n u m ) signals the importance of this shape. It is associated with the themes of ASCENSION or ASSUMPTION, and most usually forms the frame around the figure of Christ, but is used as a frame for the Virgin Mary as well. It is said that originally the almond-like shape represented the cloud which carried the saintly figures into heaven, but it gradually assumed the role of an aura or kind of "glory" [HI: 197]. The first use of the vesica in art appears in the Byzantine art of the 5th or 6th century, when it was used to represent the incarnation of Christ in the womb, in a figure known as a Platytera [HI: 337]. From there the symbol passed into western Europe in the middle ages. The Gothic may be the single most geometric style of architecture. John Harvey made a strong case that the rise of Gothic architecture was directly related to the new availability of Euclid's E/ements, as well as Arabic astronomical texts, in western Europe in about 1120. "It can be no mere accident," Harvey has noted, "that this placing of the world of thought within a strictly scientific framework parallels the sudden rise of the new Gothic art and architecture. [ . . . ] The study of practical geometry was indeed the essence of architectural design" [H2]. When we encounter a geometrical form in Gothic architecture, we are likely encountering a symbol. The iconographic

THE MATHEMATICALINTELLIGENCER9 2001 SPRINGER-VERLAGNEWYORK

content of the vesica, created from~two intersecting circles, could have been the reconciliation of opposing duals, such as terrestrial-celestial and human-divine into a harmonious unity [CG: v.2, 59]. The geometrical properties follow from the simplicity of the construction [K: 54]. The vesica is also important because it could form the basis for a cohesive proportional system (Fig. 1). In Romanesque architecture, Christ is found in a vesica in an altar apse fresco of the Chapel at Berzd-la-Ville, France (ca. 1100 AD); in the tympanum over the entrance portal of Ste. Madeleine, V~zelay, France (ca. 1120 AD); and in the tympanum of the entrance portal of Sant'Trophine, Aries (ca. 1170). In Gothic architecture, Christ appears in a vesica on the west facade of Notre-Dame-la-Grande, Poitiers (ca. 1162), and in the tympanum of the central portal of the west facade of Chartres Cathedral (ca. 1145) (Fig. 2). The Virgin Mary appears in the vesica of the tympanum of the north entrance to the Cathedral of Santa Maria del Fiore in Florence, the socalled "Porta della Mandorla," in a re-

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Figure 1. The vesica piscis, and root-n c o n structions

(drawing b y S. H i s a n o ) .

Figure 2. Christ in the vesica. Central portal, west facade, Chartres Cathedral, France. Photograph by the author.

lief executed by the artist Nanni di Banco in 1418 (Fig. 3). The Virgin also appears in a vesica in the glittering glass mosaics that decorate the medieval faqade of the Cathedral of Orvieto (Fig. 4). One needn't travel further than a good research library to find other medievai examples of vesicas. The armchair tourist will find them in many a medieval illuminated manuscript. As the Gothic age gave way to the Renaissance, ideals and concepts changed, and the vesica appears to have fallen out of favor. One late example appears in the tomb slab commemorating Cosimo de' Medici, designed by Verrocchio in 1467, and laid in the pavement under the crossing of the Basilica of San Lorenzo in Florence [W] (Fig. 5). The tomb slab is noted by historians for its abstract design, because it features a veritable vocabulary of shape, but no figurative imagery. Two vesicas of green porphyry flank a central 3:4:5 rectangle. Cosimo had wished to be eternally present at the celebration of the Eucharist, hence the location of the tomb slab at the foot of the altar (though the actual tomb is in the crypt

Figure 3 (Top). Mary in the vesica. Tympanum, north entrance, Cathedral of Santa Maria del Fiore, Florence. Photograph by the author. Figure 4 (Bottom). Mary in the ves/ca. Mosaics, main facade, Cathedral of Orvieto, Italy. Photograph by the author.

Figure 5. Verrocchio's tombslab for Cosimo de' Medici, Basilica of San Lorenzo, Florence. Drawing by the author from Italian Pavements. Patterns in Space (Houston: Anchorage Press, 1998). Reproduced by permission.

Figure 6. Christ in the vesica, intrados of the entrance portal, Certosa of Pavia, Italy. Photograph by the author.

below), and the appearance of the vesica, the fish-shaped symbol of Christ. However, note that the proportions of these vesicas differ from the "classic" vesica. Where the traditional vesica is circumscribed by a root-3 rectangle, Verrocchio's vesica is circumscribed by a root-2 rectangle. I do not know the reason for this. A last late example of Christ in the vesica is found in the reliefs decorating the intrados (the inner surface) of the arch over the main entrance portal of the Certosa of Pavia (ca. 1497) (Fig. 6). Though I have personally tracked down vesicas in Italy and France, I have not yet done so in other countries.

I hope that other tourists will let me hear of more examples. REFERENCES

[A] Artmann, Benno, "The Cloisters of Hauterive," Mathematical Intelligencer, vol. 13, no. 2, pp. 44-49. [CG] Chevalier, Jean and Alain Gheerbrant, Dizionario dei Simboli, 2 vols. Milan: Biblioteca Universale Rizzoli, 1986. [H1] Hall, James, Dictionary of Subjects and Symbols in Art. London: John Murray, 1974. [H2] Harvey, John, The Medieval Architect. London: Wayland Publishers, 1972. [H3] Horowitz, David, "The English Hammerbeam Roof," Mathematical Intelligencer, vol. 18, no. 4, pp. 61-64.

[K] Kappraff, Jay, Connections: The Geometric Bridge Between Art and Science. New York: McGraw Hill, 1991. [W] Williams, Kim, "Verrocchio's Tombslab for Cosimo de' Medici: Designing with a Mathematical Vocabulary," in Kim Williams, ed. Nexus: Architecture and Mathematics. Fucecchio, Florence: Edizioni dell'Erba, 1996, pp. 191-205.

Kim Williams Via Mazzini 7 50054 Fucecchio Florence, Italy e-mail: [email protected]

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