TIBOR TARNAI
The lion is a symbol of power, but the meaning of the ball under the paw of the male lion is not clear enough. Some people think that the ball symbolizes the globe or the world, but others think that it represents a god. In many cases, the lions’ ball looks like a winded thread decorated with embroidery and bows (Figure 1b). In Japan, such decorated spherical artifacts are called temari, and in the past, a temari was worshipped as an image of a god. At lion statues in Taiwan, the size of balls sometimes is so big that the lion climbs on it with two paws. Decoration of the balls has a great
variety, and very often its geometry is analogous to that of geodesic domes.
The aim of this paper is to survey the different ball patterns from the
point of view of structural morphology.
(a) (b) Figure
1. (a) Chinese lion in the Summer Palace, Beijing. Eighteenth
or nineteenth century.
Geodesic domes in a broader sense are spherical domes composed of smaller units where the spherical surface is subdivided into triangles not too different from each other (Tarnai, 1996). If the vertices of the triangles are considered as the basis of the construction, then four classes of geodesic domes can be defined. Interestingly enough, most lions’ balls also belong to these four classes. (a) Spherical polyhedra with triangular faces The spherical triangular network directly serves as a base of the geodesic dome where the structure is composed of bars running along the sides of the triangles. A wellknown classical example of these structures is the external layer of the US pavilion at the 1967 Montreal Expo designed by R. Buckminster Fuller and Shoji Sadao. Similar to this type of geodesic domes, the lion’s ball in Figure 1 also provides a triangular subdivision of the spherical surface. It shows clearly the wellknown fact that it is impossible to make a triangular subdivision using only 6valent vertices. We need some vertices with valency less than six. And indeed, we can identify some 5valent vertices in Figure 1b. (b) Spherical polyhedra with trivalent vertices The edge network of this sort of geodesic domes is the topological dual of a spherical triangular network. The structure is usually composed of hexagons and pentagons. The internal layer of the US pavilion at the Montreal Expo has this property. Pavlov (1987) has designed a number of domes of this type. Among the lion statues, for instance, the goldplated male lion in front of the Gate of Heavenly Purity in the Forbidden City, Beijing has a ball belonging to this class. This is a remarkable example, and because of fullerenes, it is of interest to chemists (Hargittai, 1995). (c) Packing of circles on a sphere If in a spherical triangulation, the vertices of the triangles are considered as centres of nonoverlapping equal circles, then a packing of equal circles is obtained on the sphere. Fuller’s fly’s eye domes are based on circle packings where the equal circles appear as circular openings on the dome (Tarnai, 1996). The idea of fly’s eye domes comes from the American sculptor Kenneth Snelson’s artistic atom model which looks like Chinese nested spherical shell carvings, where each layer encased by the others has equal circular holes [K. Snelson’s personal communication to T.T.]. We found lion’s balls with circle packing decorations mainly in Taiwan. Sometimes, the circles are drawn on the surface of the sphere, sometimes they appear as circular holes similar to that in the nested shell carvings. There are examples where the circles are at the vertices of a regular icosahedron or a rhombdodecahedron, or are arranged along parallel small circles of the sphere. (d) Covering the sphere with circles If in (c), the radius of the circles is big enough, then we can obtain a system of circles covering the sphere without gaps. In this case a geodesic dome can be composed of equal spherical caps. Such a scaly dome was designed by Pavlov (1987). At lion’s balls, this kind of decoration is the most common. In Japan, a covering system of equal circles fitted to a square lattice in the plane is called shippo. If this system is intended to be used on the sphere, then the arrangement will be distorted or the size of the circles will not be equal any more. There are known many examples of both cases. (e) Other configurations There are some additional patterns on the lion’s balls which are not directly related to geodesic domes: "melon" configuration defined by meridians, "bamboo framework" where the strips are running along longitudinal and latitudinal circles, and probably many more we are not yet familiar with. References Miyazaki, K. (1986) An Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes, New York: Wiley. Pavlov, G. N. (1987) Compositional formshaping of crystal domes and shells, In: Tarnai, T., ed., Spherical Grid Structures, Budapest: Hungarian Institute for Building Science, 9124. Tarnai, T. (1996) Geodesic domes: Natural and manmade, International Journal of Space Structures, 11, Nos. 12, 13–25.
