PRINT February 1986


Abracadabra: the magic space of supple geometry.

IT HAS BEEN TWO years now since Benoit Mandelbrot’s ideas of “fractal geometry” swept Japan’s computer-graphics circle. Interest in the field, a geometry of geometrically irregular shapes, has decreased recently, not because it has failed to perform in computer-graphics work, but because of its converts’ lack of insight into the connection that must be made between the image to be presented and the meaning of Mandelbrot’s concept. One of fractal geometry’s main weapons is the notion of “infinite self-embeddedness,” the analysis of the infinite number of structures that can be created through geometric repetition of a single form. Consequently, the idea as applied to computer graphics involves a risk of producing no more than a monotonous multiplying pattern. At least in Japan, many computer-graphics designs rendered through fractal geometry are trapped in such patterns; they are completely devoid of the “suppleness” underlying the technique. Japanese computer artists have not been able to grasp the method as well as Americans have—our computer graphics are five years behind the USA’s, certainly in terms of expertise, and more importantly in terms of "supple geometry.”

One of the three Japanese participants in the M. C. Escher Congress held in March of 1985, in Rome, was a young man called Yasushi Kajikawa. Kajikawa calls himself a “science designer,” but to me he is an artist, one whose geometry-inspired ideas make him heir to both M. C. Escher and Buckminster Fuller. Born in Hiroshima, in 1951, Kajikawa was in his 20s a leading leftist radical in the area. When the radical left disintegrated, in the ’70s, he began to explore—or, rather, to pioneer —a new field of design based on applied mathematics. ”If social transformation is impossible,” he believes, "the only alternative is scientific transformation.” Today he is involved with the development of children’s toys incorporating mathematical strategies.

Kajikawa does represent the supple geometry. He makes models that put the 5 Platonic solids—the regular polyhedrons (tetrahedron, cube, octahedron, dodecahedron, and icosahedron)—and the 13 Archimedean, semiregular solids through a variety of serial transformations. The most basic kind of model he designs involves a framework of tubes outlining a polyhedron. Using one of Fuller’s terms, this basic model is entitled “Vector Equilibrium” The tubes are held in place by a single thread that loops through each of them twice, in an arrangement of Kajikawa’s invention (fig. 1). By twisting and turning these models properly one can make a cube first into an octahedron and then into two adjacent tetrahedrons (fig. 2), or a dodecahedron into a tetrahedron, passing through three variant stages of tetrahedrons on the way Moreover, one can make an Archimedean semiregular polyhedron into a Platonic regular one. How is this magic possible?

The principle of suppleness was developed by Fuller, who devised a joint to link four facets of a polyhedron at a vertex. (I still remember that cheerfully omnidisciplinary scientist/artist explaining the principle to me in person.) Each vertex in one of Kajikawa’s polyhedrons is supple, but unlike Fuller’s joint it can adapt to any number of facets (fig. 3). Topology possesses a magic by which the relationship between points in a given space may be set free by doing away with the concept of the distance between them—in other words, the space may retain its topological integrity even while the relationship of one point to another changes within it. Kajikawa has gone farther, finding a magic that changes the shape of a space by freeing its angles while still maintaining the same distances between its given points. He has created a new type of topology and this is why he was besieged with questions from other participants in the Escher Congress.

Like much of Fuller’s work, Kajikawa’s new topology may well develop into a “synergetic” science, drawing on more than one scientific field for its ideas—not only geometry, for example, but also physics or biology It has already discovered a synergy between Platonic and Archimedean forms. Until now, Platonic polyhedrons were appreciated as “solid” concepts, and nobody had thought of them as supple. But this preconception has now been refuted by an obscure designer of children’s toys. The tide is high: scientists have gotten together to develop the ideas of a print artist like Escher, while computer-graphics artists, whether successfully or not, are studying the work of a geometrician, Mandelbrot. Kajikawa has a place in this tide of exploration, and may influence which way it goes.

Seigow Matsuoka is the editor-in-chief of objet magazine YU and a freelance editorial director who lives in Tokyo. He contributes a column on philosophies of craft to Artforum.