Computing Science Pleasures of Plication

mong all the forms of sculpture, the most interesting from a mathematical point of view may well be paper fold ing. In support of this assertion I would point out that paper folding requires a distinctive leap of the imagination from Flatland into Roundworld, from two dimensions to three. A sculptor who carves stone or models clay begins with a medium that is already three-dimensional, but the paper folder must envision the solid object in the flat and formless sheet. This calls for strong geometric intuition. A computer program now under development offers assistance in making the leap into a higher dimension. Given a three-dimensional object, the software attempts to generate a "folding net," a two-dimensional diagram that you can color or otherwise decorate, then print on paper, cut out and fold up to create a three-dimensional model. The program, called HyperGami, is the creation of Michael Eisenberg of the University of Colorado at Boulder and Arm Nishioka, a doctoral student at Colorado. So far it is being used mainly to explore paper models of polyhedra, which are objects that hold much mathematical interest of their own. In addition to serving as a bridge between di mensions, HyperGami spans two worlds in an other way as well. Part of its user interface relies on the direct manipulation of objects—the pointand-click paradigm common in most graphics software today. But there is also a programminglanguage interface, which allows operations on the solids and on their folding nets to be ex pressed algorithmically. Exploring ways of com bining these two styles of computing was one of the original motivations for the project.