Analogies from 2D to 3D - Exercises in Disciplined Creativity

Carlo H. Séquin
Bridges: Mathematical Connections in Art, Music, and Science (1999)
Pages 161–172


Human creativity relies to a large part on our ability to recognize and match patterns, to transpose these patterns into different domains, and to find analogies in new domains to known facts in old domains. In the realm of geometrical proofs and geometrical art, such analogies can carry concepts and methods from spaces that are easy to deal with, e.g. drawings in a 2-dimensional plane, into higher dimensions where model making and visualization are much harder to carry out. Students in a graduate course on geometric modeling are challenged with open-ended design exercises that introduce them to this analogical reasoning and, hopefully, enhance their creative thinking abilities. Examples include: constructing a Hilbert curve in 3D-space, finding an analogous constellation to the Borromean rings with four or more loops, or developing 3D shapes that capture the essence of the 2D Yin-Yang figure or of a logarithmic spiral. The proffered solutions lead to interesting discussions of fundamental issues concerning acceptable analogies, the role of symmetry, degrees of freedom, and evaluation criteria to compare the relative merits of the different proposals. In many cases, the solutions can .also be developed into attractive geometrical sculptures.