O’Rourke on Geometric Reductionism

Editor’s note: This article was initially published in The Daily Gazette, Swarthmore’s online, daily newspaper founded in Fall 1996. As of Fall 2018, the DG has merged with The Phoenix. See the about page to read more about the DG.

As part of this year’s Math Department lecture series, Philadelphia-native and Smith College professor Joseph O’ Rourke returned this Tuesday to present his lecture entitled “Geometric Folding Algorithms: Linkages, Origami, and Polyhedra.” The talk focused on the computational geometry behind the deconstruction of one, two, and three-dimensional figures.

This type of mathematics, in essence, involves reducing complicated structures into a simpler form. O’ Rourke, a member of both math and computer science departments at Smith, specializes in this sort of geometric reductionism. As author of several texts on computational geometry, O’Rourke says he has had “plenty of practice” using specific algorithms and programs to deduce (for instance) the unfolded flat paper model of a complex, irregular polyhedron and illustrate that work.

Over the course of the talk, O’Rourke used models as well as technological demonstrations to discuss different algorithmic techniques, modes of representation, and mathematical methods pertaining to all aspects of computational geometry. The mathematical origami “one-cut theorem”, for example, states that any straight-line figure drawn on a sheet of paper can be made with just “one cut” of a properly folded paper. O’Rourke first demonstrated this theorem with an ordinary square cut-out then moved on to the tougher scalene triangle and finally to the increasingly complex square animal figures and words. It is clear that even fairly complicated one-dimensional linkages can take many time-consuming, difficult periods of calculation to unravel.

So, is computational geometry useful? Mainly yes, says O’Rourke: this sort of experimental mathematics is not necessarily banished to the realm of theory either; the transformation of complex proteins to their primary linkage amino acid sequences, for instance, was one of several interdisciplinary examples O’Rourke used throughout the talk. Computational geometry has wide impact on fields like architecture, mechanical and biological engineering, and graphic design. In fact, businesses like the German Lundström Design company exists solely for 3D modeling, rendering, and image-unfolding purposes that aid in the manufacture of a wide variety of products, including airbags, cell phone covers, sheet metal boat parts, and air conditioning ducts among other things. Despite this vast applicability, O’Rourke still believes further progress and research in computational geometry is needed before practical application of the field can really be maximized.