3D Graphics Illustrating Folding and Unfolding

    Michiko Charley

    Being able to display geometric objects in a way that is both clear and visually pleasing is important when describing the mathematics of three dimensions. Joseph O’Rourke and Erik Demaine (MIT) are currently writing a monograph entitled Folding

    and Unfolding in Computational Geometry which contains many examples that need 3D

    illustrations.

    In order to create the images to be included in the book, graphical objects were computed in Mathematica, and then copied into Adobe Illustrator to be manipulated and enhanced. Several of the more complex 3D shapes were difficult to capture accurately in 2D drawings, and for these I and other students (Monta Lertpachin and Melody Donoso) constructed physical models out of paper, string, straws, and tape to help us visualize the shapes. Some of the more complex figures I designed are shown below.

     Bricard’s Flexible Octahedron Steffen’s flexible polygon Cylinder surrounded by

    triangular mesh

     Dodecagons and geodesic line octagon geodesic lines

    As these images are ultimately to be printed on paper in black-and-white, care with size and shading was needed. Other concerns were the format of the files, because different applications were used to create different objects, and not all the formats were compatible.

    A website I started this summer is intended to serve as a supplement to the book for GIF animations, Java applets, and other dynamic images. My role was to design and organize the Web site to accommodate the individual applets developed by a number of different students working in our group.

Advisor: Joseph O’Rourke; Funding: NSF and Praxis.