Design of 3D printed mathematical art

Geometry Topology Seminar
Wednesday, March 30, 2016 - 5:05pm
1 hour (actually 50 minutes)
Skiles 006
University of Oklahoma
When visualising topological objects via 3D printing, we need athree-dimensional geometric representation of the object. There areapproximately three broad strategies for doing this: "Manual" - usingwhatever design software is available to build the object by hand;"Parametric/Implicit" - generating the desired geometry using aparametrisation or implicit description of the object; and "Iterative" -numerically solving an optimisation problem.The manual strategy is unlikely to produce good results unless the subjectis very simple. In general, if there is a reasonably canonical geometricstructure on the topological object, then we hope to be able to produce aparametrisation of it. However, in many cases this seems to be impossibleand some form of iterative method is the best we can do. Within theparametric setting, there are still better and worse ways to proceed. Forexample, a geometric representation should demonstrate as many of thesymmetries of the object as possible. There are similar issues in makingthree-dimensional representations of higher dimensional objects. I willdiscuss these matters with many examples, including visualisation offour-dimensional polytopes (using orthogonal versus stereographicprojection) and Seifert surfaces (comparing my work with Saul Schleimerwith Jack van Wijk's iterative techniques).I will also describe some computational problems that have come up in my 3D printed work, including the design of 3D printed mobiles (joint work withMarco Mahler), "Triple gear" and a visualisation of the Klein Quartic(joint work with Saul Schleimer), and hinged surfaces with negativecurvature (joint work with Geoffrey Irving).