Geometry and Topology Seminar

When visualising topological objects via 3D printing, we need a
3-dimensional geometric representation of the object. There are three
broad strategies for doing this: "Manual" - using whatever design software
is available to build the object by hand; "Parametric" -
generating the desired geometry using a parametrisation of the object; and
"Iterative" - numerically solving an optimisation problem.

The manual strategy is unlikely to produce good results unless the subject
is very simple. In general, if there is a reasonably canonical
geometric structure on the topological object, then we hope to be able to
produce a parametrisation of it. However, in many cases this seems
to be impossible and some form of iterative method is the best we can do.
Within the parametric setting, there are still better and worse
ways to proceed. For example, a geometric representation should
demonstrate as many of the symmetries of the object as possible. There
are similar issues in making 3-dimensional representations of higher
dimensional objects.  I will discuss these matters with many examples,
including visualisation of 4-dimensional polytopes (using orthogonal
versus stereographic projection) and Seifert surfaces (comparing my
work with Saul Schleimer with 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 with Marco Mahler), "Triple gear" (joint work with Saul Schleimer),
and hinged surfaces with negative curvature (joint work
with Geoffrey Irving).