ACMS Postdoctoral Research Associate Daniel Brake participated in the 7th Annual Research Symposium. Dr. Brake was awarded first place in the Science category.
3D Printing Mathematical Surfaces
Visualization of mathematical concepts is paramount to public understanding, to problem solving in general, and to communicating abstractresults to other mathematicians. The new technology of 3D printing offers novel opportunities for visualization, and with the right hardware, software, and materials, one can print arbitrarily complex geometric shapes corresponding to mathematical concepts. In particular, by utilizing state of the art software for decomposing mathematical surfaces, one can directly `print a system of polynomials'. In a process not dissimilar to how we see, the program called Bertini_real computes a set of points describing the boundaries and interiors of surfaces defined by polynomials in any number of variables. The result is the barest skeleton of the surface, and can be refined arbitrarily to produce a smooth representation. The underlying computational method is called `Homotopy Continuation', and has found application in all areas of science and engineering. When combined with 3D printing, one obtains a new design tool for mathematically optimal tools, scientific apparatus, and abstract surfaces. The most immediate application of printing mathematical surfaces is in the classroom, as a visual aid for geometry and calculus classes, where good visual intuition is critical to comprehension and success. Whereas a paper or projected diagram, even if animated, can only partially describe a 3D figure, the student who can touch and hold the surface over which they are integrating, or whose gradient they are trying to compute, will truly have their imagination sparked.