If you walk down any given hallway in a place where STEM (Science, Technology, Engineering, Math) happens, it is becoming increasingly likely that you will see colorful 3D printed props adorning offices and cubicles. Affordable 3D printing and 3D printers are taking the world by storm, enabling home workshops, public schools, and, yes, even math departments to enjoy this new and exciting technology. If you have ever wanted to be surrounded by your math (in a literal sense), this can easily happen. Mathematical constructs are leaving the dimension of the mind and preparing for a full-scale invasion of the dimension of the desks! Here we provide some highlights from our experience with 3D printing in research and teaching over the past year, specifically in regards to partial differential equations (PDEs) with time-dependent solutions. The article is separated into 3 sections: (1) Printing for teaching PDEs, (2) How to make 3D prints of PDEs, and (3) Printing PDEs for research.
1. 3D Printing for PDE Pedagogy
In math, multivariable calculus is the course that typically comes to mind for incorporating 3D printing into the classroom. Another good fit is Boundary Value Problems, an undergraduate course full of beautiful solutions, both mathematically and physically. The course can be taught from several perspectives, but it is safe to say that all approaches incorporate the study of solutions to PDEs describing diffusion (heat) and/or convection (waves).
Although we seek time-varying solutions for heat and wave-like PDEs, we unfortunately have not yet figured out a way to make 3D printed objects that animate in your hand (which would be awesome, by the way), although N. Gulley's 3D Zoetrope comes close . Instead, we have mostly been printing solutions to 1D PDEs (i.e., 1 independent variable in space) by setting the space variable to be one dimension of the base, time to be the other dimension of the base, and the dependent variable (e.g., heat, wave displacement, etc.) to be the height of the printed object. An example is given in Figure 1 for the solutions to two hyperbolic 1D PDEs.
Figure 1: Solutions to (a) a modified 1D wave equation  (Eq. 22) with homogeneous Neumann conditions on each side and (b) the standard 1D wave equation with a homogeneous Dirichlet condition on the left side and a homogeneous Neumann condition on the right side. These pieces are part of a set used in a workshop where students match up boundary conditions to 3D prints. The idea is that, early in the course, students see the effect of these boundary conditions on the solution.
Analytical solution methods for PDEs (if given in detail) can be rather lengthy, and the payoff of a beautiful solution can be a welcome sight. For time-varying PDEs, computer animations of a melted or propagated shape are useful for visualizing solutions. Taking this one step further, imagine—as a student—holding an object in your hand that represents a time-history of the solution to a PDE. Consider the 3D print in Figure 1a with homogenous Neumann boundary conditions (i.e., reflective walls). Here, one can make horizontal scans with the index finger from the middle to the edges of the object and feel that the slope at the boundaries approaches zero for all time (shifting the horizontal scan up and down), even though the solution itself (height of the printed object) varies wildly with time. A comparison can then be made for prints of the same (or similar) PDE with different boundary conditions (see Figure 1b). This exercise is part of a 20-minute workshop that we run in class, in which we provide a set of 3D prints to each group of 3-4 students. This type of activity promotes discussion among students as they try to pair boundary conditions with the printed objects.
An example of the 1D heat equation with non-homogeneous boundary conditions is given in Figure 2, where one may track with their fingers in the direction of time from the initial condition (bottom of figure) to steady state (top of figure) and feel the loss of time dependence but not space dependence on the solution.