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How I Printed a Mathematical Function

As you may already know, the list of disciplines that are finding new and interesting applications for 3D printing is growing rapidly as new fields discover uses for the technology.  Professionals and hobbyists alike are finding ways to implement 3D printing as a means of creating things that were once prohibitively expensive or even simply not possible previously.

We here at MadeSolid thought we would share one such case where something was created with 3D printing that wouldn’t have been possible or at least practical by any other means.

The Function

Last year, we were approached by scientists from the National Center for Electron Microscopy (NCEM) at the Lawrence Berkeley National Lab who were interested in creating a hanging sculpture, the pieces of which would be mathematical functions used in their work.  As described in their words, “The functions describe the standard distortions (also called aberrations) produced by electromagnetic lenses which are used to focus charged particles like electrons.  The microscopes at NCEM have advanced correction lenses that nearly eliminate these distortions (aberrations) by applying magnetic fields with the shape and symmetry of these math functions.  To give a scale of the complexity, the microscope has 20 lenses, but the aberration corrector has an additional 150!”

Knowing that creating these precise shapes would be incredibly difficult to create by any other means, 3D printing was the only practical method for creating these objects.  The shapes needed to be simultaneously both delicate and precise.  To make them by hand might make shapes that closely resemble the functions they are meant to represent, but they would be imprecise.

By 3D printing the shapes, we would be able to create objects derived directly from computer-generated functions.  As one of the scientists we worked with said with respect to having the models 3D printed, “We can say, ‘these are the functions,’ rather than ‘these are close approximations of the functions we use.'”

Another of the scientists we worked with had written a program in MATLAB that would turn their functions into zero-thickness .stl models, a familiar format for anyone with any 3D printing experience.  This was a great starting point for us, but the work was only just beginning. Eventually we wound up pushing the limits of our machines and seriously testing our problem solving abilities.

Here are two of the files you can try yourself:

The Challenge

Through a lengthy process of trial and error, we learned that the consumer grade FDM machines we had been working with up until that point had limitations that made printing large, rounded models with essentially no flat surfaces quite a challenge.  With no level flat surface to align with the build plate, we solved this problem by utilizing large amounts of support on the models (even still, there were many, many, failed prints).

Most of the way through completing the project, we purchased our first two resin printers, a B9 Creator and a Formlabs Form1.  While these machines suddenly made it much easier to produce the curved models we had been struggling with in FDM, we ran into an unfortunate limitation with both machines – they were simply too small to fit the six to seven inch models we were printing.

Finding a Way

We eventually delivered the completed models to the scientists we were working with at LBNL.  In this time, plenty of frustration had mounted, many hours of sleep had been lost, but it all paid off for us when we were actually finished.  The models looked great, and the scientists who had commissioned the project were incredibly pleased with the final result.

I recently had the opportunity to go back and visit the Lab at LBNL that we created the models for.  In the time since we had delivered the models, the scientists had suspended them to create a large kinetic sculpture (also known as a mobile) that was then hung in the entrance to the lab.

I was blown away. For all the bumps and pitfalls in the creation process there had been, there was no arguing with the end result.