JULY 9, 2024

BRAID’S TEAM

<aside> 💡 Welcome to another article in our series of Tech Team Profiles! Here, we’ll introduce you to the team building Braid’s automated engineering design technology. Shane Scott is a Researcher at Braid with a background in mathematics and 3D printing.

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Give a short summary of your PhD research.

My research was about symmetries of surfaces. I work in a field called topology, which studies the connections in a geometry. Sometimes they call it “rubber sheet geometry” because you don’t care about the differences between the surface of the cube or the surface of a sphere, as both could easily morph into the other.

Shane hiking in the Rocky mountains near his home.

Here’s an example mathematicians call “homotopy.” Imagine you are walking on the surface of a huge donut. You want to explore without getting lost, so you tie a rope to your starting point. Upon returning, what would happen if you tried to pull the rope tight? That depends on where you went. Maybe you can pull back the whole rope. But if you walk around the donut-hole, you will not be able to pull the entire rope back to you: the rope is caught! But what would happen if you went walking on a sphere? On a sphere you could always pull the entire rope back. That tells you there’s something different about how space is connected in a sphere versus a donut. A type of symmetry where you can deform one geometry into another and back is called a “homeomorphism.” That type of symmetry keeps topological properties; properties like if your rope can get stuck or not. Homeomorphisms turn out to be very helpful in understanding other parts of math and physics.

What is the research process in mathematics?

People have this perception that the research process in math is quite different from other scientific disciplines such as biology. I don’t think it’s much different. The end-goal in math is a proof: an incontrovertible logic argument explaining why a bit of math has to be this way. However, a proof is the very last part of any research. Setting up a mathematics research project is similar to a computing problem. Let’s say you want to model some scenario. You need to find the correct way to describe the situation; you need to capture what’s relevant in exact teams. That’s perhaps the most challenging aspect.

To give a more concrete example, consider Euler’s Seven Bridges of Königsberg problem. Given the seven bridges connecting the town, is it possible to walk across all of the bridges exactly once and get back where you started? The answer is no, but it’s difficult to argue precisely why it isn’t possible. This is one of the earliest and more famous examples in topology. Think of the bridges as edges between vertices on a graph. You can ask and answer questions about the behavior of the graph, including its symmetries. You can test hypotheses about different types of symmetries that may allow you to put types of symmetries into groups. From there, you can ask questions about the rules that group has, what rules it has in common with other groups, and so on. This is akin to the data-gathering phase in other fields. Once you’ve gathered enough data, you can try to write down a proof for simpler cases and see how far you can generalize.

Shane presenting his research. In his words, “you can represent a surface symmetry by knowing what it does to curves on the surface, much like you can represent a linear transformation by knowing what it does to a basis of vectors.”

What originally inspired you to study mathematics?

To be honest, I didn’t really like math until I was halfway through my undergraduate studies. I was okay at math, but I was more interested in physics as an undergraduate student. When physics becomes sufficiently complicated, the mathematical language required to describe the physics becomes so beautiful. For me, math is about writing down an exact description, which applies to more than just physics. I eventually took a digital image processing class with Dr. Virginia Naibo, who became my mentor. Image processing led me to wavelet analysis, another example of math describing science. Those problems made me want to study higher math more seriously.

Briefly describe your work history following your PhD.

After my PhD, I’ve mostly been working in 3D printing for a few different companies. My main role was helping with geometry processing. When you’re 3D printing something, you will discover a lot of constraints on the geometry you can create. I’ve worked with two different methods of 3D printing. For the type of printing where you lay molten plastic drop by drop, gravity and vibrations influence the resulting shape, plus the properties of the plastic you’re using. There is another kind of 3D printing where you have a bed of aluminum power that is slowly fused together by a laser. Gravity matters less, but you have different challenges with fragile pieces and creating undesirable pockets that trap powder. My role was to make it easier for engineers to design parts without worrying about the limitations of the machinery.

Shane taking a meditative break from 3D printing.

Why did you decide to move all the way from New York to join Braid?