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The Twisting Ribbon
A small geometry problem that connected multivariable calculus and linear algebra for me.
In the summer of 2023, I was working to fast-track the completion of my MBA and prepare for a master’s in computer science. As a political science graduate with a couple of hobby coding projects under my belt, I had some gaps to fill, so I was taking Organizational Behavior & Leadership at the same time as Linear Algebra Foundations, Calculus III, and Data Structures.
I’m always looking for interesting ways to apply the concepts I learn about in courses. One of my favorite examples came from calculus, when I started thinking about two skew lines in three-dimensional space.
Skew lines are lines in three-dimensional space that do not intersect and are not parallel. For any pair of skew lines, there is a unique shortest segment connecting them that is perpendicular to both lines. I thought about how the shortest connecting segment could define a surface that turns uniformly along this axis while staying tangent to both skew lines as they extend outward.
The hard part is finding the two endpoints of that shortest connecting segment for arbitrary skew lines in three-dimensional space. It gives rise to a minimization problem: I needed to write the geometry algebraically, take derivatives, and solve the resulting system of equations.
This happened to intersect with another project I had developed that summer: a SymPy linear algebra calculator
. It started as a way to make row reduction and other matrix operations less tedious to work through (including rendering steps in LaTeX for assignments). It was perfect for handling the messy algebra and generating a formula I could simply plug in to Math3D.
I like this problem because it sits at the edge of several things I care about: creating tools to solve problems, connecting topics from courses to applications, and learning by applying. This is a small record of a habit I still value. The set of things I know how to build, and the problems I know how to approach, has only grown since then.
The twisting ribbon is still alive on Math3D, which makes me very happy.