We open with the first half of an interview with Draper Labs mathematician Erin Compaan. Dr. Compaan received her doctoral degree in mathematics from the University of Illinois Urbana-Champaign under the supervision of Nikolaos Tzirakis. She was a National Science Foundation Postdoctoral Fellow in Mathematics at the Massachusettes Institute of Technology prior to joining Draper Labs. In this first part, Erin retraces her route into mathematics and her specialty, which is partial differential equations.
Next, Esmé Krom and Molly M. Roughan describe results they found last spring in Path Counting and Eulerian Numbers. The two enjoyed counting paths in various street networks so much that they decided to devise their own network of one-way streets and analyze it. They restricted to paths that never visit a node more than once and succeeded in finding formulas for the number of paths from their starting node to all the other nodes. In the process, they brushed up against a well-studied sequence of numbers known as the Eulerian numbers. Primary guidance and mentorship for this mathematical investigation was provided by MIT undergraduate Adeline Hillier.
Deanna Needell continues making us wonder what computers, with the right algorithms are capable of today. This time, she asks, “Are Computers Artists?” She explains how people have gotten computers to replicate artworks in the style of other artists.
Emily and Jasmine’s investigation into the patterns created by two zigzags that bounce across the face of a rectangle reaches a climax as they discover and prove the following theorem:
The ZigZags Theorem (Emily and Jasmine). Let n and m be distinct positive integers. Let a rectangle be crisscrossed by an n-zigzag and an m-zigzag, each bouncing back and forth between the top and bottom edges. Then the region of the rectangle below both zigzags, the region above both zigzags, and the region between the two zigzags split the rectangle exactly in thirds.
As far as we are aware, this result is new. In previous installments, Emily and Jasmine analyzed all the shapes formed by two zigzags and computed their areas. Applying these formulas and working through a lot of algebra, they were able to prove their result, though it seems to be some kind of miracle that all the rational expressions simplify to 1/3. Tune in for the next installment of Zigzags for a beautiful conceptual proof of their result by Harvard mathematician Noam Elkies. (For precise definitions and details, please see Zigzags, Parts 1 through 7 in the Girls’ Angle Bulletin.)
This also explains the cover, which is a recoloring of the drawing from the cover of Volume 12, Number 2. There, the coloring was produced using a random number generator and reflected Emily and Jasmine’s understanding of these patterns at that time. But with their theorem, order has been discovered and the new color scheme reflects this order.
If you’ve ever wondered why there are the 6 trigonometric functions sine, cosine, tangent, secant, cosecant, and cotangent, Lightning Factorial provides a Meditate to the Math on the topic.
Finally, we conclude with some Notes from the Club, which are authored by our Head Mentor Grace Work.
We hope you enjoy it!
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