# Martin Weissman

I am a number theorist exploring graphic design. Mathematicians sometimes struggle to describe their notions of beauty and elegance, especially when they work in algebraic rather than geometric fields. By bringing mathematics and design into close contact, I hope that my images convey the mathematical aesthetic.

Specifically, I take abstract structures, data sets, theorems, and proofs – the most important aspects of my field of research -- as design challenges. In this way, mathematics influences the broad composition and fine detail of each image. Then, in an iterative process, problems of visualization raise mathematical questions, whose solutions feed back into design decisions.

“Epicycles modulo 37” is inspired by Zolotarev’s 1872 proof of Gauss’s Quadratic Reciprocity. The artwork portrays the cyclic group of nonzero numbers, modulo 37, under multiplication. The image consists of 36 diagrams, and each diagram displays the dynamics of multiplication by a different number (in red) modulo 37. For example, the second diagram displays the dynamics of multiplication by 2 mod 37: 1 goes to 2 goes to 4, to 8, to 16, to 32, to 27 (since 64 = 27 mod 37), etc. The full artwork illustrates Fermat’s Little Theorem, Lagrange’s Theorem, and Zolotarev’s Lemma, and asks the viewer to find her own patterns.

“Integer Triangles” is inspired by a famous theorem of Heegner, Baker, and Stark (H-B-S), which solved Gauss’s Class Number 1 Problem. Difficult to describe algebraically, the image reframes the problem geometrically. Pictured are triangles with whole-number edge lengths (a,b,c), arranged vertically by shortest side-length and horizontally by “discriminant” (the number a^2 + b^2 + c^2 – 2ab – 2bc – 2ca). A vertical thread marks each discriminant; the “class number” counts the triangles hanging on each thread. The discriminants 3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, and 163 have class number 1 – their threads have only one triangle. Gauss conjectured and H-B-S proved that these are the only discriminants of class number 1.