2021 Summer of Math Exposition results

A math-explainer contest that drew more than 1,200 submissions has produced a standout set of five winners—chosen not for polish, but for clarity, empathy, and genuinely useful mathematical insight. The event’s core idea was deliberately open-ended: any math topic, any audience, any format, as long as the work explains math. Cash prizes were added by Brilliant, but the organizer stressed that “winning” is a misnomer; the real success is expanding the amount of math material online that helps people learn and stay curious.

Selection began with peer review to narrow the field to a little over 100 entries for close review, then used about half a dozen guest judges to refine the final choices and reduce the impact of personal preferences. The organizer framed the contest’s structure—clear deadlines and small stakes—as a way to counter perfectionism and stagnation. Still, the emphasis remained on long-term value: ten years from now, arbitrary winner lists won’t matter, but the explainers themselves can keep helping learners.

The five featured winners highlight a range of approaches to mathematical exposition. Paralogical’s entry starts with a practical puzzle—why light reflected at the bottom of a mug forms a cardioid-like shape—and builds the key concept of envelopes, using carefully chosen examples and a friendly tone. The post by Matt Ferraro focuses on light redirection through an acrylic square whose shadow reveals a deliberate image; it walks through the math and algorithms behind the design, including false starts, while guiding readers toward what matters most.

Freya Homare’s “The Beauty of Bézier Curves” was a special case: it’s visually polished, but the selection rationale was that the graphics serve the math. The emphasis is on motivating why Bézier curves matter, showing multiple facets of how they’re used, and delivering clean intuitions rather than aesthetic flourishes. Another pick, “the most complicated passcode” video, turns a seemingly silly question about swipe-pattern phone locks into a rigorous problem-solving story that threads in number theory, induction, and the habit of generalizing after solving subproblems.

The final winner is a geometry proof of Pick’s theorem that multiple guest judges flagged as both clever and underappreciated. Beyond the proof itself, it reflects on what different kinds of proofs do for learners and teachers—arguing for staying power through ideas rather than production value.

Even with five winners, the organizer called the choice “absurdity,” pointing to many other entries that deserved attention. Additional favorites include an animated deep dive into Dehn twist/4D-sphere connections via Durock’s belt trick, a long explanation of the unsolvability of the quintic, and a wide spread of topics—from the two-envelope problem and Buffon’s needle to interactive pieces, Babylonian multiplication, and narrative-driven explainers like multiple entries on Hackenbush. The takeaway is less about a trophy list and more about a curated roadmap to hours of math learning, discovery, and delight—especially for underseen creators.