Circle Sector Packing (SOME4)

June 2025 - August 2025

The Summer of Math Exposition (SOME) is an annual contest that aims to encourage the creation of digital math content. SOME4 was hosted from June to August 2025, and I submitted a video based on math proof I had created demonstrating a novel method for packing circular sectors on an infinite Euclidean plane. The video, The Geometry of Packing Circle Sectors, ended up placing Rank 24 among roughly 400 other video entries.

The video idea came about in November 2023 during a text conversation about waffles, as recreated in the video's introduction. The "sector connector" packing method occurred to me during this conversation, but it took several months of on-and-off work for me to formally prove its effectiveness as compared to other more obvious packing methods. From there, the concept laid dormant until the announcement of SOME4 in June 2025. At that point, I had already derived the video's core equation (the density of the sector connector) and worked through proofs for the density of several related packing structures, so I decided to try and work my derivation into a video format.

The video is animated using Manim, a math animation Python library created by the host of SOME, and edited with DaVinci Resolve. Between script writing, animation, editing, and voiceover, the video took more than 100 hours to produce (~10 hours of work per minute of produced video).