Purple Square Atlas: a visual application of Burnside’s lemma
Author: Christopher Yamba · Co-author: GPT-6.1 Sol Extra High · 2026-10-10
This felt like a clear case for applying Burnside’s lemma. Six simple purple-and-white square tiles generate a surprisingly rich family of patterns. Once we identify patterns that differ only by rotation, the question becomes: how many distinct arrangements remain?
The essential detail is geometric. Turning a board moves its tiles and turns the diagonals inside them. A cell returning to its original position does not necessarily mean that its tile has returned to its original orientation.
For a 2 × 2 board, the six states give 1,296 oriented arrangements. The four rotations fix 1,296, 6, 36, and 6 patterns respectively. Burnside’s lemma averages these fixed counts:
(1,296 + 6 + 36 + 6) / 4 = 336 rotation classes.
The article explains where each fixed count comes from, constructs the symmetric patterns step by step, and develops formulas for every positive board size. It also considers reflections and a general rule for finite tile alphabets. The illustrations use exact geometry, and the companion explorer lets you rotate patterns and inspect their cycles.
The accompanying Lean development proves the rotation-orbit count in cycle coordinates. The bridge to explicit grid coordinates, the reflection extension, and the fully general cycle-product theorem remain written proofs rather than completed Lean formalizations.
I will explore similar patterns in the future and develop more music composition ideas. My next step is to see whether these visual transformations can be translated into musical structures: repeated motifs, cyclic shifts, inversions, and contrasting local states. The musical interpretation is an exploration to develop, rather than a result established by this article.
This is the first project prepared for Yamba Open Mathematics, a growing collection of mathematical ideas, reproducible software, and Lean proofs.
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