A Combinatorial Approach to the Inscribed Square Problem

The inscribed square conjecture, also known as Toeplitz’ conjecture or the square peg problem, asserts that every Jordan curve in the Euclidean plane admits an inscribed square. Although there exists no proof of the general conjecture, there are affirmative proofs of the conjecture subject to additional local smoothness conditions. The weakest of these local smoothness conditions include the special trapezoid criterion, due to Matschke, and local monotonicity, due to Stromquist. We develop several combinatorial approaches to the inscribed square problem, explicitly locate inscribed squares for conic sections, and explore the existence of inscribed squares in the Koch snowflake, which we prove to be not locally monotone.