Square inscribing asks whether a given curve contains the vertices of a square. The square peg problem (Toeplitz 1911) asks whether every Jordan curve in the Euclidean plane contains the vertices of a square. It remains open in this generality (Greene and Lobb 2021). Schnirelman (1944) proved that a square can be inscribed in any closed convex curve.
For smooth Jordan curves, Greene and Lobb (2021) proved the stronger rectangular peg theorem, which prescribes the similarity class of the inscribed rectangle. Choosing that similarity class to be a square gives the square peg conclusion for smooth curves. However, because the rectangular peg theorem assumes smoothness, it does not settle the square peg problem for arbitrary Jordan curves. By contrast, a square can be circumscribed about any Jordan curve (Steinhaus 1999, p. 104).
A related periodic formulation concerns four simple closed polygonal curves 
in the cylinder 
, each oriented to wind once
in the positive direction. Valerio (2026) proved that they jointly inscribe a square, in their specified order, if 
, where 
. The curves may
overlap, and the conclusion permits a degenerate square.
This area-vanishing theorem does not resolve the original square peg problem for
arbitrary Jordan curves. Valerio (2026) reports using
Gemini and GPT-6 Astra to identify errors and clarify final arguments.
