Research Article

Inscribed squares and square-like quadrilaterals in closed curves

Walter Stromquista1

a1 Daniel H. Wagner, Associates, Station Square Two, Paoli, PA 19301, U.S.A..


We show that for every smooth curve in Rn, there is a quadrilateral with equal sides and equal diagonals whose vertices lie on the curve. In the case of a smooth plane curve, this implies that the curve admits an inscribed square, strengthening a theorem of Schnirelmann and Guggenheimer. “Smooth” means having a continuously turning tangent. We give a weaker condition which is still sufficient for the existence of an inscribed square in a plane curve, and which is satisfied if the curve is convex, if it is a polygon, or (with certain restrictions) if it is piecewise of class C1. For other curves, the question remains open.

(Received November 29 1988)

Key Words:

  • 55M99: ALGEBRAIC TOPOLOGY; Classical topics.