• DOI:10.1090/S0002-9947-1961-0120604-3
• Corpus ID: 54091952

INSCRIBED SQUARES IN PLANE CURVES

@article{Jerrard1961INSCRIBEDSI,
  title={INSCRIBED SQUARES IN PLANE CURVES},
  author={Richard Jerrard},
  journal={Transactions of the American Mathematical Society},
  year={1961},
  volume={98},
  pages={234-241}
}

• R. Jerrard
• Published 1 February 1961
• Mathematics
• Transactions of the American Mathematical Society

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by x = x(t), y = y(t). These are real analytic periodic functions with period T. In the following paper it is shown that in a certain definite sense, exactly an odd number of squares can be inscribed in every such curve which does not contain an infinite number of inscribed squares. This theorem is similar to the theorem of Kakutani [1] that there exists a circumscribing cube around any closed, bounded convex set in E3. The latter theorem has been generalized by Yamabe and Yujobo [2], and Cairns [3] to show that in E. there are families of such cubes. Here, for the case of squares inscribed in plane curves, we remove the restriction to convexity and give certain other results. A square inscribed in a curve C means a square with its four corner points on the curve, though it may not lie entirely in the interior of C. Indeed, the spiral r=kO, 27r _ < 41r, with the two endpoints connected by a straight line possesses only one inscribed square. The square has one corner point on the straight line segment, and does not lie entirely in the interior. On C, from the point P at to to the point Q at t=to+s, we construct the chord, and upon the chord as a side erect a square in such a way that as s approaches zero the square is inside C. As s increases we consider the two free corner points of the square, Pc and Q, adjacent to P and Q respectively. As s approaches T the square will be outside C and therefore both P, and Qc must cross C an odd number of times as s varies from zero to T. The points may also touch C without crossing. Suppose P, crosses C when s = si, S2, , s. We now have certain squares with three corners on C. For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square. Each point on C, as a vertex, may possess a finite number of corresponding diagonal points by the above construction. To each paired vertex and diagonal point there corresponds a unique forward corner point, i.e., the corner on C reached first by proceeding along C from the vertex in the direction of increasing t. If the vertex is at to, and if the interior of C is on the left as one moves in the direction of increasing t,