The boundary of a simple closed curve has each point on it, touching two adjacent points one on each side. Can the entire boundary be represented as a series of circles of the same diameter "d" where each circle is touching the adjacent circle at exactly one point only without overlap in areas between the two circles.
This will allow us to approximate the simple closed curve with a polygon where each side of the polygon is represented by a line joining the centers of adjacent circles and at a distance equal to the diameter "d" of the circle.
Such approximation would imply that a inscribed square exists on the polygon (Toeplitz conjecture special cases) and therefore an "almost square" whose vertices are within the error range of the diameter of the circle must exist. Shrinking the size of the circles used will reduce the error of the "almost squares" to smaller and smaller values.
