  A parallelogram is conformally inscribed in four lines in the plane if it is
inscribed in a scaled copy of the configuration of four lines. We describe the
geometry of the three-dimensional Euclidean space whose points are the
parallelograms conformally inscribed in sequence in these four lines. In doing
so, we describe the flow of inscribed rectangles by introducing a compact model
of the rectangle inscription problem.

Metric Geometry