Here we have the case of 4 sides. It may be helpful to have seen first the
previous cases of prime numbers,
3 sides and
5 sides.

There are a red circle and inscribed square s, a blue circle and
inscribed square a (with reversed orientation), red and blue radius
control points, green midpoints r between corresponding red and blue
ones, and green lines making up the parallelogram that results from
the construction. The radius controls and the points
s0 and
a0
are all draggable (as in the previous cases).

Initially all four of the r points are all in a straight line. It is an artefact of the
applet that at first the green line to the right of the origin not drawn. Moving the
a radiuspoint in a little bit makes them appear.
We see that for smaller blue radius the green figure is a rhombus which becomes
a smaller square as the blue radius goes to 0. It is half the size of the original
as we are taking green midpoints bwtween the red and the blue.
Rotating
s0 or
a0 gives a more general
parallelogram.

The theory of the Discrete Fourier Transform of order 4, i.e., harmonic
analysis on the cyclic group of order 4---a.k.a. the basic Geometric Fourier
Transform that takes arbitrary quadrilaterals to their square, anti-square
and degenerate dipole
harmonic components---tells us that any quadrilateral can be constructed in the way
the mechanism here illustrates, provided its center of gravity is at the
origin. This is shown in the next interactive diagram.