### Square and Anti-Square Averaged

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.