Square and Anti-Square and Double Digon Averaged

Here we have the full 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). This time the circles do not start on top of each other. In addition, now there is a yellow circle and its radius control point, a yellow digon with the double point d13 on its left, and a final figure of black t-points and lines resulting from midpoints between r0 and d02 r1 and d13, r2 and d02, and r3 and d13.

Initially all the r and t points are nearly in a straight line. The diagram is cluttered. Moving s0 round a bit, say thirty degrees up, makes the figures clearer. Then for reducing the blue radius makes the final black figure into a kite shape, and making it smaller yet gives a reentrant quadrilateral (self-crossing). By starting from the kite and reducing the size of the yellow contribution you can make the quadrilateral convex again.

This illustrates fully 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 double digon harmonic components---. It lets us see that any quadrilateral can be constructed in the way the mechanism here illustrates, provided its center of gravity is at the origin. There is clearly a lot of opportunity for play. Enjoy!