A regular pentagon and pentagram

In this interactive diagram we have the mixed case of a pentagon with a pentagram case following the examples of two pentagons and two pentagrams. It may be helpful to have seen earlier demonstrations first. There are a red circle and inscribed regular pentagon s, a blue circle and inscribed regular pentagram 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 reentrant pentagram that results from the construction. The radius controls and the points s0 and a0 are all draggable (as in the triangle case).

Initially four of the r points are all in a straight line, but vertically, in the order (r1, r3, r2, r4) with r0 at the right. It is an artefact of the applet that at first the two green lines joining r1 and r4 to the vertex r0 are not drawn. Moving a0 a little bit makes them appear. We see that for small blue radius the green figure is actually convex, for somewhat larger blue radius we have a non-convex pentagon and for large enough blue radius we have a reentrant pentagon.

The theory of the Discrete Fourier Transform of order 5, i.e., harmonic analysis in the cyclic group of order 5, a.k.a. the basic Geometric Fourier Transform that takes arbitrary pentagons to their equilateral and anti-equilateral harmonic components tells us that any pentagon can be constructed in the way the mechanism here illustrates, provided its center of gravity is at the origin. That is constructed by taking a standard equilateral pentagon with positive orientation, rotating and dilating it (bigger or smaller), doing the same with the reverse orientation and with pentagrams and averaging four suitable parts.