## 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.