Relationships between Schläfli symbol and geometrical properties of regular concave (star) polygons

Question

In a Geometry class, we have been learning about the Schläfli symbol and how it is used to describe regular polygons with the notation {p/q}.

I am studying a special class of concave polygons known as star polygons. The definition for these polygons is that p and q must be relatively prime (in other words, they can be sketched continuously by hand without lifting the pencil off the paper, as the case would be for a pentagram, {5/2}.)

An important property of a star polygon (and any regular polygon in general) is that all of its edges are always tangent to a circle that can be inscribed inside of it. Apparently, this is called a caustic. By observation, I have found that the concave polygon with p vertices that has the smallest caustic must have a notation that satisfies the relation {$p/p-2$}, where $p ≥ 5$ and is an odd number. For a fixed p in the Schläfli notation, the size of the caustic increases with q, and in general the size of the caustic gets smaller as p increases. These patterns can be seen quite evidently in following diagram: https://en.wikipedia.org/wiki/Polygram_(geometry)#/media/File:Regular_Star_Polygons-en.svg

These are my questions:

• what is the mathematical relationship between the Schläfli symbol of a star polygon and the angle of its 'spikes'?
• what is the mathematical relationship between Schläfli symbol of a star polygon and the radius of its caustic?

Thanks.

Answer 1

It follows by definition.

It takes $q$ rotations to make a $p$ spiked star.

Each exterior rotation is $ 2 \pi q /p $

EDIT 1:

"Exterior " angle is conventional for a polygon at each corner.

Exterior

Internal angle is its supplement at any spike. So,

each spike has an angle $ \pi - 2 \pi q /p = \pi (1- \dfrac {2 q }{p}). $

You can find from this the in-radius as $ R \cos (\pi q/p) $.