I've seen a sheriff's 7-pointed star badge some place on YouTube and I thought - oh, this star is particularly ugly, geometrically speaking. The points are bulgy, rather than their edges lining up such that they are straight across in line with the next edge skipping each point. I've seen them come in both versions. I was interested in what angles are "default", and what the formulas are to produce them.
Suddenly I find myself immersed in "star polygons":
Star Polygon -- from Wolfram MathWorld
...and in Schläfli symbols. A 19th-century Swiss mathematician:
Schläfli symbol - Wikipedia
To explore "star polygons" I wrote this toy program. The stars can be stretched into more chubby or more pointy versions.
Here I'm showing the 7-pointed sheriff's badge with plus or minus 15 degree angle (internal angle of the points). The nominal angle, if there is such a thing as nominal, is given by the formula:
180 - (360 * skip / points)
Where "points" is the number of vertices. "skip" is the number of vertices that are skipped, in this case we draw to every 2nd point.
The program can produce three different renditions:
Lines - The star polygon can be constructed by using multiple overlapping polygons. Here the relationship between the N-Gon (number of points) and the number of vertices skipped to connect them is illustrated. If the numbers are coprime then the line is continuous to cover all points. If there is a GCD (greatest common divisor) then you need a number of individual overlapping regular polygons.
Solid - The interior is filled in. In this mode the angle of the point can be changed continuously by dragging the "midpoint" (the concave angles between the vertices) towards or away from the center. Any such custom angled star can not be drawn with straight lines as in the Lines version.
Color - This is similar to the Lines version. The regions between the lines is filled in with color just for fun.
Here is a short video clip demo:
And finally, here is the program:
This is just a toy recreational program. It doesn't prove much except maybe help in understanding this geometric figure a bit better.
Revised 7 Oct 2019