Eugene Mitchell (Wayne, NJ) once posed the question: In the old west, why is a sheriff's badge traditionally a five-pointed star but a deputy's star is six-pointed?
Think about how you would respond, before reading further....
My first thought was that I didn't know there was a difference. A quick check via Google's "images" showed Mitchell's claim to be usually true, but not always.
Charley Eckhardt (Seguin, TX), respected historian of the Old West, suggests the answer to the question lies in mathematics, or to be more specific, in the geometry of the situation.
That is, most badges, especially the more common deputy badges, were made by local gunsmiths or blacksmiths in a western town. And, using a simple measuring device, the radius of a circle can be marked off in six equal chords, producing the six-pointed star.
But, to produce a five-pointed star, a special device (e.g. protractor) was needed to divide the circle into the desired 72-degree arcs (or chords). Thus, the common star for a deputy had six-stars.
Now, the new quandry... why does the California Highway Patrol carry badges that have seven-points? Or, why do Zimbabwe Police carry a badge with eight-points?
Source: D. Feldman's A World of Imponderables, 1992, pp. 377-378