Polygons

Polygons of interest are usually regular polygons: those with all sides equal, like the equilateral triangle, the square (or rhombus), the regular pentagon, hexagon, heptagon and so forth.  They fit in a circle, inside (inscribed) or outside (circumscribed).

We are especially interested in those polygons which can be changed into stars by joining alternate vertices (corners), such as the pentagon.  This is because 5 is a prime number.  The six-pointed star formed from a hexagon is really two equilateral triangles.  All even polygons are out for this reason.  The heptagon has two seven-pointed stars, depending whether we join vertices two apart or three apart.  In general the higher we go, the wider the range of stars.  However in the case of composite numbers, like 9, we cannot make all the stars: if you join vertices three apart of a nonagon you only get a triangle.  So our interest focusses on primes.

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