### Polygon Metamorphosis

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A polygon is a closed geometric figure that consists of points - vertices - connected by straight line segments - the sides (or edges) of the polygon. Vertices are sequenced in a cyclic order, and sides only connect pairs of adjacent vertices. The terminology varies. In some sources (*The Harper Collins Dictionary of Mathematics*, Harper Perennial, 1991), the term *polygon* only applies to the cases where the sides do not intersect. Elsewhere, especially when *star polygons* form an object of study, sides are allowed to intersect. In the latter case, side intersections are not considered as vertices of the polygon.

Assume a polygon has n vertices and m sides. Each side connects two vertices, and each vertex belongs to two edges. (In the cyclic order of vertices, one of the edges may be called *incoming* while the other is naturally *outgoing*.) We thus have n = 2m/2. In other words, n = m. This argument shows that a polygon can be unambiguously referred to as an *n-gon*. There is no need to indicate whether n is the number of vertices or the sides.

The above argument is not, however, flawless. Imagine an 8-shaped polygon with a vertex shared by the two loops. Is it a *polygon* at all? Why not? Unless the definition explicitly precludes overlapping vertices, an 8-shaped figure fits the definition perfectly. Should the definition be restricted or not?

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