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?

A few more words

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