Everyone intuitively thinks of a polygon as being made up of points and straight line segments. But any attempt to formulate a rigorous mathematical definition runs into a host of "pathological" cases. Moving beyond geometry was necessary to arrive at the notion of an "abstract" polygon.
Early definitions
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A polygon, or n-gon, is a cyclically ordered sequence of n vertices S1, S2, ..., *Sn chosen arbitrarily. The sides of the polygon are the line segments A*i determined by pairs of adjacent vertices Si, Si+1.
This general definition was first given in 1769 by Albrecht Ludwig Friedrich Meister (1724–1788), whom we know chiefly through the historian of mathematics Siegmund Günther (1848–1923). Günther nevertheless claimed, without justification, that Albert Girard (1595–1632) had already conceived of the "polygon" in this way in a 1626 work, but he seems not always to have properly understood Meister's text.
Although this definition appears natural and extremely simple, it allows unexpected and counterintuitive configurations because it does not exclude certain pathological cases. Nothing, for example, prevents vertices from coinciding or sides from overlapping. Meister was aware of this: in his work he presented an enneagon (a nine-sided polygon) with vertices abcABCαβγ, which seems to consist of three superimposed triangles.
