Generalizations of Polygons
In a broad sense, a polygon is an unbounded (without ends) sequence or circuit of alternating segments (sides) and angles (corners). An ordinary polygon is unbounded because the sequence closes back in itself in a loop or circuit, while an apeirogon (infinite polygon) is unbounded because it goes on for ever so you can never reach any bounding end point. The modern mathematical understanding is to describe such a structural sequence in terms of an "abstract" polygon which is a partially ordered set (poset) of elements. The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.
A geometric polygon is understood to be a "realization" of the associated abstract polygon; this involves some "mapping" of elements from the abstract to the geometric. Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide. For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. So when we talk about "polygons" we must be careful to explain what kind we are talking about.
A digon is a closed polygon having two sides and two corners. On the sphere, we can mark two opposing points (like the North and South poles) and join them by half a great circle. Add another arc of a different great circle and you have a digon. Tile the sphere with digons and you have a polyhedron called a hosohedron. Take just one great circle instead, run it all the way around, and add just one "corner" point, and you have a monogon or henagon – although many authorities do not regard this as a proper polygon.
Other realizations of these polygons are possible on other surfaces, but in the Euclidean (flat) plane, their bodies cannot be sensibly realized and we think of them as degenerate.
The idea of a polygon has been generalized in various ways. Here is a short list of some degenerate cases (or special cases, depending on your point of view):
- Digon: Interior angle of 0° in the Euclidean plane. See remarks above re. on the sphere.
- Interior angle of 180°: In the plane this gives an apeirogon (see below), on the sphere a dihedron
- A skew polygon does not lie in a flat plane, but zigzags in three (or more) dimensions. The Petrie polygons of the regular polyhedra are classic examples.
- A spherical polygon is a circuit of sides and corners on the surface of a sphere.
- An apeirogon is an infinite sequence of sides and angles, which is not closed but it has no ends because it extends infinitely.
- A complex polygon is a figure analogous to an ordinary polygon, which exists in the complex Hilbert plane.
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