Short answers, pulled from the story.
A polygon is a plane figure made up of line segments connected to form a closed polygonal chain. The segments are called edges or sides, and the points where two edges meet are the vertices or corners. An n-gon is a polygon with n sides, so a triangle is a 3-gon.
A simple polygon does not intersect itself, allowing intersections only at the shared endpoints of consecutive segments. A self-intersecting polygon has a boundary that crosses itself, producing star polygons and other figures. A simple polygon forms the boundary of a solid polygon, a region of the plane.
The sum of the interior angles of a simple n-gon is (n minus 2) times 180 degrees, or (n minus 2) times pi radians. This works because any simple n-gon can be divided into n minus 2 triangles, each with an angle sum of 180 degrees.
A regular polygon is both equilateral, with all edges of the same length, and equiangular, with all corner angles equal. Equivalently, it is both isogonal and isotoxal, or both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon.
The word polygon derives from the Greek polús meaning many and gōnía meaning corner or angle. Individual polygons combine a Greek-derived numerical prefix with the suffix -gon, such as pentagon and dodecagon. Triangle, quadrilateral and nonagon are exceptions, and beyond the decagon and dodecagon mathematicians often use numerical notation like 17-gon.
Polygons appear in rock formations as the flat facets of crystals, with angles that depend on the mineral. Regular hexagons form when cooling lava packs into columns of basalt, seen at the Giant's Causeway in Northern Ireland and the Devil's Postpile in California. The wax honeycomb made by bees is also an array of hexagons.
In computer graphics, a polygon is a primitive used in modelling and rendering, defined in a database of vertex coordinates, connectivity, color, shading, texture and materials. Surfaces are modelled as a tessellation called a polygon mesh. The point in polygon test determines whether a given point lies inside a simple polygon.