The theory of poly gons as a special chapter in mathematics is chiefly concerned with the classification of unoriented and oriented polygons all of whose elements,—that is, vertices and sides,—are in the same plane. The corresponding question for space concerns polyhedrons and is taken up in the article on SOLIDS, GEOMETRIC. Polygons whose elements are not in one plane have not as yet formed the subject of any interesting theory. Such polygons, as well as plane polygons, however, serve as important aids, as in the study of continuous curves in general. This is largely because of the fact that any continuous arc contains the vertices of a poly gonal line, the length of whose sides are all less than any pre assigned positive number, and which is simple if the arc is simple. In particular, the length of an arc of a curve is defined by means of the lengths of the inscribed polygonal lines,—that is, polygonal lines which join the ends of the arc and whose vertices are on the arc and have an order which conforms to one of the two senses along the arc. In the geometry of the Euclidean plane, plane polygons,—that is, those having all of their points in one plane,— take on an added significance because of the fact that the Eu clidean plane is separated into two regions by any simple poly gon that is contained in it. This is a consequence of the basic fact that a line separates the Euclidean plane. Unless it is stated otherwise it is understood that in the following all configurations are in the Euclidean plane. A region is a set of points such that any point of the set is the centre of a circle which has only points of the set in its interior and such that the set is not composed of two sets having the latter property and also having no points in common. It follows easily that any two points of a region are joined by a simple polygonal line which is contained in the region. A precise statement of the important fact mentioned above is that if P is any simple plane polygon, then the plane is composed of P and two regions which have no points in common with each other or with P. One of these regions is of infinite
extent and the other is not. A region such as the latter is re ferred to as a polygonal region and also as the interior of the polygon concerned and the latter is called the boundary of the polygonal region. Every circle which has a point of the boundary of the polygonal region as centre contains points of the region and any point which is such that any circle having it as centre contains points which belong to the region and also points which do not is a point of the boundary of the region. This property of the boundary of any polygonal region is used as the defining property of the boundary of a region in general. As a further consequence should be mentioned the fact that any polygonal region plus its boundary is composed of a finite number of triangles and their interiors, which have no points in common, and the vertices of the triangles are vertices of the bounding polygon. A region which is not a polygonal region is, however, approximated to by polygonal regions according to the following theorem: If / is any region, then there exists a sequence of polygonal regions Zi, /2, /3, • • • , /., • • • such that (a)
and its boundary is contained in I and also in
for all (positive integral) values of n and (b) each point of / is contained in all but a finite number of the polygonal regions /„. This theorem is easily proved by using as the polygonal regions regions which are composed of congruent squares which are formed by two sets of parallel lines, the lines of each set being equally spaced and intersecting orthogonally those of the other set. These acts indicate the importance of polygonal lines and polygons in the study of more general configurations.