Another method of classification of unoriented and oriented polygons uses the notion of the continuous deformation of such polygons. Any one of two unoriented polygons with the vertices A2, , An and B1, B2, . . B respectively is deformable continuously into the other so that the vertex corresponds to the vertex Bi and the side to the side To obtain subclasses of unoriented polygons continuous deformations of such polygons, which satisfy any or all of the following conditions, are used: (I) no intermediate polygon of the deformation has two consecutive sides which lie in the same line and which have no points in common; (2) neither of two consecutive sides of any intermediate polygon of the deformation is contained in the other; and (3) no point is common to more than two sides of any intermediate polygon of the deformation. In the case of oriented polygons it is also required that sense be preserved by the deformation. Two oriented polygons that are transformable one into the other by a continuous deformation satisfying the first condition have the same value for a. If instead of the first condition the second is satisfied the oriented polygons have the same value for a'. If the deformation satisfies both the first and second conditions then the two oriented polygons are related so that if the magnitudes of one of two corresponding interior angles is less than 7r then the same is true of the other. Corresponding results for unoriented polygons follow easily. An interesting classi fication of unoriented polygons which satisfy the conditions on the intermediate polygons of the deformations satisfying all three conditions and which, in addition, have no vertex as the end of more than two sides and no side containing a vertex or a point belonging to more than two sides is that in which any unoriented polygon in one class is deformable into any other or into the symmetric image of any other in that class by a continuous deformation satisfying all three of the above conditions. For unoriented polygons of 4, 5 and 6 vertices there are respectively 3, II and 7o classes under this classification.
Non-metrical and Metrical Theories. Regular Polygons. Area of Polygons.It should be pointed out that the above theory of the classification of unoriented and oriented polygons holds without essential modification in a more general plane than the Euclidean for only the order relations of the Euclidean plane are essential. Between this theory and the corresponding theory in the projective plane there are, because of the different kinds of linear order, some essential differences, but both theories are non-metrical. By making use of the metric properties of the Euclidean plane the consideration of regular polygons, oriented or not, becomes possible; also the question of the area of polyg onal regions arises. An unoriented polygon is regular if any side is congruent to any other side and any angle of the polygon, i.e., the figure consisting of a vertex and the two consecutive sides having that vertex as an end, congruent to any other "angle" of the polygon. The regular polygons are convex and there exists a circle circumscribed about and another inscribed in every regular polygon. Those regular polygons that are not simple also are called star polygons. If n points which are equally spaced on the circumference of a circle and numbered in order along that cir cumference are joined by segments so that the i-th point is joined to the (i+d)-th point, where d is a fixed positive integer, then the polygon resulting is a regular polygon for which the value of a, defined above, is d. Thus the number of "types" of regular poly
gons of n vertices is half of the number of positive integers which are less than and prime to n. Other kinds of regular polygons have been studied with particular reference to their classification along the lines explained above. For instance, there are the polygons which have the property that the figure com posed of any vertex and the two sides of the polygon which have that vertex as an end is congru ent to any other such figure and also the polygons which are such that the figure consisting of any side and two adjacent angles of the polygon is congruent to any other such figure. These poly gons have an even number of vertices and a circle is circum scribable about any of those of the former kind and inscribable in any of those of the latter.
Assuming the fact that the area of any plane simple polygon or rather polygonal region is the sum of the areas of the triangles of any finite set of triangles which have no interior points in com mon and which are such that every point of the polygonal region belongs to a triangle of the set or to the interior of one and every point of any triangle of the set or of the interior of any belongs to the polygonal region or its boundary the notion of the area of any plane unoriented or oriented polygon is approached. In the case of an unoriented or oriented polygon in general there is no region uniquely determined as in the case of simple polygons. In the following only the case of oriented polygons is considered for that essentially covers the case of unoriented polygons. If the area of a triangle according to the usual meaning is then the area of that triangle with a sense assigned to it is defined as a or o according as that sense is positive or negative, i.e., the same or not the same as the counterclockwise sense along the circum ference of a circle. Using the symbol to denote the oriented polygon with the vertices A 1, A 2, , and the sensed sides A ijl the area of the oriented polygon is defined as the sum of the areas of the oriented triangles OA 0A2A3, , where 0 is any point of the plane. It is, of course, proved that the value of the area thus defined does not depend on the position of 0 and that if is simple, this definition agrees with the area of a simple polygon according to the fundamental definition. An oriented polygon P determines a finite number of polygonal regions in its plane which have no points in common and whose boundaries are composed of points belonging to P. One and only one of these regions is of infinite extent. Now the following interesting facts pertain : Let the regions, or cells, of finite extent be denoted respectively by S2, , Sk, and let the area of the cell Si according to the fundamental definition of the area of a polyg onal region be ai so that is a positive number; then there exists a set of numbers ci, c2, , ck which are either integers or zero such that the area of the oriented polygon P is . -Fckak. Further, ci is the number of plete positive revolutions minus the number of complete negative revolutions made by the radius vector, having any point 0 of Si as its initial point, as its terminal point describes once the oriented polygon P in the assigned sense. ci is called the coefficient of the cell Si.