Study on the Sum of Interior Angles or Exterior Angles of Polygon : From the View Point of the Combinatorial Properties in Figures

Abstract

　　 In this article, some combinatorial properties in figures are discussed. Here, a combinatorial property means an invariant property which has one of the following features: An invariant property obtained by adding, subtracting, multiplying, dividing or composing some of them for some plural values related to figures; an invariant property obtained by decomposing or composing figures.

　　 Practically, we consider the invariant property in the sum of interior angles or exterior angles of polygon as a typical example of combinatorial properties. By attaching importance on the invariance, we can develop combinatorial properties various way in figures, like the sum of the deficits of solid angles in a polyhedron. We show how such development proceeds, taking the case of closed polygonal lines. Then, it involves the consideration of directions and extends to the consideration of a combinatorial properties including the winding numbers of polygonal lines or closed curves.

　　 Through those cases, we extract important elements to develop such materials: To apply some kind of sums or subtractions which are significant to figures; to utilize various decompositions or compositions of figures; to consider the rotations or directions of plane figures.