



 
Bruce, Although I will address your query at the end, it is best to begin with convex polygons. Otherwise, a polygon can be a very general object — it consists of vertices and edges; the vertices do not have to lie in a plane, nor must they be finite in number. The definitions: Vertices are points that have been labeled in order 1, 2, 3, ... , and edges are the line segments that join vertex number j to vertex number j+1. A polygon with n vertices is called an ngon. In general, the edges can cross; vertices can be collinear and are even allowed to coincide. On the other hand, for an ngon to be convex each of its n edges must lie on a line that contains the successive vertices that define it, and the other We will find it convenient to use directed angles. An angle at the vertex j is the measure of the angle through which ray directed from vertex j to vertex j+1 must be rotated about vertex j in the counterclockwise direction to point in the direction from j to j1 (where we consider the last vertex n to be the same as 0): With this definition, if we label the vertices of a convex ngon in the counterclockwise direction, all the angles are positive numbers less than 180 degrees. Compare that with the middle quadrangle in the first figure: one must decide whether to call the angle at vertex 2 negative, or greater than 180 degrees. (If we define it to be negative, then the sum of the angles of that quadrangle will be zero; if the angel at 2 is defined to be greater than 180 degrees, then the angle sum will be 360.) For the nonconvex quadrangle on the right, the only interpretation that makes sense to me is for the angles at vertices 2 and 3 to be negative, and at 1 and 4 to be positive, in which case the sum of the angles of that quadrangle will be zero. should easily see how it would work for any ngon. Put your pencil on the edge EA pointing from E to A, then rotate it about A so that it lies along AB pointing toward B. Next, rotate it about B until it points toward C, then about C until it points to D, and so on until you rotate about the last vertex E until your pencil points toward A. Note that your pencil has made one complete revolution — its turned through an angle of 360 degrees. (That's why we defined convexity the way we did!) Next, observe that the angle through which the pencil rotates about each vertex is the supplement of the angle at that vertex (so that it rotates (180 – A) degrees about vertex A, (180 – B) degrees about B, and so on. Thus the pencil's total rotation is you can end up back at your starting point after an even number of turns (either right or left) as long as your last turn is taken at a point due north or due south of the starting point. Chris  


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