POLYGON and POLYHEDRON. The word polygon means figure of several angles, and polyhedron means solid of several faces : the first is used for a plane bounded by straight lines, the second for a solid bounded by planes. We shall in this article state the general properties of both kinds of figures, reserving the particular consideration of those which have equal sides or equal faces for the articles REGULAR FIGURES, SOLIDS, &c.
The Elements of Euclid confine themselves to_ convexpolygons, and to a limited number of polyhedrons. The most general propositions with respect to polygons as polygons, that is, which are true whatever the number of sides may be, are as follows : they are either in the Elements or immediately deducible from them.
1. The internal angles of a polygon of n sides are together always equal to n-2 pairs of right angles. See ROTATION for the full meaning of this proposition.
2. When a figure of an even number of sides is inscribed in a circle, the sum of the first, third, fifth, &c., angles is equal to the sum of the second, fourth, sixth, &c.,angles. But when a figure of an even number of sides is described about a circle, for angles read sides in the preceding property 3. Any one side of a polygon is less than the sum of all the others. The first-mentioned theorem remains true beyond the limits of Euclid's meaning, namely, eo long as the figure of n sides can in any way be divided into n-2 triangles : that is, hi fact, as long as no side of the figure crosses any other side. Thus the adjoining polygon of 10 sides, being divisible into 8 triangles, has the sum of all its angles 'equal to 16 right angles, four of these angles being each greater than two right angles.
To make a rule which!shall connect the angles of any polygon whatsoever, that is, of any figure, however irregular, in which a point returns by a succession of straight lines to the print from whence it set out, would be difficult in the ordinary way of measuring angles. On this subject see SION.
A polygon of a sides or edges has one face, and n angular points or corners : that ia, the number of faces and corners together exceed the number of edges by 1. On one side of the polygon let another polygon be described : it is then obvious that the two polygons have two corners in common, but only one edge, or else three corners and two edges, &c. ; that is, whatever new corners are added, one more new edge is added : or, since one face is added, the number of faces and corners is increased by the same as the number of edges. The same
may be proved of every new polygon which has one or more sides in common with any of the old ones : and since at the outset the number of corners and faces exceeds the number of edges by 1, and since every alteration adds the same to both sides of this equation, it remains true throughout. Whence the following theorem : let any number of polygons, in the same plane or not, be so connected that each has one side or more in common with one or more of the others : call each polygon one face ; each side, to how many polygons soevcr it may belong, one edge ; and each angular point, no matter how many angles may be collected there, one corner : the number of faces and corners will always exceed the number of edges by one.
Let there be a solid polyhedron, and beginning from one given face, annex the others successively : the preceding theorem willrernain true, so long as each face which is added adds one or more new edges. But it is obvious that when the polyhedron is completely finished, with the exception of the last face, the completion of the solid, by counting the last face, adds no new edge and no new corner, these having been com pletely laid down in former faces. Hence,* in every solid poly hedron, the number of faces and corners exceeds the number of edges by two.
Again, on a given face of a polyhedron as a base, let a second poly hedron be constructed, and on a given face of that a third, and so on, it being permitted to include several faces from different polyhedrons among the faces of the new one. In the part of each new polyhedron which belongs to the preceding ones, as already shown, the corners and faces exceed the number of edges by one; and the same also in the new portion. But since one new polyhedron is added at every step, it follows that the new faces and corners are the same in number as the new edges and polyhedron. But at the beginning, counting one polyhedron, the faces and corners outnumber the edges and polyhedron by one (since they outnumber the edges by 2f ; and since both Otto of this equation receive the same accession for every new polyhedron, it remains always true : that is the total number of corners and faces in any system of polyhedrons, each of which has one or more rues In common with others, exceeds the total number of edges and polyhedrons by 1.