If all the corners be four-angled, we have 2x=4 c, or whence there must be at least 8 triangles. And similarly, If all tho aides be quadrangular, there must be nt least 8 three-angled corners. If all the corners be five-angled, we have 2 a= 5 c, or F,=20+2 P, +5%44 so that there must be at least 20 triangular faces. Similarly if all the faces be pentagonal, there must be at least 20 three-angled corners. Some of the most obvious ways in which figures may be put together so as to enclose space are as follows : 1. Two n-sided faces, joined by n quadrangles. This includes the prism and truncated pyramid, and also every quadrangular hexa hedron.
2. The pyramid, with one n-sided face and a triangles.
8. The solid witha quadrangles, and 2 n triangles, the symmetrical case of which is a prism surmounted at each and by a pyramid.
4. Two faces of 11 sides, and to quadrangles, m being any whole number.
5. Twelve quadrangles so arranged that four of them are placed corner to corner, the figure being finished by four others on each side. When the quadrangles are all equilateral, this is the common rhombic dodecahedron.
6. The pentagonal dodecahedron, in which there are two pentagons, each of which has another pentagon on every side, the two figures being placed together so that the projecting angles of the one fill up the re-entering angles of the other.
7. The triangular icosahedron, the conjugate solid of the last, which may be thus imagined. Let a pentagonal prism be surmounted at each extremity by a pyramid, and let the aides of the prism which join the angles of the opposite pentagons, and also a diagonal in each quadrangle, be supposed to be formed of extensible and contractible threads. Turn one of the surmounting pyramids partly round : then the sides and diagonals of the five quadrangles will no longer continue in the same plane, but will form ten triangles, which, with the ten belonging to the pyramids, complete the number required.
When the aides of a polygon are given, the polygon itself is not given, unless it be a triangle : thus there is an infinite number of quadrangles which have the same four sides. But it is very remark
able that when a solid is formed of given faces, in a given order of juatapoaition, those faces, if they form a solid at all, can only form one. This is the reason of the stability of solid figures; were it not for this a box, for example, would require internal cross-pieces to support the sides. This remarkable property is assumed by Euclid as a part of a definition, and that improperly; since it is a new axiom.
A proof of the axiom implied in the above was given by 31. Cauchy in the article already cited, and will be found in the notes to Legendre's Geometry. It is sufficient, but depends on considerations foreign to the subject as usually considered.
For the remarkable division of equal solids into symmetrically and unsymmetrically equal, see SYMMETRICAL : for the more general view of the nature of polygons, suggested by modern geometry, see TRANS VERSAL : see also TRIANGLE, REGULAR FIGURES and REGULAR SOLIDS, TRIGONOMETRV, &e.
To explain the meaning of the term polygonal number [NtramERs, APPELLATIONS OF], let us take as an instance the pentagonal number. Take any pentagon n A b, and construct a set of pentagons, A C c, A n d, ttc., double, treble, die., of A n b In linear dimension. Divide the sides of each pentagon into parts, each equal to the corresponding side of A Lib. Then if we begin with A, which is one point, and afterwards take in all the points of the first pentagon, we have 1 +4, or 5 points. If we now add all the additional points of the second pentagon (in cluding subdivision-points), we have 1 +4+7 or 12 points. Take in the next pentagon, and we have I + 4+ 7 + 10, or 22 points. Hence the series 1, 5, 12, 22, &c., is called the series of pentagonal numbers; and a set of numbers is thus pointed out which may be as justly called pentagonal as the set 1, 4, 9, &c., may be called square. It must he supposed that the various sets of polygonal numbers were suggested by the square numbers.