We now come to the subject of regular polyhedrons, presuming the reader to know the contents of the article POLYGON AND POLYHEDRON. A great many properties of these solids have been investigated, but as they are of little use, it will be unnecessary to do more than give tables for constructing them of given dimensions. Let a solid be contained by f faces, each of which is a regular polygon of n sides. Let c be the number of corners or solid angles, e the number of edges, and m the number of angles which meet at a corner. Then since there are c corners with m angles at each, the number of edges, counting each edge as often as it meets a corner, is me ; or, as each edge meets a corner twice, m c=e, the number of distinct edges. Again, since there are f faces, of n sides each, and every edge is the union of two faces, we have It nf= e. But f + c = e + 2, or which must be a whole number. And neither at nor n can be less than 3, nor greater than 5, for there are no figures of fewer sides than 3, and [PoLicoox AND POLYHEDRON] spaces cannot be inclosed entirely by figures of more than five sides. The rest follows from the pro perties of conjugate solids in the same article.
Let n=3, or e= 6 m÷(6 —m). This is a whole • number (I) when m=2; this must be rejected : (2) when nt= 3, giving n =3, ni= 3, e =6, f.4, r=4, or four triangles ; we have here the regular tetrahedron, or
triangular pyramid : (3) when ot=4, giving = 3, »1=4, c=12,f= 8, ems% or eight triangles; we have here the regular oetohedron : (4) when so = 5, giving a =3, m=5, e=30, f=20, es...12, or 20 triangles; we have here the regular icosahedron.
Let is = 4, or e= Sas+ (8 —2,a). This is a whole number (1) when sot= 2; which reject : (2) when = 3, giving n= 4, no = 3, f=e, e=8, or six squares ; we have here the regular hexahedron, or cube, the only one of its kind.
Let si= 5, or e= 10m=(10-3m). This is a whole number (1) when ss =2 ; which reject : (2) when vs =3, giving n=5, sa =3, c= 30,f =12, c=20, or 12 pentagons; we hare here the regular dodecahedron, the only one of its kind.
We have thus the five regular solids, and have shown that there can be no others.
The centre of a regular polyhedron is obviously the point of inter section of lines drawn from the corners, each inclined at the same angle to all the edges which meet it. The radius of the circumscribed sphere is the line drawn from any corner to the centre ; that of the inscribed sphere is the perpendicular let fall from the centre upon any of the faces.
The following table answers to that for polygons, and is taken from the same source :—