IX] OF THE SEMIREGULAR SOLIDS 735 



again as a "space-filler" at the base of the bee's cell. A closely 

 related rhombic icosahedron was known to Kepler; but it was 

 left to Catalan* to discover, only some seventy-five years ago, 

 that the isohedral bodies were thirteen in number, and were 

 precisely comparable with and reciprocal to the Archimedean 

 solids. 



The semi-regular solids, both of Archimedes and of Catalan, are 

 all, like the Platonic bodies, related to the sphere |, for a circum- 

 scribing sphere meets all the corners of an Archimedean solid, and 

 an inscribed sphere touches all the faces of a solid of Catalan; and 

 while the isogonal bodies can be constructed by various simple 

 geometrical means, the general method of constructing the thirteen 

 isohedral bodies is by dividing the sphere into so many similar and 

 equal areas J. It is a matter of spherical trigonometry rather than of 

 simple geometry, and the problem, for that very reason, remained 

 long unsolved. 



The thirteen Archimedean bodies are derivable from the five 

 Platonic bodies, in most cases easily, by so truncating their corners 

 and their edges as to produce new and rpgularly polygonal faces in 

 place of the old faces, corners and edges, and the possible number 

 of faces in the new figure will be easily derived from the edges, 

 corners and faces of the old. Part of the old faces will remain; 

 each truncated corner will yield one new face; but each edge may 

 be truncated, or bevelled, more than once, so as to yield one, two, 

 or possibly three new faces. In short, if the faces, corners and 

 edges of a regular solid be F, C, E, those of the Archimedean solids 

 derivable from it {Fj) will be 



F^ = F + mC + nE, 



where m = or 1, and n = 0, 1, or 2. 



From the cube six Archimedean bodies may be derived, from the 

 dodecahedron six, and from the tetrahedron one. 



* Journal de Vecole imper. polytechnique, xli, pp. 1-71, 1865. 



t It follows that the Chinese carved and perforated ivory balls, which are 

 based on regular and symmetrical division of the sphere, can all be referred to one 

 or another of the Platonic or Archimedean bodies. 



J As a matter of fact, the Catalan bodies can be formed by adding to the Platonic 

 bodies, just as (but not so easily as) the Archimedean bodies can be formed by 

 truncating: them. 



