SEMI-REGULAR NETWORKS OF THE PLANE IN ABSOLUTE GEOMETRY. 747 
Kerppuer has also gone into some detail regarding the EKuclidean networks. He 
gives* all the developable species of nodes and some of the others, with examples of 
networks formed with them, and other patterns, containing star-polygons, which may 
be derived from them. 
It remains to notice a class of. polyhedra connected with the semi-regular 
polyhedra.t They are obtained by drawing tangent planes to the circum-sphere at 
the vertices. To a regular -gon there corresponds then a regular n-hedral angle. 
A regular polyhedron treated in this way gives the conjugate regular polyhedron, 
but in a semi-regular polyhedron none of the polyhedral angles are regular, and so 
none of the faces of the “conjugate” polyhedron will be regular polygons. The 
regular polyhedra have both a circum- and an in-scribed sphere; the semi-regular 
polyhedra have only a circumscribed sphere, while the conjugate ones have only an 
inscribed sphere. The corresponding networks are constructed simply by taking as 
new nodes the centres of the old meshes. The polyhedra conjugate to the fundamental 
varieties have their faces all congruent. This does not hold for the other varieties 
(with the exception of that corresponding to 3,°4,%, fig. 27). Two of this class are 
interesting as being the only ones which have all their edges equal, viz., the rhombo- 
hedra formed from the fundamental varieties of 3,°4,° and 3,°5,” (figs. 30 and 28). 
There is an analogous Euclidean network conjugate to T,H,, 2.c. 3,6,. 
On p. 743 there occurs a misstatement. The hexahedral network 3,°4,°*” 
(Rhombicuboctahedron), though it contains only two kinds of polygons, really corre- 
sponds to the dodecahedral network  3,°0,"4,* (Rhombicosidodecahedron), being 
obtained in a similar way from the corresponding regular network. Thus the corre- 
spondence between the hexahedral and the dodecahedral networks is complete. 
* Loc. cit., pp. 51-55. 
+ Kepruer and BanrzEr, loc, cit.; Mrrer Hirscu, loc. cit., pp. 186-196; J. H. L. MtuuEr, Trigonometrie (1852), 
p. 345. 
