SEMI-REGULAR NETWORKS OF THE. PLANE IN ABSOLUTE GEOMETRY. ‘731 
§ 9. The unique types and the fundamental varieties of T,H, and TS.H can be 
obtained from the regular networks by fairly obvious dissections. Thus, SHD (fig. 5) 
is obtained from either the triangular or the hexagonal network; for the squares, 
hexagons, and dodecagons have (1, 1) correspondence with the lines, meshes, and nodes 
respectively of the triangular network. In a similar way T,H (fig. 4) is obtained from 
the same network ; to each mesh there corresponds a triangle, to each node a hexagon, 
and to each line two triangles. And so for the others: it is only necessary to compare 
the figures, given below (§ 11), which represent the relative numbers of the various 
polygons, with the numbers of nodes, lines, and meshes in the regular networks. In 
the diagrams given for the unique types the regular network is indicated by shading. 
T,H and all the varieties of T,H, can also be obtained from the regular triangular 
network by replacing all the groups of six covertical triangles by hexagons; and TS,H 
ean be obtained from SHD by bordering every dodecagon internally with squares and 
triangles. 
§ 10. The type T;S, forms an exception to what has been said regarding the way in 
which the network may be obtained. One of its varieties, that in which no two squares 
are together, can be obtained in a simple way from the regular square network ; to each 
mesh corresponds a square, to each node a square, and to each line two triangles. But 
the other varieties cannot be obtained from this, nor, in general, in any simple way from 
the square network. The following forms may be enumerated, though the list is not 
exhaustive :—- 
(1) 1, 2, 3,.... squares always together. Hach of these is unique, and the series 
forms a general type of variety, admitting of an infinite number of mix- 
tures (fig. 8). 
(2) 2, 3,.... squares or fewer together. Here we can distinguish 
(i.) Two similar types, in which there occurs once only (a) a single triangle 
surrounded by three squares, (b) a triangular group of four triangles 
surrounded by three double squares. The network radiates from this 
figure as centre (fig. 10). 
(i1.) A general type, obtainable by a dissection of the square network, in which 
(a) and (b) are excluded (fig. 9). 
(iii.) Further, if an unlimited number of squares may be together, the groups (a) 
and (b) may occur more than once, or together. 
§ 11. From what has been said it is evident that for any of the simple types, with 
the possible exception of T;S,, the relative number of the several kinds of polygons is 
definite, and the same for all the varieties. These numbers can be found by inspection 
and a knowledge of the number of meshes, lines, and nodes in the regular networks. 
General expressions for the ratios may be found as follows. The results show that T.S, 
is not an exception in this respect. 
