SEMI-REGULAR NETWORKS OF THE PLANE IN ABSOLUTE GEOMETRY. 729 
§ 6. Collecting all the species of nodes, we can arrange them in the following 
scheme : 
Class A, 1. Tg. SSI Bp dele, 
Class B. 4. T,S,. 5, TH». Cl Eee De 28. SO) (19) |b Dee. 
Class C. 10) TSH Le ESD, 12. SHAD: 
(OIG ti 44.) -Sreuesne gis 33 
i, Ub. OMS Tee 7, 
RDI lin 1S, Se Oe 
In future we shall refer to these nodes by their numbers in this scheme. 
Of the seventeen species of nodes, only eleven are capable of development to form 
networks. 
In No. 9 the pentagon must be surrounded alternately with pentagons and decagons, 
which is impossible since 5 is odd. 
In the same way, when p,=p,=p,=1, n,, nm. and.”; must all be even, for the 
m,-gon must be surrounded alternately with n.-gons and i;-gons. Hence 13-17 cannot 
be developed by themselves; nor can they be developed in combination, for each 
contains a polygon which is not contained in any other node. Similarly, 9 cannot 
be developed in combination, hence these six species are excluded from all the 
networks. 
Again, T,SD cannot be developed by itself, for, taking the square (fig. 1, Plate IV.), 
we must have a dodecagon on one side and on the adjacent sides double triangles. At 
the free corners of the square we must now have dodecagons, but this brings two 
dodecagons at a point and introduces 7; excluding this, we must introduce 4. 
§ 7. We can now divide the types of networks into groups and classes. Five kinds 
of polygons are at our disposal, but octagons only occur in the combination SO,, hence 
there are only four classes. 
Class A. Regular networks. 
Group I. Triangles (1). 
» I. Squares (2), 
i LUD Hexacons (3); 
Class B. Two kinds of polygons. 
Group I. Triangles and Squares (1, 2, 4). 
,, I. Triangles and Hexagons (1, 3, 5, 6). 
,, IL]. Triangles and Dodecagons (1, 7). 
», LV. Squares and Octagons (2, 8). 
Class C. Three kinds of polygons. 
Group I. Triangles, Squares, and Hexagons (1, 2, 3, 4, 5, 6, 10). 
» I. Triangles, Squares, and Dodecagons (1, 2, 4, 7, 11). 
,», III. Squares, Hexagons, and Dodecagons (2, 3, 12). 
