SEMI-REGULAR NETWORKS OF THE PLANE IN ABSOLUTE GEOMETRY. 743 
of triangles and hexagons, where the subscript refers to the number of polygons at a 
point, and the index to the number of polygons in the whole network. 
I. Tetrahedral. 3,'6,*. 
Il. Hexahedral. 3.28.5, eye ee A 83, +202) 
4.,°6.", 3,°4.°. 
4,°6,°8,". 
III. Dodecahedral. 38,7°10,", 5 123 204200 | 
5 oe 3.20 5 
42°62 0,", 3,95 2A, ®. 
With two exceptions, to each hexahedral network there corresponds a dodecahedral 
one, each pair being obtained in a similar way from the regular network. Thus 
3,°8.° and 3,°°10,” are obtained by shallow sections from the cube and the dodecahedron 
respectively ; 3,°4.° and 3,°6," by sections through the middle points of the sides, and 
soon. ‘There is no dodecahedral network corresponding to 3,°4;"°, nor is there a hexa- 
hedral network corresponding to 3,°5,"4.". It may be noticed that the values of A 
for corresponding networks bear the same ratio as the values of A for the regular 
networks, viz., 1: 2. 
In representing the networks upon the Huclidean plane the method of stereographic 
projection has been employed, though in some cases, in order to avoid undue crowding 
towards the centre of the figure, strict stereographic projection has been departed from. 
For simplicity the nodes are joined by straight lines instead of ares of circles, so that 
the figures really represent the conical projections of the semi-regular polyhedra. 
Composite Types. 
§ 27. At first sight it might appear that a very large number of composite types 
could exist, for there are an infinite number of species of nodes, while on the Euclidean 
plane where there are a considerable number of composite types there are only a very 
few species of nodes. A little consideration will show, however, that it is probable 
that the number of composite types is extremely limited. 
Let us take any species of node, p,, Po, P33 Ny, M., Mg, and let a,, a,, a, be the 
angles of the different polygons, « the length of side, & the radius of the sphere. 
Then 
T T T 
cos cos — os — 
& Ny Ny Ng 
cos ok = = 
“ . a, . Ao a 
sin — sin — sin — 
2 2 2 
and 
P12 + Pot + P3043 = 27. 
These four equations determine a, , a,, a, and a. 
