730 MR DUNCAN M. Y. SOMMERVILLE ON 
Class D. Four kinds of polygons. 
Group. I. Triangles, Squares, Hexagons, and Dodecagons (1, 2, 3, 4, 5, 6, 7, 
101112), 
The numbers within the brackets denote the species ‘of nodes which the group admits. 
‘Lhe Simple Types. 
§ 8. Now let us consider the simple types. I observe, in the first place, that when 
the species of node admits of no variation, the simple type is 7 general unvaried. 
The unique nodes are the following : 
1°. Class A, 
2°. Those in which p=3, 
3°. P=l, po=4, 
while the following are varied : 
1’. p,=p,=2. Two forms, M,N, and (MN),. 
2 =p, —1, p,=2. “Two forms, LIN and ENING 
We have then the wnaque simple types. 
Class A. - eeS. abl. 
py BE Da SOs (tach Sm 
i Cy SEND Riios oe 
The type T,H is one exception to the rule stated above, for it does admit of a variation. 
The network is asymmetrical, its mirror image being different from itself. It exists, 
therefore, in two enantiomorphic forms. The one can be obtained from the other by 
turning the plane over. 
Of the other groups, C. IL. and D. I. do not possess simple types, and there remain 
the three simple types T.8,, T,H,, and TS,H, each of which is capable of infinite 
variation. 
T,H, has a variety in which there are no two triangles and no two hexagons 
together. We shall call this the fundamental variety. The opposite sides of any 
hexagon, when produced, define a strip which is capable of displacement without 
affecting the rest of the network. All the varieties can then be obtained by displacing 
any number of such parallel strips through a distance equal to the length of the side 
(fig. 6). 
TS,H has the fundamental variety in which there are no two squares together. Hach 
hexagon is surrounded by squares and triangles, forming a group whose boundary is a 
regular dodecagon. All the varieties can then be obtained by turning any number of 
such groups through “ This operation, performed upon a single group, brings two 
squares together ; performed upon two adjacent but not overlapping groups, it brings 
three squares together (fig. 7). 
