726 MR DUNCAN M. Y. SOMMERVILLE ON 
regular polyhedra. They are arranged in conjugate pairs, the number of meshes in 
one being equal to the number of nodes in the conjugate network. One is self- 
conjugate. 
n p IS NG Ny 
Tetrahedral, 3 3 4 6 4 
{ Hexahedral, 4 3 6 12 8 
Octahedral, 3 t 8 12 6 
J Dodecahedral, . 5 3 12 30 20 
| Icosahedral, 3 i) 20 30 12 
2. On the Euclidean plane there are three regular networks, all infinite. 
n p N, Nee NI, 
Square, 4 t 1 2 1 
Triangular, 3 6 2 3 1 
Hexagonal, 6 3 1 3 2 
3. On the Hyperbolic plane there are an infinite number, all infinite. 
jo= Bo 25 Dy BO, SO 
n> 6,4, 3, 3, any value 
ING oN, 2 No= 20: ep om 
§ 2. We proceed to investigate the semi-regular networks, and we shall take the 
three geometries separately. 
I. Tue EvcitipEan PLANE. 
We shall consider, first, how the space about a point can be exactly filled with 
regular polygons. Hach combination of polygons satisfying this condition determines a 
species of node, and all the semi-regular networks must be built up out of the various 
possible species of nodes. Two networks will be considered to be of the same type when 
they contain only nodes of the same species. It is obvious that there may be varieties of 
the same type. The types will be divided into Groups according to the kinds of polygons 
involved, and the groups into Classes according to the number of kinds of polygons. 
Class A. consists of the regular networks, and contains three groups with one unvaried 
type in each. The simplest type in any group is that which contains only one species 
of node. I call this the semple type; other types I call composite. A group does not 
necessarily contain the simple type. 
The Species of Nodes. 
§ 3. The angle of a regular n-gon is given by the formula 
BING en 
ay = (1-5) 180 . avbivekt soled baxiessthonnlywlgn (nn 
The following table of values of a, will be immediately useful : 
