a1 University of St Andrews
§ 1. The networks considered in the following paper are those networks of the plane whose meshes are regular polygons with the same length of side.
When the polygons are all of the same kind the network is called regular, otherwise it is semi-regular.
The regular networks have been investigated for the three geometries from various standpoints, the chief of which may be noted.
1. The three geometries can be treated separately. For Euclidean geometry we have then to find what regular polygons will exactly fill up the space round a point. For elliptic geometry we have to find the regular divisions of the sphere, or, what is the same thing, the regular polyhedra in ordinary space. The regular networks which do not belong to either of these classes are then those of the hyperbolic plane.