392 Proceedings of Royal Society of Edinburgh. [sess; 
Networks of the Plane in Absolute Geometry. By 
Duncan M. Y. Sommerville, M.A., B.Sc., University of 
St Andrews. Communicated by Professor P. R. Scott Lang. 
(Read December 19, 1904.) 
{Abstract.) 
The problem to divide the plane, without overlapping, into a 
network of regular polygons with the same length of side, has been 
completely worked out for the three geometries for the case in 
which the polygons are all of the same kind. The resulting net- 
works are called regular. 
On the Elliptic plane there are five regular networks. These 
correspond to the five regular polyhedra in ordinary space. On 
the Euclidean plane there are three, consisting respectively of 
triangles, squares, and hexagons. On the Hyperbolic plane there 
exist an infinite number. 
To investigate the extension of this problem to the case where 
the polygons are of different kinds, i.e. to find the semi-regular 
networks, I consider first how the space about a point can be 
exactly filled with regular polygons. I take the three geometries 
separately. 
I. The Euclidean Plane. — The angle of a regular polygon is 
definite. If there are wagons, w 2 -gons, etc. at a point, the 
condition that the sum of the angles at the point is 360° leads to 
an indeterminate equation which may be denoted by A = 0, A being 
an integral function of the n’s and ^’s. The solutions of this 
equation in integers give the possible combinations of polygons. Of 
these there are 17. I call them the “kinds of angles.” They are 
divided into three Classes according to the number of kinds of 
polygons involved. The development of some of the kinds of 
angles leads to impossible combinations of polygons. Rejecting 
these, there are left 11, involving triangles (T), squares (S), 
hexagons (H), octagons (0), and dodecagons (D). They may be 
denoted as follows : — 
