( 725 ) 
XXIX.—Semi-regular 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. Scorr Lane. (With Twelve Plates.) 
(The cost of the Illustrations of this Paper was defrayed by the CARNEGIE TRUST.) 
(Read December 19, 1904. Issued separately August 30, 1905.) 
§ 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, other- 
wise 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. 
2. The problem is identical with that of finding the partitions of a polygon into poly- 
gons of the same kind, with the same number of polygons at each point.* The boundary 
polygon is one of the meshes of the network. For elliptic networks the boundary is 
finite, for Euclidean networks it is wholly at infinity, and for hyperbolic networks it is 
wholly ideal. 
This method gives a convenient mode of representing the networks, viz., by their 
stereographic projections upon the Euclidean plane. This representation will be em- 
ployed throughout. 
3. The problem corresponds to a particular case of the problem of determining all 
discontinuous groups of motions in the plane.t 
It will be convenient here to collect the results. If 7 is the number of sides of each 
polygon, » the number of lines or polygons meeting at each point, N,, N,, N, the 
number of meshes, lines, and nodes respectively, the results may be summarised as 
follows :— 
1. On the Elliptic plane there are five regular networks, corresponding to the five 
* See V. ScuiEceL, “Theorie der homogenen zusammengesetzten Raumgebilde,” Nova Acta, Bd. xliv., Nr. 4, 
1883. 
+ W. Dyck, “ Gruppentheoretische Studien,” Math. Annalen, xx. 1-44 (1882), and W. Burnsipn, “ Theory of 
Groups,” ch. xii., xii. Also KuEein and Fricke, “Theorie der elliptischen Modulfunctionen.” (For these references 
_ Iam indebted to the referee.) 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART III. (NO. 29). 108 , 
