36 



NA TURE 



[May 9, 1889 



ceptions that are unfamiliar to him. He toils through 

 the solution of abstract problems whose formulation he 

 imperfectly grasps, and whose interest and importance 

 he has not been permitted to see beforehand. The very 

 beauty and logical rigour of the work is a hindrance to 

 him ; it entangles and suffocates him at the outset. And 

 yet M. Jordan's work could not be otherwise, could not 

 be better for its purpose, could not be dispensed with. 

 But let the learner first read Klein's " Ikosahedron." He 

 will there see the substitution theory first applied in 

 simple cases with concrete illustration, and will then be 

 led by degrees to see its all-embracing character. Then 

 let him return to Jordan, and he will say of the theory of 

 substitution, 



"Her loveliness I never knew, 

 Until she smiled on me." 



Under ordinary circumstances, any detailed analysis of 

 a mathematical work would be out of place in the pages 

 of Nature. The appearance of the "Ikosahedron" 

 in English is, however, an event of such importance that 

 it would be wrong to miss the opportunity of giving the 



English mathematical world some account, however 

 imperfect, of its contents. 



The proper subject of the work may be said to be the 

 general theory and applications of functions which are 

 transformable into themselves {i.e. are unaltered) by a 

 finite group of substitutions. By a group of substitutions, 

 or indeed of any operations whatsoever, is meant a 

 complete set of operations of such a nature that the com- 

 bination of any two or more of them is equivalent to 

 some one of the set. The number of operations in a 

 group is called its order, and the order may in general 

 be finite or infinite. A leading feature of Klein's work, 

 indicated by its title, is the geometrical connection which 

 he establishes between certain groupsof substitutions, and 

 the rotations which cause a regular solid to return into 

 itself. It is obvious beforehand that the totality of such 

 rotations for any given regular solid forms a group of 

 finite order, for any two successive rotations of the kind 

 may be replaced by a single one. The accompanying 

 figure will make the connection plain. It represents the 

 stereographic projection of the traces on a circumscribed 



sphere made by the planes of symmetry of a regular 

 tetrahedron, A B c D, the vertex of projection being the 

 antipodal point to A, and the plane of projection the 

 diametral plane of which A is the pole. The spherical 

 surface is obviously divided by the planes of symmetry 

 into twelve (non-shaded) congruent triangles, and twelve 

 other (shaded) congruent triangles, each of which is the 

 image of one of the former in a plane of symmetry. 

 These triangles lie in sets of v., = 2 about points such as 

 F, which correspond to mid-edges of the tetrahedron, in 

 sets of 1^2 = 3 about points such as E, which correspond 

 to centroids of the faces, and in sets of 1^3 = 3 about the 

 points which correspond to vertices of the tetrahedron. 

 It is farther obvious that any one of the unshaded 

 triangles can be transformed into each of the other twelve 

 unshaded triangles by one of the group of N = 12 rotations 

 which cause the tetrahedron to return into itself If we 

 mark one of these by inscribing i, and if S denote a rota- 

 tion of period 2 {i.e. of angular magnitude 27r/2) about G, 

 T a rotation of period 2 about F, and U a rotation of period 

 3 about A, then the rotations by which the region i is trans- 

 formed into the other twelve are — 



I, U, U-, S, SU, SU2, T, TU, T\3-, ST, STU, STU^; 



and these give the twelve rotations of the tetrahedral 

 group expressed succinctly in terms of three of them. 



The same holds for the shaded triangles. Hence, if we 

 pair each non-shaded triangle with a shaded one, and 

 thus form a fiiiidanienial domain, then we see that any 

 point within such a domain (boundary and summit points 

 excepted) is transformed by the N tetrahedral rotations 

 into N other points, one of which lies in each of the 

 N domains. (Here we count the transformation of the point 

 into itself, viz. the rotation represented by the identical 

 symbol i.) We pass over the questions that arise regard- 

 ing the composition of a group and the conception of the 

 extended group which embraces reflections in the planes 

 of symmetry as well as rotations, and merely mention 

 that a similar theory is established for the dihedron, i.e., 

 the figure composed of a great circle of the sphere 

 divided into n equal parts, the octahedron, and the 

 ikosahedron, the characteristic numbers being given by 

 the following table:- 



The next step is to connect each point on the sphere 

 with the value of a complex variable ", or with the ratio 

 of two complex variables z =. z-^lz.2. This is done, after 

 the manner of Riemann, by representing z = x -\-yi in an 

 Argand-diagram on the diametral plane of the sphere, 

 and then projecting the point {x, y) stereographically 

 upon the sphere. The point on the sphere is then spoken 

 of as the point (z) or {z^, z^. 



It is next shown that every dihedral, tetrahedral, octa- 

 hedral, or ikosahedral rotation is equivalent to a non- 

 homogeneous linear substitution of the form — 



2'= (As + B)/(C= + D), 



or to one or other of two pairs of homogeneous substitu- 

 tions of the form — 



As, + Be 



Csi -f Ds., ; 



and the proper values of A, B, C, D are calculated for 

 each case. 



If we consider the values of z corresponding to a point, 

 and the N — i other points into which it is transformed 

 by the N polyhedral rotations, we see that they are the 

 roots of an algebraical equation of the Nth degree, say 

 Z =/(z) — c, the characteristic Z of which must have the 

 property of remaining unaltered by every one of the N 



