SECTIONAL TRANSACTIONS.— A. 307 



Department of Mathematics (A*). 



Mr. H. S. M. CoxETER. — Regular Polytopes. 



Every regular polytope can be represented by a symbol of the form 



jfcj t ki , . . . Km— If » 



m being the number of dimensions. E.g. {5} is the pentagon, {|^} the pentagram, 

 |4, 3[ the cube, and {3, 4[ the octahedron. The problem before us is to find what 

 Talues of the k's are admissible. 



In this paper, it is shown that every regular polyhedron (i.e. three-dimensional 

 polytope) corresponds to a rational solution of the equation 



cos- — + COS--- + cos' — - = 1. 

 kq ki k-i 



Then this criterion is extended to more dimensions, thus leading to a complete 

 enumeration of the regular poh^topes. 



Finally, some account is given of ' density ' (i.e. the number of times the 

 boundary of a polytope encloses its interior). This quantity takes the values 1, 3, 7 

 in three dimensions; and 1, 4, 6, 20, 66, 76, 191 in four. Certain five-dimensional 

 polytopes, although they satisfy the above-mentioned (extended) criterion, have to 

 be excluded on account of infinite density. 



Mr. P. Du Val. — Some Relations between the Theory of Polytopes and 

 Algebraic Geometry. 



The insight into the structure of certain groups (namely, the groups of linear 

 fractional transformations of a complex variable) afforded bj' the remark that they 

 are simply isomorphic with the groups of rotational symmetry of the regular polyhedra 

 in three dimensional space, suggests the enquiry whether any other groups arising in 

 analysis or algebraic geometry are amenable to a similar treatment. The purpose 

 of the paper is to answer this question in the affirmative by means of a set of examples. 



The uniform polytopes of higher space have been completely enumerated by Elte 

 and Coxeter. It was remarked by Schoute that the 27 vertices of a certain polytope in 

 six dimensions correspond in a special way to the 27 lines on the general cubic surface, 

 and the group of symmetry of this polytope is simply isomorphic with the group of 

 the 27 lines. It can be found that similar results hold for the finite systems of lines 

 on all the surfaces of del Pezzo, as well as for many other finite systems of entities 

 occurring in geometrical configurations, such as : 



The 28 bitangents of the general plane quartic curve ; 



The 120 tritangent planes of the twisted sextic of genus 4, which lies on a 

 quadric cone ; 



The 15 planes and 10 nodes of Segre's cubic primal in four dimensions ; 



The 16 nodes and 16 tropes of the Kummer surface ; 



The finite systems of rational curves of various orders on the surfaces whose prime 

 sections are hyperelliptic. 



Infinite polytopes (or space fillings) can be found whose groups of symmetry are 

 simply isomorphic with certain enumerably (discretely) infinite groups of transforma- 

 tions connected with the figures of nine associated points in a plane, and of eight 

 associated points in space. 



Mr. W. V. D. HoDGE. — Topological Methods in Algebraic Geometry. 



The importance of the concept of the Riemann surface in the theory of algebraic 

 curves is well known to all mathematicians. Owing to certain superficial difiiculties 

 the use of topological methods has received comparatively little attention in discus- 

 sions of varieties of more than one dimension, though Picard has shown that such 

 methods can be very powerful in the theory of surfaces. In recent years, however, 

 Lefschetz and other workers in America have developed the theorj' of the analysis 

 situs of algebraic varieties and have shown that many problems in algebraic geometry 

 can be simplified greatly by considerations of topology. In this paper a survey is 



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