Crystallography and Cremona Transformations

Abstract

The note that follows is based essentially on some investigations which I undertook in about 1930 [4,5], in response to Coxeter’s earliest researches [1] on the pure Archimedean polytopes (PA) _n in n dimensions, later fitted into his more general notation [2] as (n − 4)_2,1 (3 ⩽ n ⩽ 9). It had been remarked that the 27 vertices of (PA)₆ correspond in an invariant manner to the 27 lines on a general cubic surface; and in the discussions that followed amongst the group of students that surrounded H. F. Baker, it soon emerged that there was a similar correspondence between (PA) _n (n = 3,4,5) and the lines on the del Pezzo surface of order 9 − n, between (PA)₇ and the bitangents of a general plane quartic curve, and between (PA)₈ and the tritangent planes of a certain twisted sextic curve. The theory I propose now to outline provides a systematic explanation of all these correspondences, as well as others that were remarked later.