Crystallography and Cremona Transformations

Patrick Du Val

A chapter in The Geometric Vein, 1981, pp 191-201 from Springer

Abstract: 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.

Keywords: Base Point; Relativity Space; Double Cone; Elementary Transformation; Opposite Vertex (search for similar items in EconPapers)
Date: 1981
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DOI: 10.1007/978-1-4612-5648-9_12

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