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Crystallography and Cremona Transformations

In: The Geometric Vein

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  • Patrick Du Val

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.

Suggested Citation

  • Patrick Du Val, 1981. "Crystallography and Cremona Transformations," Springer Books, in: Chandler Davis & Branko Grünbaum & F. A. Sherk (ed.), The Geometric Vein, pages 191-201, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-5648-9_12
    DOI: 10.1007/978-1-4612-5648-9_12
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