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H.S.M. Coxeter’s theory of accessibility: from Mario Pieri to Marvin Greenberg. (English) Zbl 1505.01044

This long, accurate paper discusses the theory of accessibility that H. S. M. Coxeter introduced in the 1960s, and its relations with M. Pieri’s work on projective geometry. The author states that in letters she exchanged with Coxeter in 1989, the latter “revealed that he used ideas from an 1898 paper of Pieri to define the relation of accessible points in a continuity-independent projective plane”. In 1979, the same relation was reinterpreted by M. J. Greenberg using incidence and order axioms inspired by Coxeter’s work. Having sketched in few lines the axiomatic development of projective geometry from von Staudt to Coxeter and Greenberg via Pieri, Veblen and Young, the author offers a detailed overview of Pieri’s paper on the “Principles of the geometry of position” [Torino Mem. (2) 48, 1–62 (1898; JFM 29.0407.01)]. Then she considers Coxeter’s theory of accessibility from both an analytic and a synthetic perspective, and its relation with Pieri’s definition of segment. Then she focuses on Greenberg’s axiomatic construction of a continuity-independent projective plane and compares some of Greenberg’s propositions and Pieri’s theorems. Eventually, she examines the origins of a statement postulated by Pieri, often credited to Pasch, and attributed to Veblen by Coxeter. Tables of comparison of axioms by Coxeter, Pieri and Veblen complete the paper.

MSC:

01A60 History of mathematics in the 20th century
01A55 History of mathematics in the 19th century
51-03 History of geometry

Citations:

JFM 29.0407.01
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Full Text: DOI

References:

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