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Zigzags with Bürgi, Bernoulli, Euler and the Seidel-Entringer-Arnol’d triangle. (English) Zbl 1439.01002

The first word of the title, i.e., “zigzag”, in a way, plays the role of a lantern whose light illuminates the path among numbers, algebraic structures, figures and graphs which are the very flesh of this work. This article sometimes recalls the fairy tales of childhood, for at each page a mathematical diamond or at least a fragment of a pearl shines and the reader, astonished as a traveler in an “undiscovered country”, falls into a kind of meditation. I shall just indicate the main steps of such voyage: (1) Jost Bürgi’s Artificium of 1586 (2) Johann Bernoulli’s successive involutes (3) Euler numbers and D. André’s alternating permutations (4) The Seidel-Entringer-Arnol’d-triangle (5) Richard Guy’s “boustrophedon transform”. These mathematicians are the stars of a constellation discovered by the authors in the sky of mathematics.
After the reading I play the game and I try to push it forth. Hence hereafter I am going to propose a challenge of a kind. For a long time it has been known that the Swiss mathematician Jost Bürgi (1552–1632) found a new method for calculating sines, but he never published his procedure. Historians of science have given up. However, in 2013, Menso Folkerts discovered, in the University Library of Wroclaw (Poland), an autograph in which Bürgi himself explains his algorithm. In modern parlance he describes a linear map between a sequence of non-negative real numbers and a new sequence, represented by a square matrix [M. Folkerts et al., Hist. Math. 43, No. 2, 133–147 (2016; Zbl 1343.01011)]. At first sight I realized that the inverse of this matrix is nothing but the so-called Cartan matrix corresponding to the symplectic Lie group of order equal or greater than 3 [J. E. Humphreys, Introduction to Lie algebras and representation theory. New York etc.: Springer (1972; Zbl 0254.17004), p. 42–72]. Obviously, Bürgi had no insight into Lie theory and this matrix appears in different contexts in mathematics which are somewhat related.
Just a coincidence? Anyway it might be interesting to discover a relationship, if any.

MSC:

01A40 History of mathematics in the 15th and 16th centuries, Renaissance
01A50 History of mathematics in the 18th century
01A55 History of mathematics in the 19th century
05A15 Exact enumeration problems, generating functions
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