Equiangular Numbers

Henry Crapo () and Claude Le Conte De Poly-Barbut ()
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Henry Crapo: Centre de Recherche Les Moutons Matheux
Claude Le Conte De Poly-Barbut: Centre d’analyse et de mathématique sociales, EHESS

Chapter Chapter 91 in Architecture and Mathematics from Antiquity to the Future, 2015, pp 649-659 from Springer

Abstract: Abstract We introduce a delightful series of real numbers starting with 0, 1.... and tending toward 2, which we call equiangular numbers, that does its best to recall the struggles along its path into existence. In analyzing these series, we note that a geometric situation gave rise to a difficult (yea, impossible) construction problem in projective geometry, then to a problem in polynomial algebra that taxes the powers of the best modern computer algebra systems, but which had a simple solution in terms of trigonometry. It is fair to ask whether these further values of σ n , n = 7,8… occur already in nature, for the simple reason that they are the natural coordinates of equiangular points. Finally, since the merits of the Golden Mean are well recognized in artistic matters, where the aspect of 5-equiangularity is thoroughly disguised, the subsequent values of s n for n > 5 can give rise to analogous aesthetic feelings in similar situations. We challenge readers to point to any instances of the use of s 7 in ancient or contemporary architecture.

Keywords: Coxeter Group; Polynomial Algebra; Fundamental Region; Geometric Situation; Permute Pair (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-00143-2_44

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DOI: 10.1007/978-3-319-00143-2_44

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