Equiangular Numbers

Some mathematical problems are resolutely geometric. No matter what you do to them, subjecting them to different sorts of manipulations and calculations, their 'geometric content' persists even in the tiniest parts of what remains, even in the numbers used to express their solution, like the parts of an image residing 'everywhere' in a hologram, or like the smile of a Cheshire cat. We want to tell you of one such problem, and of a delightful series of real numbers starting with 0, 1....and tending toward 2, that does its best to recall the struggles along its path into existence. We maintain that it is because of these ancient struggles (which are bound to recur when one tries to 'construct' them) that these numbers are of architectural and artistic significance. We call the sequence equiangular numbers.

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 sigma_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 (planning of paintings, design of building facades, or choice of relative dimensions for European paper stock), where the aspect of 5-equiangularity is thoroughly disguised, sure the subsequent values of s_n for n>5 can give rise to analogous aesthetic feelings in similar situations. Can our readers point to any instances of the use of s₇ in ancient or contemporary architecture?

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