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A unified theory of proportion. (English) Zbl 1038.52001

The authors’ conclusion: With the aid of Pascal’s triangle, the golden mean and Fibonacci sequences were generalized to a family of silver means. The Lucas sequence was then generalized with the aid of a close variant of the Pascal’s triangle. These generalized golden means and generalized \(F\)- and \(L\)-sequences were shown to form a tightly knit family with many properties of number. Perhaps it is for this reason that they occur in many dynamical systems. The numerical properties of the silver mean constants are the result of their self-referential properties which, in turn, derive from their relationship to the imaginary number \(i\). We have shown that all systems of proportion are related to a set of polynomials derived from Pascal’s triangle. These systems are related to both the edges of various species of regular star polygon and the diagonals of regular \(n\)-gons, and they share many of the additive properties of the golden mean. The heptagon was illustrated in detail.

MSC:

52A10 Convex sets in \(2\) dimensions (including convex curves)
51M20 Polyhedra and polytopes; regular figures, division of spaces
37B10 Symbolic dynamics
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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