Viète's Nested Square Root Representation of Pi

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Viète in 1543 derived a representation for involving a sequence of nested square roots. The formula is displayed in the graphic. The underbrace signifies that the expression above it contains square roots. For finite values of , the formula represents the perimeter of a regular polygon of sides inscribed in a circle of unit diameter. For a 1024-sided polygon, corresponding to , Viète computed the value , accurate to 6 significant figures. This Demonstration allows you to extend the result up to . The capability of Mathematica to compute multiply nested functions is exploited. With , the underbraced form can be computed using Nest[,, ].

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A derivation of Viète's formula is outlined in the Details section.

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Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA


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