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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[

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

In the figure below, let

be radii of a circle with unit diameter;

be the length of a side of an

-sided regular polygon inscribed in the circle; and

be the length of a side of a regular polygon with double the number of sides. With two applications of Pythagoras's theorem, you can show that

The perimeter of an

-sided regular polygon is then given by

. Starting with

, you can show that

, ⋯, building up stepwise to Viète's formula.

Reference: S. M. Blinder, Guide to Essential Math, Amsterdam: Elsevier, 2008 pp. 58-59.

Contributed by: S. M. Blinder

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Viète's Nested Square Root Representation of Pi