9772

Viète's Nested Square Root Representation of Pi

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.

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DETAILS

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.

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