I finally had time to run my program last night during the storm. It just so happens that R wouldn’t calculate pi by rounding farther than the 14^{th} digit. Here is my data produced from R, the first column is the number of sides used to approximate pi by inscribing and circumscribing the polygon and the second column is the number of digits computed accurately of pi with rounding:

[,1] [,2]

[1,] 28 2

[2,] 54 3

[3,] 76 4

[4,] 1050 5

[5,] 1748 6

[6,] 5177 7

[7,] 42805 8

[8,] 53280 9

[9,] 207163 10

[10,] 704426 11

[11,] 1911702 12

[12,] 6695507 13

[13,] 31255428 14

I next plotted the data to visualize my hypothesis of an exponentially increasing relationship of the number of sides needed to approximate pi accurately and the amount of digits found accurately.

A log transformation did an excellent job transforming the data to produce a linear relationship. I would normally use some objective approach to this transformation using something like a box-cox transformation, but for pure fun, I used a log base pi to produce the transformation which was very close to the box-cox recommended transformation. Here is the data plotted after the transformation:

I next performed some regression analysis to determine the approximate amount of sides(s) needed to approximate pi accurately to d digits. The following is the function for this estimate:

s=π^{-.37+1.1d}

Where s is the number of sides needed of a regular polygon’s perimeter being inscribed and circumscribed around a circle and averaged, and d is the number of digits of pi being computed accurately after rounding to that digit.

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