What are Some of the Coolest and Unusual Rational Fractional Approximations to Pi?

Finding an accurate approximation of pi has been one of the greatest mathematical challenges in history. The concept of pi seems to defy common sense, especially when I was a child. However, as I grew up, I realized that it is not only me who finds pi challenging. The quest to approximate pi has led to some of the most creative and unusual formulas.

Ramanujan's Pi Approximation

One of the most awe-inspiring approximations is Ramanujan's pi approximation. This mathematical wizard managed to come up with a series that computes 8 further decimals of pi with each term in the series, truly a creative feat. While this may be biased due to personal admiration, Ramanujan’s approximation is both practical and astonishing.

The Argument for Tau

Some mathematicians promote the use of tau (τ) as a better representation for the circle constant, defined as 2π. Proponents of tau argue that it represents 1/2 of a circle, making it more intuitive than pi. If pi were to be said to represent the full circumference of a circle, tau represents half of it. Thus, a fraction of pi would be a fraction of the circumference, making notation more straightforward.

My Favorite Rational Fractional Approximations

Here are some of my favorite rational fractional approximations to pi:

6098 / 2 4848 ≈ 3.141592… 8884 / 6674 ≈ 3.141592… 666 / 212 ≈ 3.141592… 666 √5 / 474 ≈ 3.141592… 697 / 222 ≈ 3.139682… 2 696 / 443 ≈ 3.141863… 7 697 / 1553 ≈ 3.141593… 2 985 / 627 ≈ 3.141592… 1595212 / 900002 ≈ 3.141592… 672 √2 √3 / 673 ≈ 3.1415893… 606 609 √2 5 / 606 617 ≈ 3.1415926… 666 - 311 / 668 - 555 ≈ 355 / 113 ≈ 3.141592654… 666 - 203 √6 / 667 - 306 ≈ 3.1415898…

Note: Some of the listed approximations are not rational. However, they are included as intriguing food for thought. While not rational, they provide an interesting perspective in the quest to represent pi.

A Sweet Approximation

A particularly sweet approximation of pi is given by the nested logarithmic expression:

π ≈ ?? ? ? log 2

This approximation is accurate up to at least four digits, showcasing the power of logarithmic expressions in approximating complex constants like pi.

Advanced Approximations

Shanks found a more precise approximation using the following expression:

π ≈ 6}{√{3502}} log 2 (7.37)10-82

where (u) is the product of four quartic units, defined as:

u asqrt{a^2-1} ^2 bsqrt{b^2-1} ^2 csqrt{c^2-1} sqrt{d^2-1}

With the given values for (a), (b), (c), and (d), this approximation significantly enhances the accuracy from (10^{-82}) to (10^{-161}).

These approximations highlight the beauty and complexity of pi, inviting mathematicians and enthusiasts to explore further.