$$\pi = 2 \times \left(\frac{2}{1}\times\frac{2}{3}\right)\times\left(\frac{4}{3}\times\frac{4}{5}\right)\times\left(\frac{6}{5}\times\frac{6}{7}\right)\times\cdots$$
Have a close look at this formula. In the numerator is 2 followed
by the squares of every even number.
In the denominator, if you shift every number one position to the left we
see the square of every odd number!
It needs to be written as displayed to actually work, but is almost as if it was
two times the squares of the even numbers divided by the squares of the odd numbers.
History
This formula comes from John Wallis (1616–1703), who has quite an interesting biography at a very turbulent time in English history. Wallis did not aim to become a mathematician, instead he studied liberal arts and theology at Emmanuel College, Cambridge to earn a B.A. in 1637 and an M.A. in 1640. He was subsequently ordained as a clergyman at 24 years of age.
This was a time leading up to the English Civil War, driven by a power struggle between the monarch, King Charles I and Parliament. The war officially broke out in 1642 between the Royalists who were loyal to the king vs the Parliamentarians, which Wallis supported. Rather unexpectedly, an encrypted message related to the war found its way to the clergyman Wallis in 1642, which he deciphered in about 2 hours. This was the catalyst for a career change for Wallis, now working as a codebreaker for Parliament. He was still not a mathematician.
By 1649, the Parliamentarians had defeated the king and put him on trial for tyranny and treason. King Charles I was convicted on 30 January 1649 and was then publicly beheaded. To many, this was unfathomable as they believed that the king inherited his rule from God himself. Wallis had supported Parliamentarians but opposed the killing of the king, which would become important later in his life.
Around this time Wallis was in his early 30s and began taking interest in mathematics. Having strong political ties and having played an important role in code breaking during the war, he remarkably was appointed Savilian Professor of Geometry at Oxford before he had published a single significant mathematical work. With no reputation for mathematics, Wallis described this next unexpected new career change: mathematics "had before been a pleasing diversion, but was now to be my serious study."
It took some time before this Professorship was justified but after about 7 years, Wallis had proved that he was worthy of the position. Two major mathematical works were published with several new discoveries including Arithmetica Infinitorum (1656), which contained the Wallis product among other results.
But the political turmoil did was not over, as the Parliament rule was not very popular and people wanted a return of the previous familiar government. In 1960, the monarchy was restored and the new king was Charles II, the son of Charles I. Upon becoming king, Charles II went on a revenge campaign and executed many of the Parliamentarians who had convicted his father. This is where Wallis's opposition to executing Charles I saved him: Wallis was allowed to keep his position at Oxford and was even appointed a royal chaplain to Charles II.
Beyond the product that bears his name, Wallis is credited with introducing the ∞ symbol for infinity, laying groundwork for the development of calculus, and making important contributions to algebra and analytic geometry.
Show the math
Wallis did not have the tools we have today, so we will present an alternative, simpler proof using the Euler infinite product for the sine function (see references):
$$\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1 - \frac{x^2}{n^2\pi^2}\right)$$
Substituting x = π/2, and remembering sin(π/2) = 1, the
left-hand side becomes sin(π/2) / (π/2) = 2/π, so:
$$\frac{2}{\pi} = \prod_{n=1}^{\infty}\left(1 - \frac{1}{4n^2}\right) = \prod_{n=1}^{\infty}\left(\frac{4n^2 - 1}{4n^2}\right) $$
Taking the reciprocal of both side yields:
$$\frac{\pi}{2} = \prod_{n=1}^{\infty}\frac{4n^2}{4n^2 - 1} = \prod_{n=1}^{\infty}\left(\frac{2n}{2n-1}\cdot\frac{2n}{2n+1}\right)$$
Writing out the first few terms of that product (n = 1, 2, 3, ...)
gives:
$$\frac{\pi}{2} = \frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdots$$
Multiply both sides by 2 to get the π-form used above.
This proof takes advantage of knowledge that Wallis did not have at the time,
as the Euler infinite product was not discovered until well over a hundred years later.
Wallis took a different approach. He was looking at areas under the
curve (1−x2)n and considered n=1/2.
He used what we now would call interpolation to calculating these areas.