Many of the entries in the Dictionary of Mathematical Eponymy have been 20th century eponyms - that’s been a deliberate choice: partly because I wanted to tread ground that was relatively new for me, partly because I wanted to find at least a few things named after women and - a few notable exceptions aside - women have only recently been allowed the opportunity to be in positions where things can be named after them.

Today, though, we’re going back to China around the start of the 14th century, where Zhao Youqin came up with a clever method for estimating $\pi$.

What is Zhao Youqin’s method?

Zhao Youqin came up with an iterative process for estimating the perimeter of a regular polygon, which can be inscribed in a circle. Suppose the radius of the circle is 1, and the $n$-gon inside it has a side length of $2L$.

  • Take the perpendicular bisector of one of the sides (this passes through the centre of the circle).
  • The segment of this line between the centre of the circle and the edge of the polygon is $d = \sqrt{1 - L^2}$.
  • The segment between the edge of the polygon and the circumference of the circle has length $e = 1-d$, so $e = 1 - \sqrt{1-L^2}$.
  • For reasons of later simplicity, I’m going to call the distance from the point where the bisector meets the circumference to an adjacent vertex of the $n$-gon, $2L_2$. Now, $(2L_2)^2 = L^2 + e^2$, and $e^2 = 2 - 2\sqrt{1-L^2} -L^2$. That means $(2L_2)^2 = 2 - 2\sqrt{1-L^2}$.
  • But $L_2$ is the side length of a regular $2n$-gon - so if we know the perimeter of an $n$-gon, we can explicitly calculate the perimeter of a $2n$-gon. If we do this multiple times, we can calculate the perimeter of something that’s practically a circle.

Why is this important?

Zhao Youqin used this iterative method with twelve steps to find the perimeter of a 16,384-gon, and hence an approximation of $\pi$ correct to six decimal places (finding a value very close to $\frac{355}{113}$, an excellent approximation.) He also determined that the approximations $3$, and $\frac{157}{50}$ were too small, $\frac{22}{7}$ too large.

I think it’s important because it’s a cool numerical method that’s 700 years old.

Who was Zhao Youqin?

Zhao Youqin (approx 1271-1335) was a Daoist philosopher, astronomer and mathematician who lived around the time of Kublai Khan’s conquest of China. He was especially well-renowned as an expert in optics.