Approximating pi: think like Archimedes!

November 21, 2017

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Archimedes of Syracuse (287-212 BC) was a Greek physicist and mathematician best known for a very famous and amusing anecdote: the king of Syracuse had a golden crown made by a blacksmith. He distrusted the blacksmith and suspected him of not having used pure gold for the crown, but having mixed silver in. He had no method of finding out, however. He decided to give the crown to Archimedes to let him devise a solution to the problem. Later, Archimedes, while stepping into a bath, noticed that the water level rose exactly by his body volume. He immediately tested the water displacement by the crown to determine its volume, which he could then divide by its weight to obtain its density. According to the story, after finding out that the density of the crown was too low to be of pure gold, Archimedes yelled “Eureka!” and immediately ran, still naked, to the king of Syracuse to report his findings.
One of the many other things Archimedes has worked on is an approximation of π, the ratio of the circumference of a circle to its diameter (approximately 3.141592654). π, as we now know, is an irrational number, meaning that it is an unending, never repeating decimal and therefore not a fraction of two integers. Many approximations of π have been given in the past. A quote from the Hebrew Bible talking about Solomon’s temple: “…a molten sea, ten cubits from the one brim to the other: it was round all about, … a line of thirty cubits did compass it round about” suggests that π is equal to 3. In this article, we are going to work on an approximation of π in the spirit of Archimedes. Let’s see if we can do better than the Hebrew Bible!

Approximating π

To approximate π, Archimedes used regular polygons. As you can see in the images, the more sides a regular polygon has, the closer it gets to a circle. We inscribe the polygons in circles and now we can use the the following ratio to approximate π.

    \begin{equation*} \pi\approx\frac{\textrm{circumference of the polygon}}{\textrm{diameter of the circle}} \end{equation*}

As the circumference of the polygons gets larger as they get more sides, the ratio we find increases with the number of sides of a polygon. As the number of sides approaches infinity, the ratio approaches π. Hence, we are approximating π from below. For a few polygons with a small number of sides we get the following approximations:

We take the diameter of the circle to be 1, meaning that the radius is 0.5. We draw bissectrices from the corners to the centre of the circle, with one bissectrice extending to another side of the triangle, creating a 90° angle. As all angles of an equilateral triangle are 60°, the bissectrice cuts it into a 30° angle. Now we have a 30°-60°-90° triangle, of which the sides are in a 1-√3-2 ratio. Hence, the part belonging to the triangle circumference has length 0.25√3, and that multiplied by 6 gives circumference=1.5√3. As the diameter is 1, the ratio is also equal to 1.5√3, or approximately 2.60.

As a square’s angles are all 90°, by bisecting an angle we get 45° angles. From the figure, it is clear that we get a 45°-45°-90° triangle, with the sides in a ratio of 1-1-√2. As the radius is 1, the sides of the square have length 0.5√2. This multiplied by 4 gives circumference=2√2. As the diameter is 1, the ratio is also equal to 2√2, or approximately 2.83.

As all angles of a hexagon are 120°, bisecting them gives angles of 60°. Now we have an equilateral triangle, of which the sides all have the same length. Letting the diameter of the circle be 1, we have that the radius and the sides of the hexagon are all equal to 0.5. Multiplying this by 6 gives circumference=3. As the diameter is 1, the ratio to approximate π is also equal to 3.

General formula

By using a hexagon, we are already doing as good as the Hebrew Bible. But we can get much closer to π by finding the approximation of π by a general n-sided polygon and taking a very large number for n.

The method goes as follows for an n-sided polygon: draw lines from every corner of the polygon to the centre of the circle. We now have n isosceles triangles. We can divide these n triangles in half by a line from the centre of the circle to the midway point of each edge, giving us 2n right-angled triangles. By taking the sine of the corner at the centre of the circle, we get the length of half of the edge of the polygon. Multiplying by 2n gives us the circumference of the polygon.
Working out the formula gives:

    \begin{equation*} \begin{split} \textrm{Ratio}&=\frac{\textrm{circumference of the polygon}}{\textrm{diameter of the circle}}\\\\ &=2n*0.5\sin(\frac{180}{n})=n*\sin(\frac{180}{n}) \end{split} \end{equation*}

I have computed a table for some values of n. Note that we correctly worked out the approximation of π for n=3,4,6:
However, Archimedes could not use this formula as he did not have an easy way to calculate the sine of an angle. Instead, he used an iterative process in which the circumference of a 2n-sided polygon could be determined by the circumference of an n-sided polygon. He stopped at a 96-sided polygon, giving an approximation of π of \frac{223}{71}<\pi<\frac{22}{7}, accurate to two decimal places.
Nowadays, computers have estimated π to over 20 trillion digits. We still use algorithms such as the circumference of an n-sided polygon sequence, but now we have much faster converging sequences such as

    \begin{equation*} \frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{k=0}^\infty\frac{(4k)!(1103+26390k)}{(k!)^4396^{4k}} \end{equation*}

Whose first iteration already gives an approximation of π correct to 6 digits! It is clear that Archimedes cannot compete with modern-day technology. However, Archimedes held the best approximation of π for around 400 years, and that’s something that ever-improving technology will never be able to do!

This article is written by Balder Stalmeier.


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