# Archimedes’ Circular Reasoning

Every geometry textbook has formulas for the circumference ($$C = 2 \pi r$$) and area ($$A = \pi r^2$$) of a circle. But where do these come from? How can we prove them?

Well, the first is more a definition than a theorem: the number $$\pi$$ is usually defined as the ratio of a circle’s circumference to its diameter: $$\pi = C/(2r)$$. Armed with this, we can compute the area of a circle. Archimedes’ idea (in 260 BCE) was to approximate this area by looking at regular $$n$$-sided polygons drawn inside and outside the circle, as in the diagram below. Increasing $$n$$ gives better and better approximations to the area.

Look first at the inner polygon. Its perimeter is slightly less than the circle’s circumference, $$C = 2 \pi r$$, and the height of each triangle is slightly less than $$r$$. So when reassembled as shown, the triangles form a rectangle whose area is just under $$C/2\cdot r = \pi r^2$$. Likewise, the outer polygon has area just larger than $$\pi r^2$$. As $$n$$ gets larger, these two bounds get closer and closer to $$\pi r^2$$, which is therefore the circle’s area.

Archimedes used this same idea to approximate the number $$\pi$$. Not only was he working by hand, but the notion of “square root” was not yet understood well enough to compute with. Nevertheless, he was amazingly able to use 96-sided polygons to approximate the circle! His computation included impressive dexterity with fractions: for example, instead of being able to use $$\sqrt{3}$$ directly, he had to use the (very close!) approximation $$\sqrt{3} > 265/153$$. In the end, he obtained the bounds $$3\frac{10}{71} < \pi < 3\frac{1}{7}$$, which are accurate to within 0.0013, or about .04%. (In fact, he proved the slightly stronger but uglier bounds $$3\frac{1137}{8069} < \pi < 3\frac{1335}{9347}$$. See this translation and exposition for more information on Archimedes’ methods.)

These ideas can be pushed further. Focus on a circle with radius 1. The area of the regular $$n$$-sided polygon inscribed in this circle can be used as an approximation for the circle’s area, namely $$\pi$$. This polygon has area $$A_n = n/2 \cdot \sin(360/n)$$ (prove this!). What happens when we double the number of sides? The approximation changes by a factor of $$\frac{A_{2n}}{A_n} = \frac{2\sin(180/n)}{\sin(360/n)} = \frac{1}{\cos(180/n)}.$$ Starting from $$A_4 = 2$$, we can use the above formula to compute $$A_8,A_{16},A_{32},\ldots$$, and in the limit we find that $$\pi = \frac{2}{\cos(180/4)\cdot\cos(180/8)\cdot\cos(180/16)\cdots}.$$ Finally, recalling that $$\cos(180/4) = \cos(45) = \sqrt{\frac{1}{2}}$$ and $$\cos(\theta/2) = \sqrt{\frac{1}{2}(1+\cos\theta)}$$ (whenever $$\cos(\theta/2) \ge 0$$), we can rearrange this into the fun infinite product $$\frac{2}{\pi} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}} \cdots$$ (which I found at Mathworld). (It’s ironic that this formula for a circle uses so many square roots!)

# Astounding “Natural” Identities

The modest sequence $$1,2,3,4,\ldots$$ can do some rather awe-inspiring things, when properly arranged. Here’s a short list of some of its many impressive feats.

There are numerous expressions for $$\pi$$ relying on the progression of integers, including the Wallis formula: $$\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots$$ (which can be derived from Complex Analysis using an infinite product representation for the sine function) and an elegant alternating sum: $$\frac{\pi-3}{4} = \frac{1}{2\cdot 3\cdot 4}-\frac{1}{4\cdot 5\cdot 6}+\frac{1}{6\cdot 7\cdot 8}-\cdots$$ (try to prove this!). Euler’s number $$e$$ has similarly surprising formulas, such as $$\frac{1}{e-2} = 1+\frac{1}{2+\frac{2}{3+\frac{3}{4+\frac{4}{5+\cdots}}}}$$ (listed at MathWorld) and $$\frac{e}{e-1} = 1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{2+\cdots}{3+\cdots}}}{2+\frac{2+\frac{2+\cdots}{3+\cdots}}{3+\frac{3+\cdots}{4+\cdots}}}$$ (which is problem 1745 in Mathematics Magazine, posed by Gerald A. Edgar).

The list doesn’t stop here! The nested square-root identity $$3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$ is attributed to Ramanujan (on this Wikipedia page). As another curiosity, the sequence $$\frac{1}{2},\qquad \frac{1}{2} \Big/ \frac{3}{4},\qquad \frac{\frac{1}{2} \big/ \frac{3}{4}}{\frac{5}{6} \big/ \frac{7}{8}},\qquad \frac{\frac{1}{2} \big/ \frac{3}{4}}{\frac{5}{6} \big/ \frac{7}{8}} \bigg/ \frac{\frac{9}{10} \big/ \frac{11}{12}}{\frac{13}{14} \big/ \frac{15}{16}},\qquad\ldots$$ (which relates to the Thue-Morse sequence) can be shown to converge to $$\sqrt{2}/2$$.

There are many pretty/unexpected/crazy formulas obtainable from the natural numbers $$1,2,3,4,\ldots$$ that could not fit in this post. What are some of your favorites?