Around The World

Intuition does not always help us getting mathematical results right. Au contraire, some very simple results are blatantly counter-intuitive. For example, the circumference of a circle of radius $r$ is given by:

$c(r)=2\pi r$

Let’s say we’re interested in the Earth’s circumference. The radius $r$ is something in the order of 6400 Km, so $c(6400\text{ Km})\approx 40200\text{ Km}$. Now, what happens to the radius if we add just 1 meter to the circumference? You’d expect the radius to vary infinitesimally. Wrong!

Indeed:

If we add 1 unit of length to the circumference, we must find the variation $\epsilon$ to the radius. We write the equation

$c(r)+1=2\pi r+1=2\pi(r+\epsilon)$

that we solve for $\epsilon$.

$2\pi r+1 = 2\pi r + 2\pi\epsilon$

We remove $2\pi r$ on both sides:

$1=2\pi\epsilon$

We isolate $\epsilon$:

$\displaystyle\epsilon=\frac{1}{2\pi}$.

Adding 1 meter to the circumference of the Earth would not add infinitesimally to its radius, but about 16 cm! (or about $6\frac{1}{4}^{\prime\prime}$, for the metrically impaired).

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Another counter-intuitive result about spheres I like very much is that spheres do not grow forever as you add dimensions. For 1, 2, 3, 4, 5 dimensions, the volume of the radius 1 sphere increases. But lo! with 6 and more dimensions, the volume decreases!

In another post, I gave the formula for the unit sphere:

$\displaystyle V(d)=\frac{\pi^\frac{d}{2}}{\Gamma(\frac{d}{2}+1)}$

It is readily verified that it peaks just after 5 (and before 6), then decreases.