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).

*
* *

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.

sphere-volume

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: