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 is given by:
Let’s say we’re interested in the Earth’s circumference. The radius is something in the order of 6400 Km, so . Now, what happens to the radius if we add just 1 meter to the circumference? You’d expect the radius to vary infinitesimally. Wrong!
If we add 1 unit of length to the circumference, we must find the variation to the radius. We write the equation
that we solve for .
We remove on both sides:
We isolate :
Adding 1 meter to the circumference of the Earth would not add infinitesimally to its radius, but about 16 cm! (or about , 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:
It is readily verified that it peaks just after 5 (and before 6), then decreases.