The Joy Of X: A Guided Tour of Math, from One to Infinity

by Steven H. Strogatz

Still, what’s missing in all this worship of Euclidean rationality is an appreciation of geometry’s more intuitive aspects. Without inspiration, there’d be no proofs—or theorems to prove in the first place. Like composing music or writing poetry, geometry requires making something from nothing.

Mathematicians and conspiracy theorists have this much in common: we’re suspicious of coincidences—especially convenient ones. There are no accidents. Things happen for a reason.

A parabola is commonly defined as the set of all points equidistant from a given point and a given line not containing that point.

An ellipse is defined as the set of points the sum of whose distances from two given points is a constant.

Parabolas and ellipses: Why is it that they, and only they, have such fantastic powers of focusing? What’s the secret they share? They’re both cross-sections of the surface of a cone.

So that’s the secret: a parabola is an ellipse in disguise, in a certain limiting sense. No wonder it shares the ellipse’s marvelous focusing ability. It’s been passed down through the bloodline.