A Disquisition on the Square Root of Three

“The Square Root of Three” is the title of at least one poem already, written by David Feinberg and recited by Kumar in the movie "Harold and Kumar Escape from Guantanamo Bay." Inspired by Feinberg's verses, and with unabashed shamelessness, I submit this “more mathematical” doggerel about the same algebraic irrational number.

But using the transcendental numbers \(\pi\) and \(e\) to approximate \(\sqrt{3}\) is very far afield of what is of interest herein. Far afield indeed, for the topic here concerns the very ancient concept of Diophantine approximations – that is, approximations of irrational numbers by rational numbers – with \(\sqrt{3}\) as the center of attention. The adjective Diophantine salutes Diophantus of Alexandria (circa 207–291 AD), whose book was entitled Arithmetica.

The first part of the exposition compares three methods of approximating \(\sqrt{3}\): Greek ladders, continued fractions, and Newton's Method. The second part addresses \(\sqrt{3}\) as the center of attention in what has become a long-standing disputation that is associated with Archimedes – and, with this paper, we enthusiastically enter that fracas.