Date: Jul 30, 2005 12:02 PM
Author: Kirby Urner
Subject: Which Polynomial?

We'd likely consider it ludicrous were NCLB administrators to put out that they had a special polynomial in mind, for all school children of eligible age to solve beginning next school year.  Anti-NCLBers would scoff that you don't "solve" a polynomial because polynomials needn't equate to anything and therefore the matter of "solving for an unknown" (perhaps several) is moot.  Others would simply decry the dictatorial sound of "mandating a polynomial."

However, *were we* to accept the above surreal development (the NCLB's mandating of an Official Polynomial), it'd be interesting to speculate just *which* polynomial (equated to zero) that'd be. I have a candidate:

p**2 - p - 1 = 0.

Why?

First, let's look at a solution. Either completing the square or the quadratic formula suggest themselves. Solve it both ways. Your answers will be (1+pow(5,0.5))/2 and (1-pow(5,0.5))/2 [Python notation]. Only one of those is positive, so if looking for a geometric length, the positive root is the most relevant. And that solves as a decimal to around 1.618, which some of you may recognize. We're talking about phi ('fie' -- not 'pi' and not 'fee').

So where does this polynomial come from? From a diagram:

p q
|--------------|-----------|

The goal here is to establish this relationship:

(q:p) :: (p:(p+q))

or

q/p == p/(p+q)

Set q = 1 and solve for p.

phi is important as the ratio between a side and a diagonal of a regular pentagon. It is known as the "golden mean" and has been called that since ancient times. That should be your cue: this number *is* important. That's why the NCLB is on board. Don't leave any child behind next year. How 'bout it folks?

Kirby