Thursday, August 31, 2006

Mathcasting About Phi

The Pentagon
Google Earth

In support of the president's NCLB initiative, I cooked up what I call the "NCLB Polynomial" over on the Math Forum. I didn't invent the polynomial itself (it's very old), but came up with the name because children should be learning about it, if wanting to keep up with their studies at all (NCLB = No Child Left Behind).

Its positive geometric solution, phi (pronounced fie or fee), is the ratio of a regular pentagon's diagonal to its edge.

A symbol dance might start with a Golden Mean derivation, with a voice saying "the smaller is to the larger as the larger is to the whole" as on screen we see: smaller:larger = larger:whole, where whole = smaller + larger.

Since it's a ratio we want, it's OK to arbitrarily set the smaller segment to 1, such that 1/L = L/(1 + L) -- L for larger. Multiplying both sides by (1+L), then L, gives (1+L) = L*L or (L*L - L - 1) = 0 (the NCLB Polynomial). The positive solution is (1 + sqrt(5))/2, which is phi.

useful trig info
ray tracing by K. Urner

Over on edu-sig (a special interest group in the Python community), we've been looking at a Python generator for 1/phi (phi's reciprocal), using Fibonacci Numbers. A generator yields an interim value in response to a next() method, while remembering its internal variables between calls.

Fibonacci Generator
Python 2.5 on Windows XP

I'm not saying we need to pack all this information into just one mathcast. The strategy is to grow a large archive of clips, so teachers can embed them within their own presentations as they see fit.

Given Fibonacci (1170-1250) depended on Iraqi intelligence for his Liber Abacci, it'd make sense to use Baghdad as a backdrop for some of these clips. Of course many of our most talented and effective math teachers are Iraqi.

Getting off the XY plane, we get to the pentagonal dodecahedron (12 pentagonal faces, 20 vertices) and its dual, the icosahedron (20 triangular faces, 12 vertices -- and buildable from three phi rectangles).

In Synergetics, we jitterbug between the icosahedron and cuboctahedron, showing both have the same number of balls in their outer shells (1, 12, 42, 92, 162...). This connects us to the geodesic spheres and domes, crystallography, virology and hexapent chemistry (see below).

Related reading:
More About Geek TV (January, 2006)
Math Wars (continued) (March, 2006)
Brute force solutions (for phi, on edu-sig)
Ayatollah of the Tetrahedron (February, 2005)