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Topic: Which Polynomial?
Replies: 28   Last Post: Jul 27, 2006 3:13 PM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Re: Which Polynomial?
Posted: Jul 31, 2005 4:27 PM
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<<JP's enhancements noted>>

OK, that's clear enough. q^n gives successive terms in
the series f[n], and is solvable for q. From the roots,
we're able to generate the additively recursive sequences,
which are also geometric, including the canonical
Fibonaccis.

My curriculum gets to the same NCLB polynomial via the
Euclidean approach of using lengths, and asking how a
smaller part to a larger part might be equivalent to
the ratio between a larger part and the whole. That
nets us Phi, which we then explore as the most irrational
number by some measures, but as the simplest continued
fraction i.e.:

phi = 1 + 1/(1 + 1/(1 + 1/...)).

We develop the Fibonacci's separately, under the more
generic heading of sequences (convergent/divergent,
oscillatory/static, chaotic, and also: arithmetic,
geometric, recursive, interpolative...).

Simple sequences (per Encyclopedia of Integer Sequences)
represent our chance to write simple computer programs
(4-10 liners), and to connect to figurate and polyhedral
numbers, ala the triangulars and tetrahedrals in Pascal's
Triangle. We do cuboctahedrals and icosahedrals as well
(no, not optionally -- too important to skip (Alexander
Graham Bell, ya know)).

What converges to phi, as the Fibonacci's diverge, is the
ratio between f[n+1] and f[n]. Whether we want to jump
from here to your generative polynomial piece would be
at student/teacher disgression. Definitely it's a
connected topic. There's also going from continued
fractions to Diophantine equations, as discussed
previously (a few weeks ago), and to interesting
representations of roots. It's a vast network, and the
trick is to pick an intelligible path that (a) hangs
together (b) tells a good story (isn't too boring) (c)
contains life-relevant skill building.

I connect the dots using a computer language approach at
my Oregon Curriculum Network website.[1] Were I to add a
piece along the lines suggested by you and VS, it might
look something like (see Plain Text for proper
indentation):

>>> def genpoly(c0, c1, roots):
def somefunc():
n = 0
while True:
yield c0*roots[0]**n + c1*roots[1]**n
n += 1
return somefunc

>>> phi = (1+pow(5,0.5))/2
>>> roots = [phi, -1/phi]
>>> c0, c1 = 1/pow(5,0.5), -1/pow(5,0.5)


>>> fibfunc = genpoly(c0,c1,roots)
>>> fibfunc # a generator

<function somefunc at 0x00C965B0>
>>> sequence = fibfunc()
>>> sequence.next()

0.0
>>> sequence.next()
1.0
>>> sequence.next()
0.99999999999999989
>>> sequence.next()
2.0
>>> sequence.next()
3.0000000000000004
>>> sequence.next()
5.0000000000000009
>>> sequence.next()
8.0000000000000018
>>> sequence.next()
13.000000000000004
>>> sequence.next()
21.000000000000004

We can deal with rounding floats to corresponding integers, plus negative powers of n, in the revisions.

Kirby

[1] http://www.4dsolutions.net/ocn/numeracy0.html



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