🚀 go-pugleaf

>The truth is, there are very few topics as narrow as the factoring of
>polynomials that one ever MUST know in mathematics or any other field
>for that matter.  I am thankful for having learned factoring.  I am thankful
>for having learned even those things I may never use.
>

I have to agree with Charles, that only a few core concepts 
are in the MUST category, with lots else more specific to
a field.  

You want kids to get some sense of mathematics as a sprawling 
network of interlinked topical areas, sometimes only tenuously 
relevant to one another.  Part of adequately equipping students 
involves supplying some roadmaps, giving them an overview of the 
areas in which mathematics is expanding (an historical approach 
is useful here, also with some attention to areas in which math 
has dead ended, or lies dormant).

In my own case, my training in polynomials came in useful 
when I was trying to derive a formula for the cumulative 
number of balls in a cuboctahedral packing.  Starting with
a nuclear ball, you pack 12 around it, each tangent to the 
central one and 3 neighbors.  A second layer packs around the 
first, containing 42 balls, a third layer has 92 balls and so 
on -- an expanding cuboctahedron.

Given that I already knew the number of balls in each
successive layer was given by 10 x F x F + 2, where 
F= layer number (e.g. 12 when L = 1, 42 when L = 2),
my problem was to find a polynomial of the third 
degree which would give the cumulative number of 
spheres out to the nth layer.

My solution was to write the polynomial:

N = AF^3 + BF^2 + CF + D

and then use actual (F,N) pairs, e.g. (1,13),(2,55)
(3,147) (4,309) to produce four linear equations.  By 
solving these, I was able to find values for A, B, C and D:

A = 10/3
B = 5
C = 11/3
D = 1

See:
http://www.inetarena.com/~pdx4d/synergetica/synergetica2.html

The cuboctahedral packing is useful to study because it's
identical to the so-called 'face centered cubic' packing
used all the time in crystallography.  It defines a lattice
of tetrahedral and octahedral volumes of ratio 1:4 and
appears in architecture as the 'octet truss' (you can 
find it everywhere), pioneered in part by the American
inventor Alexander Graham Bell, inventor of the telephone.

See: 
http://www.teleport.com/~pdx4d/bell.html


Kirby

PS:  I don't take credit for advancing mathematics as a
whole with the above derivation, merely my own appreciation
and understanding of it (which, when it comes to being a 
student, is precisely the point).  I later found out about 
Bernoulli's much earlier forays into this area -- my 
discoveries connect directly to his.  We need to empower
kids with the knowledge that 'obtaining a result already
obtained by someone else in the past' is NOT a put down,
but exciting confirmation of the power of logic to 
transcend generations and personal circumstances.  

Too often the academic culture of 'me first' is allowed 
to occlude and obscure the point Socrates wanted to make:
we each contain within direct access to a logical core,
an inner genius, which good teachers seek to awaken.

"Remember when you were young,
 You shone like the sun.
 Shine on, you crazy diamond."

            from Pink Floyd's
            Wish You Were Here


----------------------------------------------------
Kirby Urner  "ALL realities are 'virtual'" -- KU
Email: pdx4d@teleport.com
Web:   http://www.teleport.com/~pdx4d/