Re: [Phys-l] Infinite recursion, iterated maps, ...

[chuck britton <cvbritton@mac.com>]

SOMEwhat like the math Proof by Induction that used to be taught in HS.

FIRST you assume that an answer exists (and show that it works for 
some limited case.)
THEN you prove that it generalizes to the infinite.

Strange results can follow if an assumed answer DOESN't work for the 
limited case.
.
At 8:43 AM -0700 4/19/11, curtis osterhoudt wrote:
> This is also the idea between renormalization group methods, and any technique
> where one suspects _something_ is repeated many times, on different scales,
> either spatial or temporal, or in some other domain. It's powerful stuff.
>
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> ________________________________
> From: John Denker <jsd@av8n.com>
> To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
> Sent: Tue, April 19, 2011 9:19:35 AM
> Subject: Re: [Phys-l] Infinite recursion, iterated maps, ...
>
> On 04/19/2011 06:57 AM, Herbert Schulz wrote:
> >  Imagine making a vertical cut to the right of the first R3.
>
> Right.
>
> ================
>
> It is worth thinking about the _idea_ behind this method of solution,
> because the idea comes up again and again.  Indeed this idea is
> closely related to the very definition of infinity.
>
> Here's a non-electrical purely-mathematical example.  R is
> defined by in infinitely recursive formula
>
>   R = crt(6 + crt(6 + crt(6 + crt(6 + ...))))       [A]
>
> where crt stands for cube root.
>
> Equation [A] or one of its congeners tends to show up on the high
> school math contest every year.  The algebra is dead simple;  the
> only challenge is to understand what "..." means.
>
> Another word for this type of problem is "iterated map".  The value R
> is a "fixed point".  Iterated maps show up a lot in physics, including
> dynamics, including chaotic dynamics.
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