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.
________________________________
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.
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l