Re: game of life question
- From: Subluxian <cbrown@xxxxxxxxxxxxxxxxx>
- Date: Tue, 31 Jul 2007 11:47:29 -0700
On Jul 30, 4:20 pm, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Subluxian <cbr...@xxxxxxxxxxxxxxxxx> writes:
On Jul 30, 6:58 am, Allan Adler <a...@xxxxxxxxxxxxxxxxxxxx> wrote:
Let L denote the set of all points of the plane with integer coordinates
and let P denote the set of all subsets of L. Conway's game of life
determines a mapping T from P to itself. Call an element x of P timeless
if there is a sequence x0=x, x1, x2, ... of elements of P such that, for
every positive integer n, we have T(x(n)) = x(n-1).
It seems that you are asking for those elements x in P such that T(x)
= x. These are usually called "stable configurations".
No he isn't. He's asking for configurations that have a possible
ancestry of infinite length.
Yes, I misread and got the time direction wrong.
In that case, of course stable configurations have such an ancestry,
as do oscillating configurations.
If we insist that the sequence contain no "repeats", then a single
glider crossing an infinite expanse meets this requirement as well;
and there are many configurations which can be arranged as the
collision of some number of gliders which approach "from infinity".
There /is/ a term for an opposite sort of idea: Garden of Eden
configurations; which are configurations x such that there exists no y
such that x = T(y).
A "timeless" configuration is never a Garden of Eden configuration;
but the descendants of a Garden of Eden configuration may be
"timeless", because there are in general multiple possible ancestors
for any given configuration.
Cheers - Chas
.
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- From: Allan Adler
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