Owen . . .
Hmmm . . . several potential issues here . . .
If your goal is to "get to know Conway" better, then you really
ought to start with ONAG (On Numbers and Games), which is a "classic"
of sorts -- a non-standard way of developing number systems (but, if
you are thinking about going there, you'd probably have more fun
starting with Knuth's "Surreal Numbers" -- yes, that Knuth). If this
kind of stuff gets you going, then the Winning Ways (with Berlekamp
and Guy) volume(s) will give you more than you could ever want (and
still yet again more :-) . . .
On the other hand, my guess is that more interesting is likely to
be quaternions (aka hamiltonians), and their applications, in which
case Conway is probably not the best starting place. Specifically,
this would largely boil down to, are you more interested in "division
algebras", or in a "unified" framework for (-1, 1, 1, 1) signature
metrics for "minkowski space" where you can do special (and eventually
general) relativity in a "non-kludgey" way? :-)
If you want to see quaternions in action (and they are quite
fun :-), a reasonable place to look is here:
http://world.std.com/~sweetser/quaternions/ps/book.pdf
More generally, it might be worth noting that although "Conway's
Game of Life" gets lots of airplay, in Conway's intellectual life it
is almost certainly just about what it was . . . an evening's
amusement on a cocktail napkin! :-)
Remember that cellular automata basically got their start with von
Neuman's efforts to "automate" exploration of the universe :-)
Here's the problem: If you assume the universe is isotropic (the
"same" in all directions), then the "search space" grows quadratically
(at least) as you go out radially from the solar system, and given
(e.g., the Challenger example, which says we can barely put minimal
mass in low earth orbit) that there's no way we could "carry with us"
the supplies, or launch enough probes, to survey the universe, the
only "solution" is to send out a self replicating probe (it lands on
some planet, makes copies of itself, which then land on other planets,
make copies, etc.).
Around here though (actually, around 1950), we got the so-called
"Fermi paradox": "Where are they?" In other words, if the universe
is generally (locally) temporally (as well as spatially) isotropic, so
that "we" are just sort of average, then there must have been other
planets in other solar systems where intelligent life evolved long
(say millions or billions of years) ahead of us. In which case, since
"curiosity" is clearly part of "intelligence", at least one of those
species would have sent out a self replicating probe . . . and, since
exponential is bigger than quadratic (or cubic), the universe should
be "full" of copies of that original probe, so we should have seen at
least one by now!
Von Neumann asked the question, "Is there some theoretical reason
there can't be self replicating machines?" (although one might argue
that "life itself" is such a self replicating machine, so perhaps we
ourselves are just the current stage in the development of an earlier
"probe" that landed on earth long ago . . . :-). Anyway, von Neumann
set himself the task of designing a self replicating machine (and in
the process more or less invented cellular automata --where
appropriate credit should also go to Ulam). He did "solve the
problem" in the sense of a mathematical "existence proof". Von
Neumann's "machine" lived in a 2-d space, used orthogonal (4
neighbors . . .) neighborhoods, and each cell had 29 possible states,
and quite complicated "transition rules". You can look here:
http://en.wikipedia.org/wiki/Von_Neumann_cellular_automaton
Conway, at some point, said, "von Neumann's machine is way too
messy. Is there a simpler version?" The answer he found was "yes" --
2-d space, 2-state cells (although 8 neighbors rather than 4 . . .),
and very simple "transition rule". Cute little simplification of von
Neumann's original, but not particularly "deep" . . . Of course,
Martin Gardner's Mathematical Games Column in Scientific American
deserves most of the credit for the popularity of Conway's CA . . .
Oh, well . . .
tom
p.s. Some disclosure . . . my dissertation was on the (localized)
homotopy of the classical Lie groups (orthogonal, unitary, and
symplectic). The symplectic group is the group of (isometric)
rotations in n-dimensionsional (or, eventually, infinite dimensional)
quaternionic space . . . One of the nice tools I used was a
representation of quaternions as skew-symmetric 2x2 complex matrices,
and a representation of complex numbers as skew-symmetric 2x2 real
matrices, which induce mappings . . . --> Sp(n) --> U(2n) --> O(4n) --
> Sp(4n) --> . . .
hence, my longstanding enjoyment of quaternions (and, of course,
category theory, etc. . . .)
Also, it turns out I'm somewhat of a bigot -- I like my algebraic
structures to be associative, so I really don't like the octonians --
too weird for me!!! (the quaternions are non-commutative, and
that's bad enough! :-)
tom
On Oct 10, 2009, at 8:26 PM, Owen Densmore wrote:
Has anyone read this?
http://math.ucr.edu/home/baez/octonions/conway_smith/
I've not read enough Conway and I'm not sure where to start!
-- Owen
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============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
