From: Anders Sandberg (asa@nada.kth.se)
Date: Tue Jan 28 2003 - 11:12:46 MST
On Tue, Jan 28, 2003 at 09:34:11AM -0500, Artillo5@cs.com wrote:
>
> The way I look at it, all a "dimension" is really just an additional
> parameter for describing anything.
This is roughly how it is handled in mathematics (with some
complications since some spaces are weird: see
http://mathworld.wolfram.com/Dimension.html for a fuller discussion -
see the links for much fun).
> But if you extend the usage of the term "dimension" to describe *any*
> parameter describing something, then you can see where things like charge,
> spin, time, etc. come in to play as "dimensions" to describe stuff.
In mechanics it is often convenient to treat the positions of particles
or objects and their momenta as dimensions. So a single point particle
has three dimensions of location and three extra dimensions; an
N-particle system would have 6N dimensions (or "degrees of freedom" as
it is usually called). This way one can look at the state of the system
as a single point moving in a 6N dimensional state space, subject to
constraints due to its energy, forces etc.
In the same way quantum mechanics deal with things - the description of
a system is a so-called wave function that for each position in state
space tells you the probability of finding the system there (and some
other stuff like "phase"). Actually, quantum mechanics is a theory
about how these wave functions change over time or when you interact
with them - one often deals with wave functions of a system as points in
an infinite-dimensional Hilbert space. As the wave function changes over
time it traces out a path through this space, where each instant
corresponds to a particular probability distribution of finding the
system in a particular state. Abstract, but highly useful and elegant.
In my own neural network research each state of a neuron can be viewed
as a dimension, and the dynamics of my network corresponds to movement
of a point through a N-sqrt(N) dimensional space. Rather hard to
visualize, unfortunately.
-- ----------------------------------------------------------------------- Anders Sandberg Towards Ascension! asa@nada.kth.se http://www.nada.kth.se/~asa/ GCS/M/S/O d++ -p+ c++++ !l u+ e++ m++ s+/+ n--- h+/* f+ g+ w++ t+ r+ !y
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