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From: matt@oddjob.UUCP (Matt Crawford)
Newsgroups: net.astro,net.sci
Subject: Re: Dimensions
Message-ID: <777@oddjob.UUCP>
Date: Tue, 11-Jun-85 17:05:07 EDT
Article-I.D.: oddjob.777
Posted: Tue Jun 11 17:05:07 1985
Date-Received: Wed, 12-Jun-85 20:03:02 EDT
References: <2529@decwrl.UUCP>
Reply-To: matt@oddjob.UUCP (Matt Crawford)
Organization: U. Chicago, Astronomy & Astrophysics
Lines: 34
Xref: watmath net.astro:658 net.sci:345

In article <2529@decwrl.UUCP> kallis@pen.DEC (Steve Kallis, Jr.) writes:
>Steve Aldrich asks if there's a clearcut definition of a "dimension."
>There are many definitions.
>
>Primarily, a dimension is a measurement.  Or something that can be
>quantified.  . . .
>Time is a dimension, which we measure in seconds, minutes, hours, years, 
>and so on.  . . .
>Brightness or intensity is a dimension, usually measured in "magnitude."
>At least in astronomical circles.  It is also measured in foot-candles.
>Temperature is a dimension, measured in degrees Kelvin, Celsuis, Farenheit,
>or Rankine.

Steve Kallis is (perhaps understandably) mixing the two usages of the
word "dimension".  One usage is synonymous with "units", as in:
	Velocity has dimensions of length/time
	Newton's constant has dimensions (length cubed)/(mass * time squared)
	The fine structure constant is dimensionless.

The term "dimensionless units" is also employed to mean that all quantities
have been multiplied by the appropriate powers of of the speed of light,
Newton's constant, Boltzmann's constant and Planck's constant to cancel all
the units.  In effect, this sets all four of those constants equal to 1.

The geometrical meaning of "dimension" does not depend on the existence
of a set of mutually perpendicular coordinate axes.  Loosely speaking,
a manifold (such as space-time) has dimension n if each small region of
the manifold can be put into a smooth 1-to-1 correspondence with a subset
of the ordinary Euclidean space R^n.

I hope this clarifies rather than the opposite.
_____________________________________________________
Matt		University	crawford@anl-mcs.arpa
Crawford	of Chicago	ihnp4!oddjob!matt