Re: OT: baloney and cheese

On Mon, Aug 04, 2003 at 03:53:07PM -0400, John Cowan wrote:
> On the contrary!  Georgia has four times as many counties, but is only 20%
> bigger in area (50,000 mi^2 vs. 60,000).  So on average one must go about
> 1.8 times as far to reach the county seat (or other distinguished point) in
> England than in Georgia.
Well, the mean GA county has an area of ~375 sq. miles; the mean
UK county has an area of ~ 1275 sq. miles.  That's easy enough.
Finding the average distance between any two points within one of
those counties is a harder problem that depends upon the shape of the
county.

If all of the counties in question were perfect circles, then the
increase  would be proportional to the increase in radius -
which is, in this case, about a factor of 1.8, as John said.
That's a lower bound.

There is no upper bound.  If you imagine that each county is a
rectangle, you can reduce one dimension to as small as you like
while increasing the other to maintain the same area.   Then the
county becomes a very long, thin strip, and you can always make it
thinner and longer.  The limit as one dimension approaches zero is to
have the other dimension - and with it, the average distance - approach
infinity.

More realistically, if the counties were all rectangles of which
one side was exactly 1 mile, then the increase in mean distance would be
proportional to the increase in area, which is about 3.4.  The real
number lies somewhere in between and probably rather closer to 1.8 than
3.4.  I'd call it 2 and have done. :)

-Mark