[EM] Competitive Districting Rule (back on topic)

[raphfrk at netscape.net]

I was trying to think of a reasonable definition of the shape of a 
district such that it would be "reasonably contiguous"

One possibility that occured to me was using the ratio of the length of 
the border to the square root of the area

By using the sqrt, it doesn't matter the size.  2 shapes which are just 
scaled versions of each other will give the same result.

Some examples:

Circle:  sqrt(pi*r^2)/(2*pi*r) = 0.282 (I think this is the highest 
possible)

Square:  sqrt(a^2)/4a = 0.250

2:1 rectangle:  sqrt(2*1)/6 = 0.236

3:1 rectangle:  sqrt(3*1)/8 = 0.217

4:1 rectangle:  sqrt(4*1)/10 = 0.200

5:1 rectangle:  sqrt(5*1)/12 = 0.186

9:1 rectangle:  sqrt(9*1)/20 = 0.15

A good threshold might be 0.2.  This would allow districts with an 
aspect ratio of 4:1 or less.

It would also make it alot harder to have dumbbell shaped districts 
which are joined by a narrow channel.  Two equal circles that are 
touching wouldn't work.

sqrt(2*pi*r^2) / (4*pi*r) = 0.199

In fact, that is another good reason to use 0.2.

A circular district could have a "shooter" of length 1.29 times the 
radius (0 area but just adds to the border length).  However, this 
would also mess up the neighbouring district(s), which would then end 
up having to be circular too.

The rule would be something like "the length of the border of a 
district shall not be more than five times the square root of the area 
of the district"