Some of the paper is highly mathematical (the proof that an entire
class of compactness measures ranks any two redistricting plans
exactly the same). However, most of the paper is very understandable,
especially the counterexamples showing why most of the proposed
measures of compactness can be used for gerrymandering because they do
not reliably measure compactness.
Legislative districts in the 50 states are being redrawn following the
completion of the 2010 United States census. Thirty-five states
require that districts are compact, which is believed to make
gerrymandering - designing legislative districts so as to advantage
one political party - more difficult. There are now more than a dozen
proposed competing numerical measures of the relative compactness of
legislative districts. This article demonstrates that nine of the
proposed measures of compactness do not reliably measure compactness.
Pictorial counterexamples show that these nine proposed measures of
area compactness assign the exact same value to shapes having visually
distinct compactness levels. Next, this paper mathematically proves
that all area-to-perimeter or area to square-of-perimeter measures (or
their reciprocals or square roots) rank the compactness of any two
sets of redistricting plans in the exact same order. Thus, these
reliable proposed measures of district compactness are equivalent to
the simplest such measure defined as the ratio of area to the square
of the perimeter. An index of compactness is conceptually and
computationally best when it has a maximum value of one (1) when the
area is as compact as a circle, a minimum value approaching zero when
the area's perimeter is very large compared to its area, and provides
a direct comparison of any two districts' compactness regardless of
district area size. This compactness measure is 4π times the ratio of
the area of the district to the square-of-its-perimeter, known in
mathematics as the isoperimetric quotient. This paper concludes by
briefly setting the general context within which a compactness measure
is applied to compare proposed district plans meeting other crucial
The Legislative Redistricting Context of Compactness Measures
In drawing district maps, compactness only comes into play after other
considerations are met. 'Communities of interest' should be preserved
as much as possible and "neighborhoods mean a lot in redistricting ...
neighborhoods which would probably not pass a compactness test. Our
streets are said to follow the windings of cow paths and our
neighborhoods are fairly unruly in shape." A state district map
should be chosen that:
(i) has equal population districts to within 0.5%;
(ii) utilizes natural and geographic boundaries and barriers such as
vehicular impassable mountain ranges and rivers in the creation of
district boundaries; and
(iii) utilizes existing neighborhood, municipal and county government
boundaries in the creation of district boundaries.
One possible procedure for redistricting, might be to allow various
parties to propose district maps that meet the above three conditions,
and the "winning" map would be one that has a minimum sum of
compactness of districts, and has a minimum ratio of the number of
uniquely administered districts to the number of election
jurisdictions (to keep election administrative simple). These last
two conditions could be equally weighted.
Such a process for deciding on which district map to adopt might
reduce the need for subjective judgments or competing claims and
create districts that better serve voters and are easier for
representatives to serve and for election officials to administer.
 Email exchange of Sat Jun 4, 2011 on the email discussion list
LWVTopics@???. The quote is by Hollie Courage, former
President of the League of Women Voters of Rhode Island (LWV, RI).
Barbara Klein of the LWV, Arizona reminded me during the same
conversation of the need to preserve 'communities of interest'.
Please see http://www.lwvri.org/redistricting.htm and "The Law and
Drawing District Lines"
Some of the language in (ii) and (iii) is borrowed from Roz McGee,
a Utah House member who made a different redistricting proposal in the
2007 Utah legislative session.
http://electionmathematics.org Town of Colonie, NY 12304
"One of the best ways to keep any conversation civil is to support the
discussion with true facts."