Gerrymander and the Need for Redistricting Reform

by Michael D. Robbins

December 5, 2000

Revised January 2, 2007

This self-contained comprehensive article on reapportionment, redistricting, and gerrymander is suitable for both individual and classroom use.

Gerrymandering is the drawing of election district boundary lines for partisan advantage, to favor the majority party and incumbent politicians of all political parties.

Abstract    ( concise abstract )
Table of Contents
Article
Side Bar Index

Keywords: redistricting, reapportionment, gerrymander, gerrymandering, gerrymandered, gerymander, gerymandering, gerymandered, germander, germandering, germandered, jerrymander, jerrymandering, jerrymandered, jerymander, jerymandering, jerymandered, redistricting map, election, fraud, election fraud, packing, vote dilution, cracking, gerymander, gerymandering, Redistricting, Reapportionment, Gerrymander, Gerrymandering, Gerrymandered, Redistricting Map, Gerymander, Gerymandering, Gerymandered, Germander, Germandering, Germandered, Jerrymander, Jerrymandering, Jerrymandered, Jerymander, Jerymandering, Jerymandered, REDISTRICTING, REAPPORTIONMENT, GERRYMANDER, GERRYMANDERING, GERRYMANDERED, JERRYMANDER, JERRYMANDERING, JERRYMANDERED, JERYMANDER, JERYMANDERING, JERYMANDERED, REDISTRICTING MAP, compactness metric, fairness metric, algorithm, fair, unfair, Grid Gerrymander, Checkerboard Gerrymander, Chessboard Gerrymander, Checker Board Gerrymander, Chess Board Gerrymander, ballot, vote, voter, voter fraud, corruption, politics, politician, political, political corruption, reform, redistricting reform, political reform, initiative, voter initiative, redistricting reform initiative, teach, explain, describe, understand, understanding, classroom, school, class, class project, teacher, demonstration, instruction, instruct, educate, education, instructor, educator, redistrict, reapportion, census, demographic data, race, racial, race-based, preferences, affirmative action, segregated, segregation, majority-minority, minority-majority, Gerry, Elbridge Gerry, party, fair, unfair, entrenched, incumbent, safe, noncompetitive, district, legislature, Congress, representative, Constitution, electronic, voting, system, electronic voting, electronic voting system, liberal, judicial activist, judicial activism, conservative, strict constructionist, Republican, Democrat, Democratic, ACLU, Assembly Speaker, Willie Brown, Reform Party, Pat Buchanan, Pete Wilson, Arnold Schwarzenegger, Fraud, Factor, Fraud Factor, FraudFactor, Michael D. Robbins, Michael Robbins, Mike Robbins

Abstract:

The U.S. Constitution requires that every ten years a national census be taken and the results of that census be used to reapportion representatives in Congress among the states according to population. After reapportionment, each state must perform redistricting, the process of re-drawing the election district boundary lines for each type of state or federal office (e.g., state senator, state assemblyman, congress representative, etc.) so that all districts for the same type of office have nearly identical voter population. Local governments such as cities and counties also perform redistricting for their elected offices. The purpose of reapportionment and redistricting is to preserve the one voter-one vote fairness principle.

The gerrymander is a form of election fraud that misuses redistricting to violate the one voter-one vote fairness principle that redistricting is intended to preserve. Gerrymandering is the process where the majority party draws an election district map with district boundary lines that give itself an unfair and undeserved numerical vote advantage during each election. This numerical advantage is obtained by maximizing the number of districts with a majority of voters from the majority party. Here, "majority party" refers to the party with a majority of seats in the state legislature, which usually but not always corresponds to the party that received the majority of total votes in the previous election. Exceptions are possible due to gerrymanders.

A gerrymandered redistricting map concentrates minority party voters into the fewest possible number of election districts (packing), distributes minority party voters among many districts so their vote will not influence the election outcome in any one district (vote dilution), and/or divides incumbent minority party legislator districts and constituents up among multiple new districts with a majority of majority party voters (cracking). In some gerrymander cases, multiple minority party incumbents are forced to run against each other in the same district.

Bizarre election district boundaries are drawn to connect distant disjoint areas with thin strips of land running through unpopulated areas such as industrial parks and cemeteries, down highways and railroad tracks, and through bodies of water such as rivers, lakes, and the ocean. Modern computers are used to produce optimized gerrymanders. Gerrymandering is the equivalent of legally stealing an election by stuffing the ballot box with fraudulent ballots voted for the incumbent or removing legitimate ballots voted for the challengers before they are counted. However, the gerrymander is a mathematical concept and is not easily explained or understood without the use of effective visual aids.

Gerrymandered election district map boundaries give a disproportionately large amount of legislative seats and political power to the majority party, which perpetuates the gerrymander cycle decade after decade. Gerrymander also produces safe noncompetitive election districts for all incumbents of all parties, resulting in massive political corruption and nonresponsive government. Gerrymanders hurt all voters of all parties. The voters can stop this form of election fraud and the resulting political corruption by enacting redistricting reform laws through voter initiative in states where the right of voter initiative is available. Redistricting maps can also be challenged in the courts.

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Table of Contents

• Introduction
• Gerrymander Violates the Fairness Principle
• Gerrymander as a National Issue
• Gerrymander is a Mathematical Concept
• Brief History
• Tools to Teach About Gerrymandering
  • Gerrymander Analogy
  • Simple Graph System
  • Actual Redistricting Maps
  • Compactness Metric
  • Simple Graph System Example
  • Class Project and Demonstration
  • A Simple to Use Analytical Tool
• Impact on Specific Demographic Groups
• Statistical Sampling and Counting Illegal Aliens
• Electronic Voting Schemes
• The Vicious Cycle of Gerrymandering
• The Need for Redistricting Reform
• Redistricting Reform Efforts in California
  • California Governor Arnold Schwarzenegger's proposed Redistricting Reform initiative for 2005 - Proposition 77 - will be on the ballot for the special election on Tuesday, November 8, 2005.

Side Bars

• Concise Abstract 
• Definition of Gerrymander
• Definition of Elbridge Gerry
• Biography #1 of Elbridge Gerry
• Biography #2 of Elbridge Gerry
• Excerpts from the U.S. Constitution on Reapportionment
• California Constitution Article 21 on Reapportionment
• Closed Primary Elections
• Presidential Elections and the Electoral College
• Bicameral State Legislature
• The Danger of Judicial Activism
• Louisiana Demographic Map Examples
• Deriving the Formula for a Compactness Metric for an Election District
• Formulas for the Area and Centroid (Center of Mass) of a Polygon
• Mathematical Proofs
• Theoretical Grid and Checkerboard Gerrymanders
• An Example Demonstrating Unsafe, Competitive Districts
• An Example Demonstrating The Minority Party Controlling a Majority of Districts

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Have you ever wondered why many politicians engage in political corruption without concern about voter anger and rejection at the next election? Or why they engage in punitive legislation targeting their opponents and even their own constituents? Or why they cater to special interests while ignoring the needs of their voter constituency? Or why incumbent politicians abuse illegal narcotic drugs and alcohol while in office, and still get re-elected?

Gerrymander Violates the Fairness Principle

The Census and redrawing of election district boundary lines is done every ten years as mandated by the U.S. Constitution to ensure fair and proper representation among the states in Congress, and to preserve the one voter-one vote fairness principle in the lower house of Congress and the state legislatures. State Constitutions and/or Statutes specify additional, more detailed requirements for reapportionment and redistricting process, which are also supposed to preserve the fairness principle. Furthermore, electing representatives by district rather than at large (state-wide election for all candidates) provides local representation and avoids the problem of one party controlling all the legislative seats. However, gerrymander subverts this process designed for fairness into a process for unfair political corruption. Under gerrymandering, we have a one voter-no vote system in the general election, where there is at most one candidate from each party for each office.

For a Democrat in California and most other places, a 55 percent Democratic voter registration district is typically a safe district. For a Republican in California and most other places, a 50 percent Republican voter registration district is typically safe, due to greater Republican voter turnout. A 60 percent district is very safe for either party. Gerrymander often produces 65 to 80 percent safe election districts for all incumbents. Almost always, gerrymander alone is sufficient to guarantee automatic victory for the incumbent, or for a candidate of the party that "owns" the district when there is an empty seat. When combined with other incumbent advantages including name recognition, ability to use the office as a campaign platform, and the use of the influence of public official status to raise greater amounts of campaign funds, any challenger doesn't stand a chance at unseating the incumbent.

This is why there are elections where incumbent state legislators and congressman run unopposed. It does not make sense to spend much time and money to run in an election where the election has already been stolen and the outcome is predetermined before the election is held due to gerrymandered election districts.

It is important to note that not all safe election districts are gerrymandered, and not all gerrymandered districts are safe. Some election districts are naturally safe noncompetitive districts due to the local demographics. For example rural, sparsely populated areas tend to be more conservative and Republican, and urban, more densely populated areas tend to be more liberal and Democratic. Gerrymandering simply takes an imperfect situation and makes it much worse. Even if the gerrymander results in a few marginally safe districts, nearly all of the districts are extremely safe noncompetitive districts.

Some people falsely claim that gerrymander is not an important issue nor cause for concern. They claim that voter demographics, especially where voters of each party choose to live in greater numbers, is the dominant factor. These people are either misinformed or intentionally trying to deceive the public. If gerrymander had no significant impact on election outcomes and the dominant or only factor were voter demographics, then the United States would not have a long and continuing history of gerrymander. There would not be political battles including court challenges between the parties and various other political organizations over redistricting maps.

Even though voter demographics makes some election districts naturally safe and noncompetitive, gerrymander makes all or all but a few election districts artificially extremely safe and noncompetitive. Also, gerrymander differs from voter demographics by giving the majority party an artificially large number and proportion of election districts and legislative and congressional seats that it controls. Thus, a party that receives a small majority of all votes in an election is guaranteed an artificially large majority of seats. This leads to single party control in what is supposed to be a multi-party representative system of self-government.

Some people argue that gerrymandering does not constitute election fraud and corruption, and is not the equivalent of stealing elections, because it is legal and is made possible by a long-established political process. However, gerrymandering should be considered a form of election fraud and corruption, even though it is legal until outlawed by legislation, a voter initiative, or a court decision. The artificially wide vote count spreads resulting from gerrymandering show that gerrymandering is significantly more effective at changing election outcomes than other methods of election theft, which are usually effective only in close elections.

In gerrymandered election districts, the general election outcome is already determined by the district boundary lines before the election takes place. The citizens' votes only make a difference in the primary election where the voters of each party select their candiate to run in the general election. Furthermore, in closed primary elections, only voters of the majority party for a given district have any influence in the primary election outcome and therefore in the general election outcome.

The results and intentions of laws, policies, and practices should be considered when assessing whether they constitute fraud or corruption. The words "fraud" and "corruption" have specific legal meanings and there is legislation addressing these subjects. However, there are forms of fraud and corruption that a reasonable person would agree are "fraud" and "corruption", even though they are permissible under the law. In fact, anti-fraud and anti-corruption laws have been enacted to prohibit forms of fraud and corruption that were previously legal but widely recognized as fraud and corruption.

Conversely, laws have been enacted as a result of corrupt influences, up to and including bribery, and with corrupt intentions. Such laws, and the results of such laws, can be reasonably described as corruption even though they were enacted by elected officials.

Therefore, gerrymanders should be considered a significant form of election fraud and corruption, and the equivalent of stealing elections. Gerrymandering should be prohibited in order to restore the citizens' voting rights.

During the post-election contest of the 2000 presidential election, Democrats and Republicans exchanged accusations of election fraud and election theft. The news media widely reported and amplified these accusations of the theft of a single election. In contrast, the news media has ignored and continues to ignore the real and successful election thefts that take place throughout California and the U.S. every two years through gerrymandering by one party or the other, or by both parties in a bipartisan gerrymander as was done in California.

If the news media had any competence and integrity at all, and if they cared at all about the citizens and tax payers, they would continuously report on these election thefts through gerrymandering and on the need for redistricting reform.

The president, on the other hand, is elected by electors in the Electoral College, who are in turn elected by all the voters in each state. Thus, presidential elections do not involve election districts and are not subject to gerrymandering. Presidential elections more closely reflect the true will of the people. However, presidents have been elected with less than a majority of the popular vote, and in very few cases, with less popular votes than the leading challenger. This is a result of third party candidates splitting the popular vote and the "winner takes all" system used by most states to select their electors in the Electoral College.

After the 2000 election, I predicted that the Republican Party would probably lose their majority in the House of Representatives because they neglected to provide sufficient and effective support for Redistricting Reform in California on a high priority and ongoing basis. It should have been obvious to the Republican political analysts and strategists that the future of the Republican Party and all voters depends on solving the gerrymander problem in California, as well as the problem of massive illegal immigration and election fraud. Gerrymandering is a problem wherever it exists no matter which political party benefits the most. This is because all voters lose and all incumbents of all parties win with safe districts. However, California is critically important because of its exceptionally large congressional delegation.

This prediction did not come true, because President George W. Bush convinced California's Republican leadership in the state legislature to make a deal with the Democrats. They agreed that the Republicans would not challenge or attempt to block the Democrat redistricting plan even though the state legislature was heavily gerrymandered in favor of the Democrats, as long as the gerrymander of California's Congressional districts preserved the number of safe Republican controlled districts. President Bush wanted sufficient numbers of Republicans in Congress to provide the necessary votes for his programs and political agenda. As it turned out, the Republican legislators in California did not achieve the best possible deal when negotiating with the Democrats. As a result of this deal, the California election districts became hopelessly gerrymandered against all the voters, with even greater imbalance and safer districts.

At the national level, each of the major political parties keep score, calculating which states will gain or lose congressional seats due to reapportionment, and how many seats will be gained and lost in states where redistricting will be controlled by a Democratic or Republican majority state legislature and governor. This score is used to determine whether there will be a net gain or loss of representatives in Congress for each of the major parties, based on which party will get to gerrymander which states, and based on the new numbers of representatives for each state. Notice that this calculation does not take into account the will of the voters. The will of the voters becomes irrelevant in states with gerrymandered election districts.

Each of the major parties can rationalize, and to some extent actually justify, gerrymandering the states where they have a majority control of the legislature. After all, the other major party will gerrymander every state where they have control, and there must be some attempt to offset their gerrymandering to maintain some balance in the Congress. And if it is possible to have a net gain over the other party in Congress due to gerrymandering, then this can be rationalized as a "punishment" for the other party for gerrymandering the states that they control. But the real winners are all incumbents of all parties, and the real losers are all voters of all parties.

Gerrymandering has been around for a long time, long before electronic computers were invented. However, it is difficult to explain gerrymandering to the public without using special tools and techniques, because gerrymandering is a mathematical concept. Modern day "redistricting" computer programs are used to determine the optimal gerrymander district map boundary lines. These programs use linear programming techniques to solve minimization/maximization problems involving many variables. Such programs can use voter registration data, voter turn-out data, and vote count numbers for each party and candidate with resolution down to the voter precinct level.

The redistricting programs can also make use of other demographic data, including income level, race, religion, age, and gender, in addition to political party affiliation to fine-tune which voters are included or excluded for each new election district and its various precincts. Projected future demographic trend data can be used to produce a robust gerrymander that will endure for ten years until the next census and redistricting cycle. A set of Louisiana demographic maps is provided in the section, Louisiana Demographic Map Examples, to demonstrate some of the different types of demographic data available and how it looks when displayed as map overlays. Gerrymandering can have an even greater adverse impact on specific targeted demographic groups.

Current election district boundary lines are drawn with only a few constraints. These constraints, which are described in detail below, are as follows:

1. One representative per district;
2. Approximately equal population for each district of the same type;
3. Contiguous districts; and
4. Additional constraints, such as respecting geographical integrity.

First, there must be one representative per district.

Second, each district for the same type of office must contain approximately the same number of people. Early gerrymanders created election districts for the lower house of Congress and the State Legislature with substantially different populations. The U.S. Supreme Court held in a series of cases in the 1960's that the Fourteenth Amendment guaranteed "equality" of voting power and that the electoral systems in states which failed to allocate voting power on the basis of population were unconstitutional. One of these cases coined the phrase "one person, one vote". The Court held in 1964 that all members of a state legislature must be elected from districts containing substantially equal populations. Note that counting illegal aliens violates the one voter-one vote fairness principle, and may also violate Article XIV, Section 2 of the U.S. Constitution.

Prior to this ruling, California and other states had a bicameral legislature modeled after the Congress and the balance of powers created in the federal Constitution. Yet the Supreme Court ruled that a legislature modeled after the federal Constitution was unconstitutional, and violated that very same Constitution. This ruling destroyed the balance of power, favoring the Democratic Party and densely populated urban areas, and subjected the state Senate to gerrymandering. This ruling also subjected the state Assembly to a greater degree of gerrymandering because redistricting legislation must be approved by both the state Senate and Assembly, and signed by the governor. Because the state Senate was not subject to gerrymandering, there was a greater chance that it would block legislation to gerrymander the state Assembly and congressional districts. Could this ruling have been the work of a liberal judicial activist majority on the Court?

Third, each district must be contiguous. That is, all voters in a given district must reside within a single contiguous geographical area that can be enclosed in a single continuous boundary line that has no breaks and does not cross itself. This requirement is supposed to ensure that election districts contain land areas and populations that are close to each other, that are connected to each other, and that have issues and concerns in common.

However, as the examples below illustrate, this requirement is easily circumvented. For example, a square within a larger square, or a circle within a larger circle, is not permitted, but can be effectively achieved with a narrow strip or path of land that joins the outer area to the inner area, as in the letter "C". This example contains three contiguous election districts, where each district contains portions of the other two districts. You can verify that each of the three (red, blue, and green) districts is in fact contiguous by tracing its boundary, starting from any point on its boundary, without reversing direction along your path, and you will eventually arrive at your starting point. Notice how the red district in the example joins two distant, disconnected areas with a thin strip of land. Much more complex examples of carving specific areas in or out of an election district can be created, where there are hooks on hooks that are on hooks. Also, real election district maps use all types of curves in addition to straight lines, which provides even greater flexibility.

As another example, it is even possible, at least in theory, to have a Checkerboard Gerrymander (or Chessboard Gerrymander, if you prefer) as shown below, as a special case of the theoretical Grid Gerrymander. In a Checkerboard Gerrymander, two "adjacent" contiguous districts are created in a checkerboard pattern where any two adjacent squares adjoining along an edge belong to different districts. The theoretical Grid Gerrymander and Checkerboard Gerrymander were devised by the author, Michael D. Robbins, to demonstrate in a simple manner how extreme a gerrymander can become in spite of the contiguous district requirement. This example shows how a region can be divided up into a grid pattern, and each of the squares or rectangles in each row of the grid can be arbitrarily assigned to either one of two contiguous districts by opening up its boundary line adjoining a thin strip above or below it, to form a Grid Gerrymander. This example is explained in greater detail below in the section, Theoretical Grid and Checkerboard Gerrymanders. You can verify that each of the two (red and blue) districts is in fact contiguous using the boundary tracing method from the previous example.

Fourth, an additional constraint may be imposed, as in California, to respect the geographical integrity "to the extent possible" of any city, county, or city and county, or of any geographical region. This constraint is vague, allowing legislators more than enough flexibility to create even the most extreme gerrymanders. This constraint should be either relaxed or eliminated to allow or even require more compact election districts. One approach would require that districts are approximate rectangles with some deviations allowed to follow roads, waterways, and state boundary lines.

A description of possible virtual election districts that violate these rules is given below.

Gerrymander Analogy

Actual Redistricting Maps

The original gerrymander cartoon below is a useful and humorous instructional aid. It is provided here in two sizes.

Although the Democratic-Republican party was in power in Massachusetts in 1812, it had little hope of retaining its control in the approaching elections. To save something for the party Governor Elbridge Gerry signed a reapportionment bill to construct new senatorial election districts that consolidated the Federalist vote. An exasperated editor hung a map showing one of these districts. Gilbert Stuart, the painter, added head, wings, and claws to the outline, noting, "That will do for a salamander." "Better say Gerrymander," the editor responded. The Boston Weekly Messenger brought the term gerrymander into common usage when it printed an editorial cartoon illustrating the district in question with a monster's head, arms, and tail and named the creature a gerrymander. The Democratic-Republican party broke apart in 1824 and most of its supporters moved to the Democratic Party under the leadership of Andrew Jackson in 1828. Text Source: First paragraph: Compton's Encyclopedia Online v3.0, 1998  - http://www.Comptons.com/encyclopedia/ARTICLES/0050/00734720_A.html Second paragraph: About.com - search on Gerrymandering http://geography.MiningCo.com/library/weekly/aa030199.htm Third paragraph: Online Highways,  definition for Democratic-Republican Party  - http://www.OHwy.com/us/d/drp.htm

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The Illinois 4th Congressional District is a Hispanic majority district in parts of the North Side and the southwest side of Chicago. It surrounds a black majority district, the 7th district. It is ten miles wide, and runs along railroad tracks, forest preserves, and cemeteries. It contains significant immigrant populations and is majority Democratic but has low voter turnout. The Supreme Court declined to hear a suit about the 4th district. Note that the 1st, 2nd, 3rd, 5th, 6th, 9th, and 11th districts also appear to be gerrymandered.

Illinois 1990 Hispanic majority 4th Congressional District wraps around the black majority 7th Cong. District

North Carolina's black-majority 12th Congressional district is another example of a racial gerrymander, as demonstrated by the following maps. Note the distant disjoint areas connect by thin strips of land. There was litigation over this district. (Maps provided courtesy of Adversity.net, http://www.Adversity.net/special/gerrymander_1.htm).

North Carolina black majority 12th Congressional District Gerrymander

North Carolina black majority 12th Congressional District Gerrymander

North Carolina black majority 12th Congressional District Gerrymander

North Carolina black majority 12th Congressional District Gerrymander

The Louisiana Congressional election district maps shown below, with the disputed fourth congressional district shown in solid black in the small map and in yellow in the large map, help illustrate the general appearance of the boundaries of gerrymandered election districts. Note the lack of compactness, i.e., the very long boundary line relative to the area enclosed by the boundary. For comparison, imagine the much shorter boundary of a square enclosing the same area. Also note the hooks along the boundary, and the areas outside the fourth district that are almost completely surrounded by the fourth district. Also note how the fourth district boundary connects otherwise distant and disjoint areas of the state using narrow strips of land. These are all characteristics of gerrymandered election districts. The larger map is of poor quality and difficult to read. However, it does show all of the election districts in the state.

A formula can be derived for a Compactness Metric to determine whether an election district lacks compactness and is likely to be gerrymandered. One such compactness metric is the value CM = B/Sqrt(A) = the total length of the boundary of an election district divided by the square root of the area of the district. An equivalent but more useful compactness metric is the value CM² = B²/A = the square of the total length of the boundary of an election district divided by the area of the district. This compactness metric is more useful because it provides a greater range of values and greater differences between the values to be compared.

This compactness metric has several desirable qualities. It is a pure dimensionless number. It is unambiguous, intuitive, and straightforward to compute, and its value is invariant for the same shape with different sizes and areas.

Thus, two districts with the same shape and (approximately) the same number of voters will have the same (or nearly identical) Compactness Metric values even though the less densely populated district has a larger area. The boundary length of an election district is easily computed using the distance formula for each pair of adjacent boundary points and adding all of the boundary segment lengths.

The distance formula is d = SQRT(x² + y²) where SQRT is the square root function. It is derived from the Pythagorean theorem to compute the length of the hypotenuse of a right triangle (contains a right or 90 degree angle) from the lengths of the sides forming the right angle. The area of an election district is easily computed using a simple formula for the area of an arbitrary polygon that does not intersect itself.

Another possible compactness metric uses the distance between the two furthest points in the district divided by the square root of the area of the district. Variations of this compactness metric divide the diameter of the minimum bounding circle or the length of the diagonal of the minimum bounding rectangle by the square root of the area of the district. The minimum bounding rectangle is easily computed using the minimum and maximum latitude and longitude (x and y) of all map coordinates for points on the district boundary line. However, furthest distance and bounding circles and rectangles do not completely account for the amount of twists and turns, or cumulative curvature, of the district boundary line.

A third possible compactness metric computes the average distance from all boundary points, all boundary and interior points, or the centers of small patch areas in the district to the computed "center of mass" of the district and divides this average length by the square root of the area of the district. Calculations involving standard deviation may also be useful.

Yet another possible compactness metric could measure the number of times the boundary changes direction or the count or sum of the angles in the boundary that are greater than some threshold number.

Multiple compactness metrics can be used simultaneously to judge the compactness of the election district boundary lines and to define mathematical criteria for either accepting or rejecting a proposed redistricting map. These compactness measures were conceived by the author, Michael D. Robbins, but may have also been independently conceived by others. Undoubtedly, other Compactness Measures exist or can be conceived.

Note, however, that some election districts that are not gerrymandered, but which follow the boundary lines of political subdivisions such as counties and cities, may not be compact due to the irregularities of the pre-existing county or city boundaries.

Compactness is not always a sufficient criteria, but it is a reasonable constraint to prevent the more extreme cases of election manipulation through gerrymandering. It is possible to have different redistricting maps with equally compact districts that yield different results in balance of power between the parties and in whether the districts are safe or competitive. This can be illustrated by the district maps shown using the Simple Graph System. Redistricting Map 5A can be redrawn to retain District 5, the green vertical column of four squares on the far right, and to replace Districts 1-4, the remaining four districts on the left with straight horizontal rather than straight vertical districts. This results in Redistricting Map 5B. The four new districts on the left are considerably safer for the incumbents even though they have the same shape and area, and therefore the same compactness metric, as the four districts they replaced. Thus, the compactness metric should be only one of several metrics or criteria used to objectively determine whether or not a given election district or a complete set of election districts is gerrymandered.

A measure for the entire redistricting map would compare the percentage of total votes cast across all districts in the state for members of each political party for each office type with the percentage of elected officials for each party that hold each type of office. For California, the legislative office types would be Congress (U.S. Representative), State Assembly, and State Senate. Local offices such as City Councilmember, School Board Member, and County Supervisor are considered "nonpartisan" offices by law. However, these districts are often gerrymandered along partisan or racial lines as well. One percentage can be divided into the other to form a Fairness Metric, and if this fairness metric is outside some threshold, the redistricting map is rejected. Thus, the balance between political parties in Congress and the state legislature would more closely resemble the balance of votes cast for members of each political party.

Simple Graph System Example

This and the other similar simple graph system examples can be used as the basis for a class project and demonstration. The details are given at the end of this first example.

Recall that for a Democrat, a 55 percent Democratic voter registration district is typically a safe district. For a Republican, a 50 percent Republican voter registration district is typically safe, due to greater Republican voter turnout. A 60 percent district is very safe for either party.

This example illustrates four different redistricting maps for the same geographical area and voter population. Each map distributes one fourth of the voter population in the map region into each of four election districts. Although the total voter population consists of half Republicans and half Democrats, Map 1 and Map 2 yield a 50-50 percent split of legislative seats between Republicans and Democrats, while Map 3 yields a 75-25 percent split favoring Democrats, and Map 4 yields a 75-25 percent split favoring Republicans.

In this example, each square represents a region of similar population. All squares in the same election district must touch at least one other square in the district either along a side or at a corner, and two election districts may not cross over each other at a square corner. The R squares have a majority of Republican voters and the D squares have a majority of Democratic voters.

For simplicity, assume that there are only Republican and Democratic voters, and further assume that 70 percent of the voters in each R square are Republicans, and 70 percent of the voters in each D square are Democrats.

Under these assumptions for the percentage of voters from each party in a square, we have the following: An election district with 2 R squares and 1 D square contains 57 percent Republican voters and 43 percent Democratic voters. This is a very safe 57 percent Republican district. Similarly, an election district with 2 D squares and 1 R square contains 57 percent Democratic voters and 43 percent Republican voters. This is a safe 57 percent Democratic district.

The calculations for a 2 R - 1 D election district with 70 percent squares are as follows:

The calculations for a 2 D - 1 R election district with 70 percent squares are as follows:

The calculations for a 3 R - 0 D election district with 70 percent squares are as follows:

The calculations for a 3 D - 0 R election district with 70 percent squares are as follows:

The following material provides details about a live classroom demonstration, and may be skipped.

A class project and demonstration involving student participation can be developed from these examples. A suggested class project involves grouping students into election districts and political parties (or candidate supporters) using the groupings shown in the redistricting maps in the examples. By repeating this process for different redistricting maps with the same numbers of student voters supporting each party (or set of candidates), the different election outcomes for the different district maps are demonstrated.

The numbers of students used must be equal or proportional to the numbers of map squares used in the examples. Extra students can play the role of election officials, ballot collectors and counters, observers ("poll watchers"), candidates, or news reporters to report on the reason for the different election outcomes using different election district maps. Actual paper ballots should be cast according to the party registration or the support for a candidate running in the voter's district.

However, it should be made clear that it is not necessary for individual voters to vote strict party line in a real election, but in this exercise each student is representing a block of voters in an election district with a much larger voter population than the number of students participating in the exercise. The exception to this rule is primary elections in states that have closed primaries, where voters choose among one or more candidates for each office from their own party to determine which candidates receive the party nomination to run in the general election. As an option, you may also point out that given the current voting patterns, about two-thirds of all voters and 80 to 90 percent of some demographic groups (e.g., Jewish and black voters) tend to vote straight party line. You may also discuss whether or not this is in the best interest of voters in those demographic groups and provide explanations. For more information, see the section titled "Impact on Specific Demographic Groups" below.

These instructions may be modified as appropriate given the situation. For example, if paper ballots are too time consuming, you may substitute a vote by show of hands or by roll call vote.

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The following material provides detailed information on the calculations, and may be skipped.

These calculations only require the appropriate election vote count and voter registration data, and a simple calculator or preferable, a computer with a spreadsheet program. This data should be available from the Secretary of State's office in your state capitol. It may be available for immediate download over the internet. Search for your Secretary of State's web site. You may need to search for the "Election Division" within the Secretary of State's web site.

The vote count data should include, for each election district for the given legislative house, the number of votes that each candidate received. Optionally, obtain the data indicating which candidates were incumbents, and also obtain the number of registered voters for each party for each election district. For simplicity, you may prefer to perform the calculations using only the vote count and party voter registration data for the main parties and for the entire state and the individual election districts as a whole.

The procedure is as follows:

1. For a given general election and legislative house, for each political party, add up all of the votes that were cast in all election districts for candidates of the given party. Also compute the total number of votes cast statewide for all of the parties combined for all seats in the given legislative house.
   
   
2. Next, compute the percentage of the total vote that each party received, and multiply this percentage by the number of seats in the legislative house. For example, California has 80 State Assembly members and 40 State Senators. Round these numbers up or down as appropriate to the nearest integer. You now have the number of legislative seats each party would have when we completely ignore the effects of election district boundary lines.
   
   
3. Now compare the computed percentage and number of seats for each party with the actual numbers resulting from the election being analyzed. But remember that some of these differences may be due to demographic differences and not only gerrymandering.

As an added exercise, you can perform the following computations:

1. Compute the percentage of the vote that each candidate received in each of the election districts, and compute the percentage point spread between the winner and the candidate with the second largest number of votes.
   
   
2. Compute, among all incumbents that ran for reelection, the percentage of incumbents that won.
   
   
3. Obtain the number of registered voters for each of the parties and the total number of registered voters for each election district for the legislative house. Compute the percentage of registered voters for each party in each election district. For each of the candidates, compute the candidate's performance relative to voter registration demographics by subtracting the percentage of registered voters of the candidate's party from the percentage of votes the candidate actually received. A candidate that outperforms demographics attracted more cross-over voters than he lost voters of his own party. Conversely, a candidate that underperforms demographics lost more voters of his own party than he attracted cross-over voters.

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Statistical Sampling and Counting Illegal Aliens

Statistical sampling and counting illegal aliens violate the one voter-one vote fairness principle. Statistical sampling has been proposed by Democratic Party officials including President Clinton and liberal Democratic organizations such as the ACLU as a replacement for performing a real census, or "enumeration", as required by the U.S. Constitution. Statistical sampling would substitute "estimates" for actual counts of difficult to count people such as homeless people and people who distrust the government and do not wish to be identified and counted. The use of statistical sampling would create new opportunities for election fraud and would unfairly benefit the Democratic Party and its candidates.

The Democratic Clinton Administration sought a law from Congress to require the Census Bureau to use statistical sampling rather than the "enumeration" required by the Constitution. The Republican majority Congress refused to pass such a law. President Clinton then issued an Executive Order directing the Census Bureau to use statistical sampling. This order was challenged before the U.S. Supreme Court, which ruled that "enumeration" means exactly what its definition in the dictionary indicates: an actual count of the population. The Court directed that the reapportionment of Congress must be based on the actual count released by the Census Bureau and not on statistical sampling. However, the Court left open the question of whether the Census Bureau could provide, in addition to the actual count, a second count based on statistical sampling to be used by the states when drawing new election districts for all state and local officials.

Statistical sampling would allow partisan Democratic government officials in cities with corrupt Democratic political machines to violate the fairness principle by using subjective and undefined or poorly defined methods of estimation, including the counting and estimation of illegal aliens. Introducing census errors by over-estimation and counting illegal aliens creates election districts that have fewer legal voters, citizens, and legal residents compared to other election districts. Thus, voters in election districts where sampling is used and large numbers of illegal aliens are counted in the census have more voting power than voters in districts without sampling and with few illegal aliens. This violates the one voter-one vote fairness principle.

Counting illegal aliens for reapportionment purposes would also violate the Equal Protection clause of the U.S. Constitution, Article XIV (14), Section 1, which states, "... nor shall any State ... deny to any person within its jurisdiction the equal protection of the laws." On December 12, 2000, the U.S. Supreme Court ruled in a 7-2 decision, in Bush v. Gore, that manual hand vote recounts that use different counting methods and different standards, or no standards at all, for different voters in the same election violates Equal Protection. After two statewide machine counts in Florida determined that George Bush narrowly won the Florida Presidential Election, and therefore the U.S. Presidential Election, several Florida counties with Democratic election officials initiated manual hand recounts of the "under counted" ballots. These ballots did not register a vote for any candidate for President, either due to a vote for "none of the above", or due to voter neglect or voter error. The recount process involved interpreting the intent of the voter without clear or rational standards, and with standards that changed whenever more votes were needed for Democratic candidate Al Gore. Based on the 2000 Florida Presidential Election and other similar experiences, it is certain that sampling would result in significant election fraud and violations of the fairness principle.

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Electronic Voting Schemes

There are many other potential problems with electronic voting schemes that are worth discussing, even though they are not directly related to gerrymandering. ( skip )  These include new opportunities for election and voter fraud, denial of service attacks that prevent individuals or groups of people from voting, and breach of confidentiality. If instead electronic voting turns out to be more efficient and more anonymous for the voter, then election and voter fraud involving people voting multiple times may also be more efficient and less detectable.

Other problems include a possible restructuring of our voting system to give different voters different amounts of votes, including fractional votes, based on dubious criteria; allowing voters to give fractions of their vote to different candidates or even to other voters; allowing voters to sign over electronic proxies to various special interest groups; and giving blocks of votes to special interests such as labor unions, large corporations, or other organizations, with or without the knowledge of the public. 

Although encryption and electronic signature and authentication systems may be devised to preserve the integrity and confidentiality of the voting system, it would be difficult for the public to have sufficient confidence in the integrity of the computer software involved and the inability for determined individuals or organizations to compromise that software. There is already some concern with the software in the existing vote counting machines.

Those who support electronic voting schemes would argue that some existing sources of errors and fraud would be eliminated, and recounts would either be eliminated or more accurate and efficient. However, electronic voting systems provide inadequate public insight into the workings of the system, and there is no tangible proof of how the votes were cast. And new opportunities for errors and fraud would more than offset the elimination of some of the existing sources of errors and fraud. Because the existing government is in control of the election process, it is crucial that the public and all candidates and their representatives be allowed to observe the voting and vote counting process in a way that is meaningful.

Even if the source code listings were made available to independent computer programmers for analysis, which probably would not be the case, complex software can be difficult to analyze, and it would be difficult or impossible to confirm that the source code listings match the actual compiled and linked object code being run on the computers. Furthermore, computer failure for any reason, including a hacker's denial of service attack or any other form of attack, would create serious questions about the legitimacy of the election. Even if we assume that an electronic election can be held a second time with minimal additional cost and time impact, a re-vote for the same election may not give the same results as the original election, especially for a close election. Reform Party presidential candidate Pat Buchanan said, regarding claims of a confusing ballot by Democratic voters in Palm Beach Florida in the 2000 Presidential Election, "An election is a snapshot of a twelve hour period in time, and there is no way to reproduce that snapshot once everyone knows the election results."

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One of the most difficult problems with gerrymandering is that the party that has an unfair and artificially large majority of seats and power due to gerrymandering controls the next redistricting cycle. This creates a difficult to break vicious cycle of gerrymandering with its associated corruption that continues from each decade to the next. Another difficult problem is that a small window of opportunity to break the gerrymandering cycle opens up only once every ten years, with one possible exception.

Under the current system, most or all politicians of the majority party, and many from the minority party, will support a gerrymander because they benefit from the safe districts it creates. Those honest politicians who refuse to cooperate by providing needed votes to implement the gerrymander may become the victim of a hatchet job done on their own district. In this case, the voters of their own district and party are divided up into several new districts where entrenched incumbents will have most of their original voter base. Each of these incumbents will be impossible to defeat in the next election. The honest politicians are effectively either retired or exiled by the leadership of the majority party. Their only hope of winning an election is to move into some other district where there is an open seat, i.e., the incumbent is not re-running for that same seat. But this involves starting over again, with little name recognition and less ability to raise needed campaign funds in the new district.

Many politicians cave in to the pressure when confronted with only two options: either vote in support of the gerrymandered redistricting plan or be axed out of their district when the plan eventually passes without there vote. The plan will pass, because the majority party leaders will offer various bribes and inducements to get the remaining votes that they need. The bribes and inducements may include powerful and lucrative committee appointments, office upgrades and increased staff budgets, promises of large sums of campaign funds for future elections, passage or defeat of specific legislation, or outright cash. In the end, this system of reward and punishment works and the necessary votes are secured. Those who did not make a deal become the losers.

The Need for Redistricting Reform

As a minimum, any proposed redistricting map should be validated, ranked, and either accepted or rejected by a computer program using various compactness and fairness metrics computed for each election district and for the entire collection of districts. The best of multiple proposed district maps should be selected based on which has the best values for these metrics.

The contiguous single-member district requirement should be retained. However, the requirement to respect the geographical integrity of any city, county, or geographical region should be relaxed or eliminated entirely. It gives politicians too much flexibility to manipulate district boundaries by selectively following odd-shaped city, county, and region boundary lines. This requirement has not provided fair, unbiased, competitive election districts or better representative government in California, where some of the more extreme gerrymanders have been accomplished.

If the goal of using computers and a mathematical algorithm cannot be met, then the redistricting maps should be drawn by an independent nonpartisan or bipartisan commission. The maps should then be validated by computer using the compactness and fairness metrics, and subjected to public review and criticism, and ultimately to judicial review under strict guidelines. The best approach would also use a redistricting computer program to provide a neutral or more competitive than neutral redistricting map. A neutral, unbiased, redistricting map provides more accurate voter representation than one that creates unnaturally competitive districts. Election districts should accurately reflect voter demographics without bias or manipulation in any direction.

Thus, the same redistricting software that is used to create gerrymanders would instead be used to create neutral or more competitive than neutral district maps. Like firearms, redistricting software can be used for good or bad purposes, depending on the exercise of free will by the users. However, the last twenty-five years of scientific criminological research shows that firearms are used at least five times more often for self defense than they are misused in all crimes, suicides, and accidents combined. In contrast, I am inclined to believe that redistricting software is used more often for gerrymandering than for fair redistricting.

The new redistricting system will have to be enacted into law by the voters, or adjudicated by the courts, to prevent the legislature from subverting the system at a future time. However, one risk of relying on the courts is the danger of judicial activism, where judges act as political partisans and ignore the Constitution, the law, the previous court precedents, and even common sense, and instead decide as they please to further their political agenda.

Some redistricting reform initiatives would use a panel of retired judges to draw the district maps. This approach has been criticized by those who oppose reform. However, even if this approach is imperfect, it is no worse than, and likely much better than the existing system. For example, the current situation in California is the worst possible under the 2001 Democratic Party gerrymander. A panel of retired judges could produce a redistricting map that is neutral, slightly gerrymandered, or extremely gerrymandered, which is already the case. California's 1991 redistricting was performed by judges, and was much closer to neutral than the current gerrymander.

There are only four ways to break the ten-year gerrymandering cycle:

1. Voters must elect sufficient members of the minority party (the Republican party in California) to the state legislature, which is nearly impossible under an existing gerrymander;



2. Voters must elect a Governor from the minority party and sufficient minority party legislators to sustain a Governor's veto of redistricting legislation;



3. Individual voters, a government watchdog group, and/or minority party legislators must sue in court to overturn a specific gerrymandered redistricting plan; or



4. Voters must qualify and pass a Redistricting Reform initiative.

A useful guide to redistricting litigation is the Outline of Redistricting Litigation - the 1990's, a project of the Redistricting Task Force of the National Conference of State Legislatures, which can be found on the internet at the following address:

http://www.senate.leg.state.mn.us/departments/scr/REDIST/redout.htm

Passage of a Redistricting Reform Initiative is the most effective and the only long term solution. In California, a voter initiative can only be repealed or amended by another vote of the people, and not by the legislature. Such an initiative would implement some combination of the above reforms. It would set fair, reasonable, and effective rules and constraints for drawing new election district maps boundaries. It might specify an algorithm that could be implemented by computer and independently verified as meeting the specified criteria. It might also take control of the process away from the legislators and give it to a nonpartisan or bipartisan committee with proper oversight and checks and balances including an appeals process. Also, it may be legally possible for such an initiative to require that new maps be drawn before the next two-year election rather than wait until the next ten-year window of opportunity after the next census.

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Proposition 77, a new redistricting reform initiative, will be on the ballot for the California special election on Tuesday, November 8, 2005. More information on Proposition 77 is provided below.

A number of redistricting reform initiatives have been qualified for the ballot in California. However, these measures were defeated due to a lack of understanding by the voters, incompetent and biased media coverage, and massive spending on deceptive advertising by corrupt incumbent politicians. Typically, the news media launches a massive propaganda campaign to tell members of the majority (Democratic) party that the Redistricting Reform Initiative is not reform at all, but that it is an underhanded trick to improperly steal political power away from the majority party. Never mind the truth that Redistricting Reform actually restores the true and fair balance of power in the State Legislature and Congress, and restores the voting power of all voters of all parties by creating competitive election districts where politicians actually have to compete for votes.

One Redistricting Reform Initiative in California was defeated by a massive television false advertising campaign. TV ads showed politicians sitting at a table. All of a sudden, the wall, the table, and the politicians rotate into another room, a hidden smoke filled back room where they make secret deals. The TV ad claimed that the reform initiative would instead create political corruption in hidden smoke filled back rooms! This ad campaign also claimed that there would be environmental disasters including oil slicks on California beaches if the voter initiative won a majority of votes and became law! On election night, in front of the news media TV cameras, former California Democratic Party political boss and Assembly Speaker Willie Brown boasted, "I sure pulled the wool over the voters' eyes this time!"

Out of frustration with the inability to enact Redistricting Reform, California voters eventually broke Willie Brown's stranglehold on the California Legislature and ousted many corrupt entrenched politicians by passing a Term Limits Voter Initiative. The politicians tried to overturn the Term Limits Initiative in the courts but lost. However, term limits is not a substitute for Redistricting Reform. It does not solve the problem of safe, noncompetitive districts and the associated political corruption. Also, term limits throws the baby out with the bath water, prematurely ending the political careers of those elected officials who are honest, competent, effective, and principled representatives of the people. But term limits is a simple concept compared to the mathematical concept of gerrymandering. A majority of the voters understood term limits but did not understand gerrymandering.

California had been gerrymandered for many decades by the Democratic party machine. In 1990, California voters elected Republican Governor Pete Wilson, who vetoed the Democratic gerrymander redistricting legislation. When the Democratic majority state legislature and the Republican Governor could not reach agreement on enacting a fair redistricting plan, the matter was settled in the courts, resulting in a somewhat fair redistricting plan. Under this more fair redistricting plan, the Republicans actually obtained a minimal majority in the State Assembly for the two year term from 1994 to 1996.

In 1999, California voters qualified another Redistricting Reform Initiative for the ballot. One problem was that there were two competing initiatives and lack of agreement on which initiative was better and should receive full support. Another problem was that the structure of the initiative that eventually prevailed left it open to attack in the courts. This vulnerability involved additional provisions that were added as an inducement, or sugar coating, for voters who do not understand gerrymandering. The provisions included limits on legislators' salaries and penalties for failure to pass a budget on time. Unfortunately, this initiative was successfully challenged in a lawsuit, and on December 13, 1999, the California Supreme Court removed it from the March 2000 ballot. The reason cited was violation of the restriction that an initiative may only address a single subject. One would think that government integrity and efficiency is a single subject. Perhaps this same standard should be applied to legislation enacted by the legislature.

A better approach would have been to put the easy to understand inducement in a separate companion initiative that would be packaged and promoted together with the Redistricting Reform Initiative. Also, a different set of voter inducements may be more appropriate, such as increased penalties for political corruption, abuse of government powers, and misusing government records on private citizens and businesses. Or would that require three more initiatives? In any case, there is some concern that paying elected officials too little may lead to more political corruption or a situation where only those who are retired or independently wealthy can afford to run for public office.

California is now due for another redistricting following the 2000 Census. This will lead to another massive gerrymander, because there is now one party control in California State Government. The Democratic Party machine has full control of the State Assembly and Senate, the Governorship, and the Attorney General. One might wonder how the Democrats have large majorities in both state houses under a reasonably fair redistricting plan from the Wilson administration.

There are multiple answers to this question. The California Republican Party lacks the necessary organization skills, party discipline, and ability to consistently run effective political campaigns. Also, the California Republican Party has fielded candidates that behave and campaign more like liberal Democrats, and this alienates the Republican core constituency and grass roots activists, as well as cross-over Democratic voters. These problems are compounded by an incompetent and biased liberal media, massive amounts of illegal immigration, and organized election and voter fraud that exploits illegal aliens.

It is now time for everyone who has integrity to help educate voters about gerrymandering and the need for redistricting reform. We must agree upon, vigorously support, and successfully pass an effective Redistricting Reform Initiative in California that will withstand all legal and political challenges. We must begin our work immediately rather than wait until a few months before the 2002 elections. A failure to do so will adversely affect all Americans for many decades to come.

AUGUST 19, 2005 UPDATE:

Proposition 77, a new redistricting reform initiative, will be on the ballot for the California special election on Tuesday, November 8, 2005. This proposition is supported by California Governor Arnold Schwarzenegger, and is opposed by entrenched incumbent politicians, lobbyists, and other special interests who want to keep the current system of corruption, fraud, and waste that results from safe, non-competitive gerrymandered election districts.

MARCH 24, 2005 UPDATE:

California Governor Arnold Schwarzenegger is proposing a redistricting reform ballot initiative that would redraw California's legislative and Congressional districts by mid-decade. The redrawing would be done by an independent panel of retired judges rather than by the state legislators. The effectiveness of the proposal will depend on how the retired judges are selected, on the approval and appeal process, and on whether effective mathematical or algorithmic constraints such as compactness metrics and computational drawing of the district boundaries are included in the plan.

More information on California redistricting initiatives can be found at the following links. Keep in mind that phony redistricting reform initiatives may be used by incumbent politicians and other opponents in an attempt to cause confusion and undermine or outright defeat real redistricting reform initiatives. Each initiative must be carefully analyzed and evaluated.

http://www.joinarnold.com/en/press/pressdetail.php?id=599

http://caag.state.ca.us/initiatives/

http://caag.state.ca.us/initiatives/activeindex.htm

SA2005RF0003 (PDF 410kb/ 7 pages)
Cleared for petition circulation
District Boundary Changes. Initiative Constitutional Amendment.

SA2005RF0004 (PDF 477kb/ 8 pages)
Cleared for petition circulation
Changing District Boundaries. Initiative Constitutional Amendment.

SA2005RF0005 (PDF 431kb/ 8 pages)
Cleared for petition circulation
Reapportioning District Boundaries. Initiative Constitutional Amendment.

SA2005RF0006 (PDF 222 kb/ 5 pages)
Cleared for petition circulation
Reapportionment of Districts. Initiative Constitutional Amendment.

SA2005RF0016 (PDF 356kb/ 8 pages)
Cleared for petition circulation
District Reapportionment. Initiative Constitutional Amendment.

SA2005RF0035 (PDF 241kb/ 4 pages)
Cleared for petition circulation
Reapportioning Election Districts. Congressional Exception. Initiative Constitutional Amendment.

SA2005RF0062 (PDF 387kb/ 7 pages)
Cleared for petition circulation
Mid-Decade District Reapportionment. Congressional Exception. Initiative Constitutional Amendment.

• Concise Abstract  ( reference )
• Definition of Gerrymander   ( reference )
• Definition of Elbridge Gerry  ( reference )
• Biography #1 of Elbridge Gerry  ( reference )
• Biography #2 of Elbridge Gerry  ( reference )
• Excerpts from the U.S. Constitution on Reapportionment  ( reference )
• California Constitution Article 21 on Reapportionment  ( reference )
• Closed Primary Elections   ( reference )
• Presidential Elections and the Electoral College   ( reference )
• Bicameral State Legislature  ( reference )
• The Danger of Judicial Activism  ( mathematical concept reference )  ( redistricting reform reference )  ( bicameral legislature reference )
• Louisiana Demographic Map Examples  ( reference )
• Deriving the Formula for a compactness metric for an Election District  ( reference )
• Formulas for the Area and Centroid (Center of Mass) of a Polygon   ( reference )
• Theoretical Grid and Checkerboard Gerrymanders   ( reference )
• An Example Demonstrating Unsafe, Competitive Districts  ( reference )
• An Example Demonstrating The Minority Party Controlling a Majority of Districts  ( reference )

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The following definition of gerrymander is adapted from The American Heritage® Dictionary of the English Language: Fourth Edition. 2000.  http://www.Bartleby.com/61/62/G0106200.html

SYLLABICATION:	ger·ry·man·der
PRONUNCIATION:	jr´-mn´´dr, gr´-
TRANSITIVE VERB:	Inflected forms: —dered, —der·ing, —ders To divide (a geographic area) into voting districts so as to give unfair advantage to one party in elections.
NOUN:	1. The act, process, or an instance of gerrymandering. 2. A district or configuration of districts differing widely in size or population because of gerrymandering.
ETYMOLOGY:	After Gerry, Elbridge + (sala)mander from the shape of an election district created while Elbridge Gerry was governor of Massachusetts.
WORD HISTORY:	The May 12, 1813, edition of the Massachusetts Spy reported "An official statement of the returns of voters for senators give[s] twenty nine friends of peace, and eleven gerrymanders." The word gerrymander was created by combining the word salamander, a small lizard-like amphibian, with the last name of Elbridge Gerry, a former Massachusetts governor. Note that Gerry's name was pronounced with a hard g, but gerrymander is now commonly pronounced with a soft g. Elbridge Gerry was immortalized with this word because an election district created by members of his party in 1812 looked like a salamander. According to one version of how gerrymander was coined, the shape of the district attracted the eye of the painter Gilbert Stuart, who noticed it on a map in a newspaper editor's office. Stuart decorated the outline of the district with a head, wings, and claws and then said to the editor, "That will do for a salamander!" "Gerrymander!" came the reply. The word is first recorded in April 1812 in reference to the creature or its caricature, but it soon came to mean both "the action of shaping a district to gain political advantage" and "any representative elected from such a district by that method." Within the same year gerrymander was also recorded as a verb.

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Definition of Elbridge Gerry

The following definition of Elbridge Gerry is adapted from The American Heritage^® Dictionary of the English Language: Fourth Edition. 2000. http://www.Bartleby.com/61/61/G0106100.html

See also: Grolier Encyclopedia:  http://gi.Grolier.com/presidents/ea/vp/vpgerry.html

Gerry, Elbridge

DATES:	1744–1814
SYLLABICATION:	Ger·ry
PRONUNCIATION:	gr´
	Elbridge Gerry was an American politician, a signer of the Declaration of Independence (1776), and a delegate to the Continental Congress (1787). He served as governor of Massachusetts (1810–1811) and as Vice President of the United States (1813–1814) under James Madison. The word gerrymander was coined in his name.

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Biography #1 of Elbridge Gerry

The following biographical information is adapted from Grolier Encyclopedia, The American Presidency, Elbridge Gerry Biography, Russell W. Knight, Editor of Elbridge Gerry's Letterbook: Paris 1797-1798.

See also: Grolier Encyclopedia:  http://gi.Grolier.com/presidents/ea/vp/vpgerry.html

ELBRIDGE GERRY
Biography

Elbridge Gerry, (1744-1814), ger'e, was an American patriot and political leader. He was a signer of the Declaration of Independence and a delegate to the Constitutional Convention, a U.S. congressman, governor of Massachusetts, and a U.S. Vice President to President James Madison.

Gerry was born in Marblehead, Mass., on July 17, 1744. He graduated from Harvard College in 1762 and entered his father's mercantile and shipping business. It is likely that he adopted the patriot cause as a result of grievances over Britain's attempt to tax colonial commerce.

In 1772, Gerry was elected to the General Court of Massachusetts. He wrote in 1773 in a letter to his friend Samuel Adams, "I humbly conceive that the People ought ... to be apprized of their Situation & to have the Opportunity of Choosing their Submission to Slavery, or of righteously supporting with their Lives, their Rights and Liberties." Gerry was then elected to the Massachusetts provincial congress, where he provided for the needs of the patriot forces.

Gerry was elected a delegate to the Continental Congress in 1776 and he signed the Declaration of Independence and the Articles of Confederation. He withdraw from the congress in 1780 because of actions he felt violated states' rights. He resumed his seat in Congress in 1783 and remained active there until 1785.

In 1786, Gerry married and returned to Massachusetts where he served a year in the state legislature. He was a delegate and an active participant in the Constitutional Convention of 1787. However, he refused to sign the completed document because he objected to certain provisions as inadequate, ambiguous, or dangerous.

Gerry served in Congress from 1789 to 1793.

Gerry won the election as the Republican candidate for governor of Massachusetts in 1810. He was reelected in 1811, but was defeated the following year after supporting a redistricting bill that was responsible for the coining of the word "gerrymander." Gerry was elected U.S. vice president in 1812 on the Republican ticket with James Madison as president. He died in office, in Washington, D.C., on November 23, 1814, at age 70.

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Biography #2 of Elbridge Gerry

The following biographical text was copied from the National Archives and Records Administration "The Founding Fathers" Page.

http://www.NARA.gov/exhall/charters/constitution/mass.html#gerry

Elbridge Gerry

Massachusetts

Image: The National Portrait Gallery, Smithsonian Institution

Gerry was born in 1744 at Marblehead, MA, the third of 12 children. His mother was the daughter of a Boston merchant; his father, a wealthy and politically active merchant-shipper who had once been a sea captain. Upon graduating from Harvard in 1762, Gerry joined his father and two brothers in the family business, exporting dried codfish to Barbados and Spain. He entered the colonial legislature (1772-74), where he came under the influence of Samuel Adams, and took part in the Marblehead and Massachusetts committees of correspondence. When Parliament closed Boston harbor in June 1774, Marblehead became a major port of entry for supplies donated by patriots throughout the colonies to relieve Bostonians, and Gerry helped transport the goods.

Between 1774 and 1776 Gerry attended the first and second provincial congresses. He served with Samuel Adams and John Hancock on the council of safety and, as chairman of the committee of supply (a job for which his merchant background ideally suited him) wherein he raised troops and dealt with military logistics. On the night of April 18, 1775, Gerry attended a meeting of the council of safety at an inn in Menotomy (Arlington), between Cambridge and Lexington, and barely escaped the British troops marching on Lexington and Concord.

In 1776 Gerry entered the Continental Congress, where his congressional specialities were military and financial matters. In Congress and throughout his career his actions often appeared contradictory. He earned the nickname "soldiers' friend" for his advocacy of better pay and equipment, yet he vacillated on the issue of pensions. Despite his disapproval of standing armies, he recommended long-term enlistments.

Until 1779 Gerry sat on and sometimes presided over the congressional board that regulated Continental finances. After a quarrel over the price schedule for suppliers, Gerry, himself a supplier, walked out of Congress. Although nominally a member, he did not reappear for 3 years. During the interim, he engaged in trade and privateering and served in the lower house of the Massachusetts legislature.

As a representative in Congress in the years 1783-85, Gerry numbered among those who had possessed talent as Revolutionary agitators and wartime leaders but who could not effectually cope with the painstaking task of stabilizing the national government. He was experienced and conscientious but created many enemies with his lack of humor, suspicion of the motives of others, and obsessive fear of political and military tyranny. In 1786, the year after leaving Congress, he retired from business, married Ann Thompson, and took a seat in the state legislature.

Gerry was one of the most vocal delegates at the Constitutional Convention of 1787. He presided as chairman of the committee that produced the Great Compromise but disliked the compromise itself. He antagonized nearly everyone by his inconsistency and, according to a colleague, "objected to everything he did not propose." At first an advocate of a strong central government, Gerry ultimately rejected and refused to sign the Constitution because it lacked a bill of rights and because he deemed it a threat to republicanism. He led the drive against ratification in Massachusetts and denounced the document as "full of vices." Among the vices, he listed inadequate representation of the people, dangerously ambiguous legislative powers, the blending of the executive and the legislative, and the danger of an oppressive judiciary. Gerry did see some merit in the Constitution, though, and believed that its flaws could be remedied through amendments. In 1789, after he announced his intention to support the Constitution, he was elected to the First Congress where, to the chagrin of the Antifederalists, he championed Federalist policies.

Gerry left Congress for the last time in 1793 and retired for 4 years. During this period he came to mistrust the aims of the Federalists, particularly their attempts to nurture an alliance with Britain, and sided with the pro-French Democratic-Republicans. In 1797 President John Adams appointed him as the only non-Federalist member of a three-man commission charged with negotiating a reconciliation with France, which was on the brink of war with the United States. During the ensuing XYZ affair (1797-98), Gerry tarnished his reputation. Talleyrand, the French foreign minister, led him to believe that his presence in France would prevent war, and Gerry lingered on long after the departure of John Marshall and Charles Cotesworth Pinckney, the two other commissioners. Finally, the embarrassed Adams recalled him, and Gerry met severe censure from the Federalists upon his return.

In 1800-1803 Gerry, never very popular among the Massachusetts electorate because of his aristocratic haughtiness, met defeat in four bids for the Massachusetts governorship but finally triumphed in 1810. Near the end of his two terms, scarred by partisan controversy, the Democratic-Republicans passed a redistricting measure to ensure their domination of the state senate. In response, the Federalists heaped ridicule on Gerry and coined the pun "gerrymander" to describe the salamander-like shape of one of the redistricted areas.

Despite his advanced age, frail health, and the threat of poverty brought on by neglect of personal affairs, Gerry served as James Madison's Vice President in 1813. In the fall of 1814, the 70-year old politician collapsed on his way to the Senate and died. He left his wife, who was to live until 1849, the last surviving widow of a signer of the Declaration of Independence, as well as three sons and four daughters. Gerry is buried in Congressional Cemetery at Washington, DC.

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Excerpts from the U.S. Constitution on Reapportionment

Article I, Section 2 of the U.S. Constitution specifies the method of apportionment of Representatives among the states, including a Census ("Enumeration") at least once every ten years. Article XIV (14), Section 1 defines a U.S. citizen and a State citizen. Article XIV, Section 2 specifies the method of apportionment of Representatives among the states following the prohibition of slavery. The following are excerpts from the U.S. Constitution pertaining to reapportionment and therefore to redistricting. State Constitutions and/or legislation provide more detailed requirements for the redistricting process.

Article I

Section 2. ... Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons. The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct. The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative; and until such enumeration shall be made, the State of New Hampshire shall be entitled to chuse three, Massachusetts eight, Rhode-Island and Providence Plantations one, Connecticut five, New-York six, New Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North Carolina five, South Carolina five, and Georgia three. ...

Article XIV
[Proposed 1866; Ratified Under Duress 1868]

Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws. 

Section 2. Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed. But when the right to vote at any election for the choice of electors for President and Vice President of the United States, Representatives in Congress, the Executive and Judicial officers of a State, or the members of the Legislature thereof, is denied to any of the male inhabitants of such State, being twenty-one years of age, and citizens of the United States, or in any way abridged, except for participation in rebellion, or other crime, the basis of representation therein shall be reduced in the proportion which the number of such male citizens shall bear to the whole number of male citizens twenty-one years of age in such State.

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California Constitution Article 21 on Reapportionment

Articles 4, 13, and 21 of the California State Constitution specify the requirements for reapportionment and redistricting, with Article 21 Section 1 defining all of the constitutional requirements except for the number of elected officials and corresponding election districts for each type of office.

ARTICLE 4:	http://www.LegInfo.ca.gov/.const/.article_4
ARTICLE 13:	http://www.LegInfo.ca.gov/.const/.article_13
ARTICLE 21:	http://www.LegInfo.ca.gov/.const/.article_21
TABLE OF CONTENTS:	http://www.LegInfo.ca.gov/const-toc.html

CALIFORNIA STATE CONSTITUTION

ARTICLE 4  LEGISLATIVE

SEC. 6.  For the purpose of choosing members of the Legislature, the State shall be divided into 40 Senatorial and 80 Assembly districts to be called Senatorial and Assembly Districts.  Each Senatorial district shall choose one Senator and each Assembly district shall choose one member of the Assembly.

ARTICLE 13  TAXATION

SEC. 17.  The Board of Equalization consists of 5 voting members: the Controller and 4 members elected for 4-year terms at gubernatorial elections. The State shall be divided into four Board of Equalization districts with the voters of each district electing one member.  No member may serve more than 2 terms.

ARTICLE 21  REAPPORTIONMENT OF SENATE, ASSEMBLY, CONGRESSIONAL AND BOARD OF EQUALIZATION DISTRICTS

SECTION 1.  In the year following the year in which the national census is taken under the direction of Congress at the beginning of each decade, the Legislature shall adjust the boundary lines of the Senatorial, Assembly, Congressional, and Board of Equalization districts in conformance with the following standards:
   (a) Each member of the Senate, Assembly, Congress, and the Board of Equalization shall be elected from a single-member district.
   (b) The population of all districts of a particular type shall be reasonably equal.
   (c) Every district shall be contiguous.
   (d) Districts of each type shall be numbered consecutively commencing at the northern boundary of the State and ending at the southern boundary.
   (e) The geographical integrity of any city, county, or city and county, or of any geographical region shall be respected to the extent possible without violating the requirements of any other subdivision of this section.

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Closed Primary Elections

The general election determines which political party's candidate will represent each election district for each type of office. In the general election, at most one candidate from each party is allowed to run in each district.

The primary election is used by each political party to determine which candidate will represent the party in each district in the general election. In closed primary elections, only voters registered as members of a given party can vote to determine which candidate will represent that party in the general election.

In contrast, open primary elections allow voters to determine which candidate will represent a party other than their own party. Those who promote open primary elections claim they want the ability to influence the outcome of primary elections when their party has only one candidate who is running unoposed.

However, open primary elections are inherently unfair, corrupt, and irrational. They promote sabatage of one party by voters of another party, who vote for the less popular candidate that is less likely to win the general election against the candidate from their own party. Only registered members of a political party should be allowed to influence which candidate represents the party in the general election.

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Presidential Elections and the Electoral College

Under the Constitution the Electoral College gives, by design, some additional political power to the less populous states in electing the president, as does the U.S. Senate in legislative matters. Each state has two senators, and one or more representatives depending on state population. Each state has one elector in the Electoral College for each of its senators and representatives.

The Electoral College together with the "winner takes all" system used by most states to award electors to presidential candidates keeps national politics in a stable equilibrium near the center by creating a two-party system that minimizes the political power of fringe party extremists.

United States Constitution Article II, Section 1, Clause 2 "Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector." http://www.house.gov/Constitution/Constitution.html http://www.archives.gov/national_archives_experience/ charters/constitution_transcript.html#2.0 http://www.archives.gov/national_archives_experience/ charters/constitution.html

For example, Democrat Bill Clinton became president in 1992 with only 43 percent of the popular vote when Henry Ross Perot ran as an independent and split the vote, receiving 18.9 percent of the popular vote but no electoral votes. In 1996, Bill Clinton was reelected with only 49.2 percent when Henry Ross Perot ran as a Reform party candidate and split the vote again, receiving 8.4 percent of the popular vote but no electoral votes. Green Party candidate Ralph Nader received 2.7 percent of the popular vote, but no electoral college votes. In 2000, Republican George Walker Bush became president with only 47.9 percent of the popular vote compared to Democrat Al Gore's 48.4 percent, after winning 271 electoral votes compared to Al Gore's 266 electoral votes. (See http://presidentelect.org/index.html, http://presidentelect.org/e1992.html, http://presidentelect.org/e1996.html, and http://presidentelect.org/e2000.html.)

"In a multi-candidate race where candidates have strong regional appeal, as in 1824, it is quite possible that a candidate who collects the most votes on a nation-wide basis will not win the electoral vote. In a two-candidate race, that is less likely to occur. But it did occur in the Hayes/Tilden election of 1876 and the Harrison/Cleveland election of 1888 due to the statistical disparity between vote totals in individual State elections and the national vote totals. This also occurred in the 2000 presidential election, where George W. Bush received fewer popular votes than Albert Gore Jr., but received a majority of electoral votes." http://www.archives.gov/federal_register/ electoral_college/faq.html#popularelectoral

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Bicameral State Legislature

Prior to a 1964 U.S. Supreme Court ruling, California and other states had a bicameral legislature modeled after the Congress and the federal Constitution. The state Senate, the upper house, provided representation based on geographical and political subdivisions of the larger entity, namely, equal representation for each county just as there is for each state in the U.S. Senate. Therefore, the state Senate was not subject to gerrymandering.

The state Assembly, the lower house, provided representation based on population. Thus, there was an additional balance of power modeled after the federal Constitution, where the upper house would be a safeguard against the "tyranny of the majority", where large population counties (or states) would out-vote and dominate over small population counties (or states). Yet the Supreme Court ruled that a legislature modeled after the federal Constitution was unconstitutional, and violated that very same Constitution.

This ruling required that the state Senate representation also be based on population, making it somewhat redundant with the state Assembly, except for some differences including number of seats and length of terms. There are forty Senators serving four year terms, compared to eighty Assemblymen serving two year terms.

The ruling subjected the state Senate to gerrymandering. It also subjected the state Assembly to a greater degree of gerrymandering because redistricting legislation must be approved by both the state Senate and Assembly, and signed by the governor. When the state Senate was not subject to gerrymandering, there was a greater chance that it would block legislation to gerrymander the Assembly and congressional districts.

This ruling destroyed the balance of power, favoring the Democratic Party machine and densely populated urban areas. But the ruling was a benefit to all future incumbent Senators and Assembly members, because the state Senate was now also subject to gerrymandering and the Assembly could be more severely gerrymandered. Could this ruling have been the work of a liberal judicial activist majority on the Court?

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The Danger of Judicial Activism

One risk of relying on the courts is the danger of judicial activism, where judges act as political partisans and ignore the Constitution, the law, the previous court precedents, and even common sense, and instead decide as they please to further their political agenda.

Judicial activism is more of a problem among elitist liberal judges. Liberal judges claim that the Constitution is a "living document" that must bend and change with the times. They re-interpret, i.e., misinterpret, and effectively rewrite the Constitution as they please through their court decisions, operating outside their legitimate authority and the existing system of extensive checks and balances. This approach directly violates Article V of the U.S. Constitution, which clearly defines the lengthy and difficult process for amending the Constitution, and which is the only legitimate method for making changes. Article V establishes a system of checks and balances that is absolutely necessary to ensure the continuing existence of the Constitution and the protections and freedoms it affords the people.

"On every question of construction (of the Constitution) let us carry ourselves back to the time when the Constitution was adopted, recollect the spirit manifested in the debates, and instead of trying what meaning may be squeezed out of the text, or invented against it, conform to the probable one in which it was passed." - Thomas Jefferson Letter to William Johnson, June 12, 1823 The Complete Jefferson, p. 322

Under the "living document" approach, the Constitution is in fact a dead document that can no longer protect freedom and stability from the excesses, oppression, and persecution of an out of control government. Conservative judges, on the other hand, tend to be strict constructionists. They adhere strictly to the original construction and intent of the founders and framers of the Constitution, relying on the written document itself, the writings of the founders including the federalist papers and the proceedings of the constitutional debates, and previous court precedents. They honor the entire Constitution, including Article V and the entire Bill of Rights, and they respect the limits on their own authority. Judicial activism is another form of fraud and corruption that is at least as dangerous as gerrymandering.< /p>

Judicial activism is one of the reasons why it is important whether conservatives or liberals are elected as presidents and governors, because presidents and governors appoint justices to the federal and state supreme courts.

Article V of the U.S. Constitution reads as follows:

Article. V.

The Congress, whenever two thirds of both Houses shall deem it necessary, shall propose Amendments to this Constitution, or, on the Application of the Legislatures of two thirds of the several States, shall call a Convention for proposing Amendments, which, in either Case, shall be valid to all Intents and Purposes, as Part of this Constitution, when ratified by the Legislatures of three fourths of the several States, or by Conventions in three fourths thereof, as the one or the other Mode of Ratification may be proposed by the Congress; Provided that no Amendment which may be made prior to the Year One thousand eight hundred and eight shall in any Manner affect the first and fourth Clauses in the Ninth Section of the first Article; and that no State, without its Consent, shall be deprived of its equal Suffrage in the Senate.

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Louisiana Demographic Map Examples

The Louisiana demographic map examples below demonstrate some of the different types of demographic data available and how it looks when displayed as map overlays. Darker areas represent larger numbers and lighter areas represent smaller numbers. You can click on each map or the title below it to see a larger image in a new browser window. Simply close that window to return to this window. These map images are also available from the Louisiana state census information web page:

http://www.state.la.us/census/demographic_themes.htm

1990 Population Land Area Population Density 1999 Population 1990 Pop >= 65 French Ancestry %French Ancestry % French Pop Cajun Population % 1990 Pop >= 65 Female HH w/Child Household Income % Pop in Poverty Children in Poverty 1990 Pop 18-49 Black Population White Population Asian Population Hispanic Pop % 1990 Pop 18-49 Sex Ratio Mobile Homes % Mobile Homes 1987 Nbr Farms 1987 Farm Size

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Deriving the Formula for a Compactness Metric for an Election District

One approach to deriving a Compactness Metric is to compare an election district shape with a compact reference shape. In addition, a useful Compactness Metric should yield the same numeric result for the same shape with different areas. As mentioned earlier, this has the desired property that two districts with the same shape and (approximately) the same number of voters will have the same (or nearly identical) Compactness Metric values even though the less densely populated district has a larger area. Thus, the Compactness Metric should be normalized to remove differences in population density.

A formula can be derived for a Compactness Metric to compare an election district's compactness with that of a square containing the same area, which we know to be compact. In fact, if we draw a series of increasingly wider and shorter rectangles that have the same area as the square, we see that the total boundary length of the rectangle increases while the area remains constant, and the ratio of the boundary to the area (B/A) increases as the rectangle becomes less compact. Therefore, a square is a good reference shape for such comparisons. The total boundary length (B) and area (A) of the election district can be computed using a computer program and digitized map data for the election district boundary.

For those who prefer more rigor, we provide a mathematical proof that the boundary to area ratio of a square is less than that of any rectangle of equal area. We state without proof that the boundary to area ratio of a circle is less than that of any other geometric shape in a plane with equal area. After deriving the formula for the Compactness Metric, we also prove that using a square is equivalent to using a circle as a reference shape. However, it is possible to completely cover a given area with a finite number of non-overlapping squares, but not with non-overlapping circles.

We derive the Compactness Metric formula as follows. The ratio of the boundary to the area (B/A) of a square is four times the length of a side (4s) divided by the square of the length of a side (s²), or:

B / A = 4s / (s2) = 4 Sqrt(A) / A = 4 / Sqrt(A)

where Sqrt(A) is the square root of A, or s. Then for any election district, the Compactness Metric (CM) is the ratio of the B/A ratio of the district to the B/A ratio of the equivalent area square, which is

CM = (B / A) / (4 / Sqrt(A)) = (1/4) B / Sqrt(A)

For simplicity, we can simply omit the 1/4 coefficient and adjust the valid range of CM values used for comparison accordingly, by multiplying them by 4:

CM = B / Sqrt(A)

An equivalent but more useful Compactness Metric is obtained by squaring the value of the B / Sqrt(A) Compactness Metric to provide a greater range of values and greater differences between the values to be compared:

CM = B2 / A

The following discussion is based on the B / Sqrt(A) Compactness Metric although it applies equally to the B² / A Compactness Metric.

This Compactness Metric has several desirable qualities. First, the simplified formula is quite intuitive, because the length of the boundary is a linear measure whereas the area is a square measure, and the square root of the area brings us back to a linear measure. Second, it is unambiguous and straightforward to compute. And third, its value is invariant for the same shape with different sizes and areas. That is, B/Sqrt(A) yields the same numeric result for the same shape with different areas whereas B/A does not. This is because scaling the size of a shape by a constant multiplier C multiples the boundary length by C but multiplies the area by the square of C, or C².

For example, B/Sqrt(A) is the same for a square with side of length S and another square with side of length 2S and does not depend on the size or area of the square, but B/A is not the same because it depends on the area:

Small Square:		B/Sqrt(A) = 4S / Sqrt(S2) = 4S/S = 4
Large Square:		B/Sqrt(A) = 4(2S) / Sqrt((2S)2) = 8S/2S = 4
Small Square:		B/A = 4S/(S2) = 4/S = 4/Sqrt(A)
Large Square:		B/A = 4(2S) / ((2S)2) = 8S / 4 S2 = 2/S = 2 / ( Sqrt(A) / 2 ) = 4/Sqrt(A)

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If it is obvious that scaling the size of a shape by a constant multiplier C multiples the boundary length by C and multiplies the area by C², then you may skip the following material.

If this result is not obvious, it can be understood by using some techniques from calculus. We approximate the shape's boundary with a piecewise linear closed curve, and the shape's area with a fine grid of squares that fit entirely within the shape's boundary. We reduce the granularity of these approximations until they approach the actual boundary and area. Then we examine what happens when we scale the approximations by a constant multiplier.

The boundary can be approximated by a sequence of line segments where each segment connects to two other segments at its endpoints. As the number of line segments approximating curved nonlinear sections of the boundary increases, the sum of the lengths of the line segments approaches the length of the boundary. Scaling the shape will also scale the coordinates of each line segment by the same multiplier. Multiplying a line segment's endpoint coordinates by a constant multiplies the length of the line segment by the same constant.

To prove this, we use the distance formula for the distance between two points in a plane, which is based on the Pythagorean Theorem for the length of the hypotenuse (longest side) c of a right-triangle (one with a 90 degree right angle) with sides of length a, b, and c:

c2 = a2 + b2

The distance formula for the distance between two points in a plane,
(x₁, y₁) and (x₂, y₂), is:

d( (x1, y1), (x2, y2) ) = Sqrt( (x2 - x1)2 + (y2 - y1)2 )

After scaling a line by multiplying its endpoint coordinates by the constant C, we have:

d( (Cx1, Cy1), (Cx2, Cy2) )  = Sqrt( (Cx2 - Cx1)2 + (Cy2 - Cy1)2 )  = Sqrt( [C(x2 - x1)]2 + [C(y2 - y1)]2 )  = Sqrt( C2 (x2 - x1)2 + C2 (y2 - y1)2 )  = Sqrt(C2) * Sqrt( (x2 - x1)2 + (y2 - y1)2 )  = C * Sqrt( (x2 - x1)2 + (y2 - y1)2 )  = C * d( (x1, y1), (x2, y2) )

For the area, we approximate the shape with a fine grid of squares that fit entirely within the shape, and omit squares outside the boundary or near the boundary that do not fit entirely within the boundary. As the length of the side of each square in the grid approaches zero and the number of squares in the grid approximation approaches infinity, the combined area of the grid of squares approaches the area of the shape. Scaling the shape by the multiplier C will also scale each square in the grid by the same multiplier. Thus, each square in the grid will have its sides along both dimensions scaled by the constant multiplier, which will multiply the area of each square in the grid, and the entire area of the shape, by the square of the multiplier which is C².

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The Compactness Metric of a district will have a value close to one (or four for the simplified form) if its compactness is close to that of a square with the same area. It will have a value greater than one (or four) if it is less compact than the equivalent square. Compactness Metric values can be computed for election districts that are known to be fair and for other districts that are known to be gerrymandered. These values can be compared to determine what range of values is reasonable for the Compactness Metric.

Note, however, that some election districts that are not gerrymandered, but which follow the boundary lines of political subdivisions such as counties and cities may not be compact due to the irregularities of the pre-existing county or city boundaries. Thus, the Compactness Metric should be only one of several metrics or criteria used to objectively determine whether or not a given election district or a complete set of election districts are gerrymandered. One modification to the Compactness formula would be to exclude or adjust the length of boundary lines of the election district that follows county or city boundary lines from the total boundary length. In this case, the range of valid values may need to be adjusted.

Another type of Compactness Metric can be defined to measure the total accumulated amount of turns, curves, and changes in direction that occur during the course of tracing the boundary of an election district. Threshold values for minimum distance and other criteria can be defined to accumulate only amounts that exceed the thresholds and are therefore considered significant. Portions of the boundary that conform to the pre-existing boundary of a county, city, or some other political subdivision or natural boundary can be excluded from the calculation. Then the total accumulated amount can be compared with some threshold value to determine in an objective manner whether the district boundary is gerrymandered.

Yet another compactness metric might be the maximum distance between any two map locations in the district relative to the square root of the area of the district. Gerrymandered districts often connect otherwise disjoint and relatively distant areas of land with thin strips that run along highways, through bodies of water including even the ocean, and through other unpopulated or sparsely populated areas. A compactness metric based on maximum distance would detect this type of gerrymander. However, it may be the case that a state with distant densely populated areas separated by sparsely populated areas may require such geometry to create districts with approximately equal voter population. Clearly, several types of compactness metrics and validation criteria can be defined and used together in the same mathematical model of a fair election district versus a gerrymandered district.

As a goal, a computer program would be developed that would use an objective mathematical algorithm to draw compact contiguous election district boundaries that meet specific criteria to ensure that the districts are not gerrymandered. Such a program would also ensure that the district maps generated meet other criteria such as preserving the integrity of county and city boundary lines where possible. Metrics can be computed and analyzed to verify that the redistricting maps are not gerrymandered. If the goal of using computers and a mathematical algorithm cannot be met, then the redistricting maps should be drawn by an independent bipartisan commission. The maps should then have metrics computed and analyzed, and the maps should be subject to public review and criticism, and ultimately to judicial review.

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The boundary to area ratio (B/A) of a square is less than that of any rectangle of equal area.

Here is a figure and an outline of the proof:

Proof:

This is equivalent to proving that for any positive ( > 0 ) real numbers x, y, and z such that x times y is equal to the square of z and x is not equal to y, i.e.,
xy = z² and x <> y,

z < Average(x,y) = (x+y)/2

To see this, assume that we are comparing the boundary to area ratio (B/A) of a square with sides of length S and a rectangle with length L and width W, both having equal area A. Let x=L, y=W, and z=S. Then the B/A ratio for the square is 4S/A, and the B/A ratio for the rectangle is 2(L+W)/A. Then using the <==> symbol to represent "if and only if" (iff), we have the following:

4S/A < 2(L+W)/A <==> 2S < L+W <==> S < (L+W)/2 <==>  z < (x+y)/2 = Average(x,y)

Proof 1:

We will prove the latter statement, using simple algebra.

Let x, y, and z be positive real numbers ( > 0 ) such that x times y is equal to the square of z, and x is not equal to y, i.e., xy = z2 and x <> y. Then z = Sqrt(xy) < (x+y)/2 = Average(x,y) <==> x + y > 2 Sqrt(xy) <==> (x+y)2 > (2 Sqrt(xy))2 <==> x2 + 2xy + y2 > 4xy <==> x2 + (2xy - 4xy) + y2 > 0 <==> x2 - 2xy + y2 > 0 <==> (x-y)2 = (y-x)2 > 0 <==> Sqrt((x-y)2) = Sqrt((y-x)2) > Sqrt(0) <==> +/- (x-y) > 0 <==> x > y or y > x <==> x <> y, which is true because it is one of the assumptions. Note that if x = y, then Sqrt(xy) = Sqrt(x2) = x = 2x/2 = (x+x)/2 = (x+y)/2

We will restate this proof, using the variables S, L, and W in place of z, x, and y to retain the geometric relationships.

Let L, W, and S be positive real numbers ( > 0 ) such that L times W is equal to the square of S, and L is not equal to W, i.e., LW = S2 and L <> W. Then 4S = 4 Sqrt(LW) < 2 (L+W) = 4 ((L+W)/2) = 4 Average(L,W) where 4S = perimeter of a square with area = s2 and with sides of length S, and 2 (L+W) = perimeter of a rectangle with equal area LW and with unequal sides of length L and W <==> S = Sqrt(LW) < (L+W)/2 = Average(L,W) <==> L + W > 2 Sqrt(LW) <==> (L+W)2 > (2 Sqrt(LW))2 <==> L2 + 2LW + W2 > 4LW <==> L2 + (2LW - 4LW) + W2 > 0 <==> L2 - 2LW + W2 > 0 <==> (L-W)2 = (W-L)2 > 0 <==> Sqrt((L-W)2) = Sqrt((W-L)2) > Sqrt(0) <==> +/- (L-W) > 0 <==> L > W or W > L <==> L <> W, which is true because it is one of the assumptions. Note that if L = W, then Sqrt(LW) = Sqrt(L2) = L = 2L/2 = (L+L)/2 = (L+W)/2

Proof 2:

We will prove the latter statement, using the variables S, L, and W in place of z, x, and y to retain the geometric relationships, although the proof only requires algebra. We will prove two lemmas as aids, or stepping stones, to proving the original statement.

Lemma 1: If L, W, and S are positive ( > 0 ) real numbers such that L > W and LW = S2, then L > S and W < S.

Proof of Lemma 1: Define the auxiliary variable d such that L = W+d. Then S2 = LW = L(L-d) = L2 - Ld <==> (iff) L2 = S2 + Ld <==> L = Sqrt( S2 + Ld ) > Sqrt( S2 ) = S (because L > 0, d >  0 ==> (implies) Ld > 0). Therefore, L > S. Similarly, S2 = LW = (W+d)W = W2 + Wd <==> W2 = S2 - Wd <==> W = Sqrt( S2 - Wd ) < Sqrt( S2 ) = S (because W > 0, d > 0 ==> Wd > 0). Therefore, W < S.

Lemma 2: For any square with sides of length S and any rectangle of the same area with sides of length L and width W (i.e., L > W), L - S > S - W.

Proof of Lemma 2: Define the auxiliary variables a and b such that a = L - S, b = S - W. Then equal area ==> S2 = LW = (S+a)(S-b) = S2 + (a-b)S - ab <==> 0 = (a - b)S - ab <==> ab = S(a - b) <==> a - b = (ab)/S > 0 (because of Lemma 1, L > S and W < S; therefore, L > S ==> a = L - S > 0; W < S ==> b = S - W > 0; and S > 0) ==> a - b > 0 ==> a > b Therefore, L - S > S - W.

The B/A ratio of a square is 4S / A. The B/A ratio of a rectangle with the same area is 2(L + W) / A. 4S / A < 2(L + W) / A  <==>  L + W > 2S  = S + S  <==>  L - S > S - W , which we know to be true from Lemma 2.

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Deriving the formula for the Compactness Metric (CM) using a square is equivalent to using a circle as the reference shape.

To prove this, we derive the formula for the Compactness Metric using a circle and show that it is equivalent to the formula derived above using a square except for a constant multiplier. This multiplier can be eliminated from the formula and the threshold values can be adjusted accordingly, as was done above.

Proof:

For a circle as the reference shape, where C is the circumference of the circle and A is the area of the circle, and r is the radius, we have:

Pi = 3.141592654... = (approximately) 22 / 7 ; Circumference of a circle = C = 2 * Pi * r ; Area of a circle = A = Pi * (r2) ==> r = Sqrt(A/Pi). CM = (B/Sqrt(A)) / (C/Sqrt(A)) = B/C = B / (2 * Pi * r) = B / ( 2 * Pi * Sqrt( A / Pi ) ) = B / ( 2 * Sqrt(Pi)  * Sqrt(A) ) = ( 1 / (2 * Sqrt(Pi)) )  *  ( B / Sqrt(A) ) = 0.2820947942 * ( B / Sqrt(A) ) = (approximately)  0.2821 * ( B / Sqrt(A) )

Thus, using a circle as the reference shape leads to the same Compactness Metric formula as using a square, except for a different constant multiplier coefficient of  0.2821 for the circle instead of 0.25 for the square.

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Formulas for the Area and Centroid (Center of Mass) of a Polygon

There is a simple formula for computing the area of a polygon. Similarly, there is a simple formula for computing the centroid (also known as the center of mass or center of gravity) of a polygon of uniform density.

Consider an arbitrary polygon that does not intersect itself and that is made up of line segments between N vertices (x_i,y_i), i=0 to N-1. The last vertex (x_N,y_N) is assumed to be the same as the first, i.e., the polygon is closed.

Using Green's theorem for the plane, the polygon area is given by

A = (1/2) x Summation for i = 0 to N - 1 of (xiyi+1 - xi+1yi)

The sign of the area expression above is positive if the polygon vertices are ordered counter clockwise about the normal, and negative otherwise.

This formula can be visualized by drawing a polygon above the x axis and to the right of the y axis in the x-y plane. Draw a vertical auxiliary line from each polygon vertex point down to the x axis. Compute the area of the trapezoid formed by each adjacent pair of polygon vertex points and the intersection points of the x axis and the two auxiliary lines drawn from these points to the x axis. The area of each trapezoid is the area of the base (x_i - x_i+1) times the average of the lengths of the two auxiliary lines ((y_i + x_i+1)/2). The polygon area is computed as the sum of the area of the trapezoids formed using the top polygon vertex points minus the area of the trapezoids formed using the bottom polygon vertex points. That is, the areas outside the polygon (between the polygon and the x axis) eventually cancel as the polygon loops around to the beginning.

When applying this formula to real world absolute map coordinates, it is important to use sufficient precision and to avoid roundoff errors. Roundoff errors may become noticeable when the coordinate values exceed 10⁶ and the coordinate products in Green's formula exceed 10¹². Roundoff errors are reduced (but not eliminated) by using relative coordinates in place of real coordinates for the polygon vertex points. The relative coordinates are relative to an arbitrary origin point that is close to the polygon vertex points (boundary points) rather than far away outside the polygon. The chosen origin point may be any arbitrary polygon vertex point, or the center point of the minimum bounding rectangle. The coordinates of this origin point are subtracted from all polygon vertex points used in the polygon area formula.

The center point of the minimum bounding rectangle is easily computed as

cx = min(xi) + (1/2)(max(xi) - min(xi)) ,   for i = 0 to N - 1 ; cy = min(yi) + (1/2)(max(yi) - min(yi)) ,   for i = 0 to N - 1 .

Then the polygon area formula becomes

A = (1/2) x Summation for i = 0 to N - 1 of ( (xi - cx)(yi+1 - cy) - (xi+1 - cx)(yi - cy) )

Note that simplifying or optimizing the above formula by pulling the constants c_x and c_y outside the Summation operator will reintroduce the roundoff errors we are trying to avoid. However, these constants can be subtracted from all boundary points before applying the Polygon Area formula to avoid redundant calculations.


The centroid is given by the following formulas for its x and y coordinates, c_x and c_y:

cx = (1/(6A)) x Summation for i = 0 to N - 1 of ( (xi + xi+1)(xiyi+1 - xi+1yi) ) cy = (1/(6A)) x Summation for i = 0 to N - 1 of ( (yi + yi+1)(yiyi+1 - yi+1yi) )

When using these centroid formulas, real world absolute map coordinates should be converted to relative coordinates to reduce roundoff errors, as was described for the Polygon Area formula. Note, however, that the c_x and c_y notation is used differently for the centroid formulas.


Additional information on computing Polygon Area and Centroid can be found at the following web pages:

Calculating the area and centroid of a polygon
http://astronomy.swin.edu.au/~pbourke/geometry/polyarea/

Polygon Area
http://mathworld.wolfram.com/PolygonArea.html

efg's Graphics -- Polygon Area and Centroid
http://homepages.borland.com/efg2lab/Library/Graphics/PolygonArea.htm

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Theoretical Grid and Checkerboard Gerrymanders

It is possible, at least in theory, to have a Grid Gerrymander where a region is divided up into a grid of squares or rectangles, and each square or rectangle of the grid is arbitrarily assigned to either one of two "adjacent" contiguous districts. The Checkerboard Gerrymander (or Chessboard Gerrymander, if you prefer) is a special case of the Grid Gerrymander. In a Checkerboard Gerrymander, two "adjacent" contiguous districts are created in a checkerboard pattern where any two adjacent squares or rectangles adjoining along an edge belong to different districts. The theoretical Grid Gerrymander and Checkerboard Gerrymander were devised by the author, Michael D. Robbins, to demonstrate in a simple manner how extreme a gerrymander can become in spite of the contiguous district requirement.

The Grid Gerrymander is constructed by dividing a region up into multiple rows of squares or rectangles; one thin strip each along the top, the bottom, the left, and the right boundaries of the region; and a pair of thin horizontal strips between each row of squares or rectangles. The thin strips can be arbitrarily narrow relative to the dimensions of the squares or rectangles, and in absolute numbers.

The top and left boundary strips and the bottom strip of each pair of horizontal strips are assigned to the first of the two districts as shown below in red. The remaining strips (i.e., the bottom and right boundary strips and the top strip of each pair of horizontal strips) are assigned to the second district of the pair as shown below in blue.

Then each of the squares or rectangles in each row can be arbitrarily assigned to either the first or the second district by opening up its (thin black) boundary line adjoining the thin strip above or below it.

To form a Checkerboard Gerrymander, we perform a two-step process. First, we assign each square or rectangle to either one of the two contiguous regions or districts, in a checkerboard pattern where any two adjacent squares or rectangles adjoining along an edge belong to different districts. Second, we open up the top or bottom boundary line of each square or rectangle as appropriate, depending on the district assignment for the square.

The following example shows arbitrarily selected squares assigned to either one of two separate contiguous regions or districts. The first image shows the assignment of each square to a district, either the red district or the blue district. The second image shows the result from the first image after we open up the top or bottom boundary line of each square as appropriate, depending on the district assignment for the square.

As an optional third step, we may remove the black side boundary lines between portions of squares that have the same color and belong to the same region or district, as shown in the left image below.

As an optional fourth step, we may remove portions of the horizontal connecting strips that are not used to connect squares or rectangles to a region or district, as shown in the right image below. Here we have removed the unused rightmost portion of the third and fourth horizontal red connecting strips from the top, and the entire fifth horizontal red connecting strip from the top because its entire length is unused.

The theoretical Modified Grid Gerrymander is obtained from the standard Grid Gerrymander by modifying the boundary and shape of one or more grid squares or rectangles to modify and have greater control over the effective shape and boundary of the two adjacent regions or districts, as shown in the lower left image below. Note that curves can be used in addition to straight lines. We can remove additional unused horizontal and vertical connecting strips, and we can extend a connecting strip upward, to minimize the number and linear distance of connecting strips required, as shown in the right image below. We can even replace the long vertical connecting strip on the left with a shorter one closer to the center of the image.

The Grid Gerrymander and the Checkerboard Gerrymander demonstrate how contiguous regions can be crafted that do not conform to, but stray far from, our intuitive notion of contiguity. For each of the examples above, you can verify that each of the two (red and blue) districts is in fact contiguous by tracing its boundary, starting from any point on its boundary, without reversing direction along your path, and you will eventually arrive at your starting point.

It is possible to use the grid approach to gerrymandering, at least in part, even where election districts must respect the geographical integrity of political subdivisions to the extent possible. However, it is not necessary to use the grid approach because similar results can be achieved with significantly less complex boundaries. The theoretical Grid Gerrymander simply provides extreme flexibility when carving up a region into two districts that are safe for different political parties.

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An Example Demonstrating Unsafe, Competitive Districts

This 4-by-5 example illustrates three different redistricting maps for the same geographical area and voter population. Each map distributes one fifth of the voter population into each of five election districts. Map 5 yields five fair and desirable, unsafe, competitive election districts. Although the total voter population consists of half Republicans and half Democrats, Map 6 yields a 60-40 percent split of Legislature seats favoring Democrats, and Map 7 yields a 60-40 percent split of Legislature seats favoring Republicans.

In this example, as in the previous example, each square represents a region of similar population. All squares in the same election district must touch at least one other square in the district either along a side or at a corner, and two election districts may not cross over each other at a square corner. The R squares have a majority of Republican voters and the D squares have a majority of Democratic voters.

For simplicity, assume that there are only Republican and Democratic voters, and further assume that 60 percent of the voters in each R square are Republicans, and 60 percent of the voters in each D square are Democrats.

Under these assumptions for the percentage of voters from each party in a square, we have the following: An election district with 3 R squares and 1 D square is a safe 55 percent Republican district. Similarly, an election district with 3 D squares and 1 R square is a safe 55 percent Democratic district.

The calculations for a 2 R - 2 D election district with 60 percent squares are as follows:

Republican Percentage:	(60% x 2/4) + (40% x 2/4) = 50%
Democratic Percentage:	(40% x 2/4) + (60% x 2/4) = 50%

The calculations for a 3 R - 1 D election district with 60 percent squares are as follows:

Republican Percentage:	(60% x 3/4) + (40% x 1/4) = 55%
Democratic Percentage:	(40% x 3/4) + (60% x 1/4) = 45%

The calculations for a 3 D - 1 R election district with 60 percent squares are as follows:

Democratic Percentage:	(60% x 3/4) + (40% x 1/4) = 55%
Republican Percentage:	(40% x 3/4) + (60% x 1/4) = 45%

The calculations for a 4 R - 1 D election district with 60 percent squares are as follows:

Republican Percentage:	(60% x 4/4) + (40% x 0/4) = 60%
Democratic Percentage:	(40% x 4/4) + (60% x 0/4) = 40%

The calculations for a 4 D - 1 R election district with 60 percent squares are as follows:

Democratic Percentage:	(60% x 4/4) + (40% x 0/4) = 60%
Republican Percentage:	(40% x 4/4) + (60% x 0/4) = 40%

Election Map Before Districts are Drawn: Contains a population with equal numbers of Republican and Democrat voters, divided into twenty square regions of equal population, where half the square regions contain a majority of Republican voters and the other half contain a majority of Democrat voters: D D D D D D D D D D R R R R R R R R R R A new election district map is to be drawn as an overlay on this map, by drawing the boundary lines for five contiguous election districts each made up of four square regions where no two districts overlap or cross each other. A district is contiguous if every square region in the district is adjacent to another square region in the district. Two square regions are adjacent if they have an edge in common or a point in common where their corners meet and the point is not also used to establish contiguity of another district. Thus, a district is contiguous if every square region in the district "touches" another square region in the district.

For the following election maps, each set of squares with the same color represent a single election district. For example, District 1 is white; District 2 is yellow; District 3 is blue; District 4 is purple, and District 5 is green, as follows:

District 1 - WHITE

District 2 - YELLOW

District 3 - BLUE

District 4 - PURPLE

District 5 - GREEN

Redistricting Map 5A: Yields five desirable less safe, more competitive districts, where the winner of the election may be either a Republican or a Democrat: D D D D D D D D D D R R R R R R R R R R Actually, these 50 percent districts will be fairly safe for Republican incumbents due to higher voter turnout among Republican voters.

Redistricting Map 5B: Yields one desirable unsafe competitive district (District 5) and four undesirable extremely safe, noncompetitive districts (Districts 1-4), even though the districts in Maps 5A and 5B have the same size and shape and therefore the same compactness metric: D D D D D D D D D D R R R R R R R R R R Actually, the 50 percent Districts 5 will be fairly safe for Republican incumbents due to higher voter turnout among Republican voters.

Redistricting Map 6: Yields five districts total, with two Republican and three Democratic majority districts, all safe 55 percent or better districts for incumbents of the respective parties: D D D D D D D D D D R R R R R R R R R R In this scheme, the Republican voters are grouped and concentrated in the smallest possible number of election districts. This results in District 1 being an extremely safe 60 percent Republican district. Thus, the Democrats who organized this gerrymander created even safer districts for their opposition, the Republicans.

Redistricting Map 7: Yields five districts total, with three Republican and two Democratic majority districts, all safe 55 percent or better districts for incumbents of the respective parties: D D D D D D D D D D R R R R R R R R R R In this scheme, the Democratic voters are grouped and concentrated in the smallest possible number of election districts. This results in District 4 being a very safe 60 percent Democratic district. Thus, the Republicans who organized this gerrymander created even safer districts for their opposition, the Democrats.

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An Example Demonstrating The Minority Party Controlling a Majority of Districts

This example demonstrates that, at least in theory, gerrymandered maps can be created where more than 60 percent of the registered voters in the map region belong to the majority party, but 60 percent of the election districts and legislative seats are safe districts for the minority party. Thus, the minority party has significantly more representation and political power in the State Legislature (or the Congress) than is fair under the one voter-one vote principle, at the expense of the minority party and all voters who suffer under safe noncompetitive election districts.

In actual practice, all that is necessary to have a majority control of the legislature is to have a solid voting block that consistently provides the 50 percent plus 1 simple majority vote required to approve most if not all legislation. A sixty percent majority is not necessary.

This 5-by-5 example illustrates three different redistricting maps for the same geographical area and voter population. Each map distributes one fifth of the voter population into each of five election districts. In this example, 16 of the 25 squares (more than 60 percent) are R squares, and 9 of the squares (less than 40 percent) are D squares. More than 55 percent of the voters in the map region are registered Republicans (or, if you prefer, are voters who vote Republican).

All three maps yield five safe noncompetitive election districts. Map 8 yields 3 Democratic and 2 Republican majority districts, for a 60-40 percent split favoring Democrats, even though more than 60 percent of the voters in the map region are Republicans. Map 9 yields 5 Republican and zero Democratic majority districts, for a 100-0 percent split favoring Republicans, even though nearly 40 percent of the voters in the map region are Democrats. Map 10 yields 3 Republican and 2 Democratic majority districts, for a 60-40 percent split favoring Republicans, which is close to the actual percentages of voter registration.

In this example, as in the previous example, each square represents a region of similar population. All squares in the same election district must touch at least one other square in the district either along a side or at a corner, and two election districts may not cross over each other at a square corner. The R squares have a majority of Republican voters and the D squares have a majority of Democratic voters.

For simplicity, assume that there are only Republican and Democratic voters, and further assume that 80 percent of the voters in each R square are Republicans, and 80 percent of the voters in each D square are Democrats.

Under these assumptions for the percentage of voters from each party in a square, we have the following: An election district with 3 R squares and 2 D squares is a very safe 56 percent Republican district. Similarly, an election district with 3 D squares and 2 R squares is a safe 56 percent Democratic district.

The calculations for a 3 R - 2 D election district with 80 percent squares are as follows:

Republican Percentage:	(80% x 3/5) + (20% x 2/5) = 56%
Democratic Percentage:	(20% x 3/5) + (80% x 2/5) = 44%

The calculations for a 4 R - 1 D election district with 80 percent squares are as follows:

Republican Percentage:	(80% x 4/5) + (20% x 1/5) = 68%
Democratic Percentage:	(20% x 4/5) + (80% x 1/5) = 32%

The calculations for a 5 R - 0 D election district with 80 percent squares are as follows:

Republican Percentage:	(80% x 5/5) + (20% x 0/5) = 80%
Democratic Percentage:	(20% x 5/5) + (80% x 0/5) = 20%

Election Map Before Districts are Drawn: Contains a population with 64 percent Republican and 36 percent Democrat voters, divided into twenty-five square regions of equal population, where sixteen square regions contain a majority of Republican voters and the other nine contain a majority of Democrat voters: R R R R R R R R R R R R R R R R D D D D D D D D D A new election district map is to be drawn as an overlay on this map, by drawing the boundary lines for five contiguous election districts each made up of five square regions where no two districts overlap or cross each other. A district is contiguous if every square region in the district is adjacent to another square region in the district. Two square regions are adjacent if they have an edge in common or a point in common where their corners meet and the point is not also used to establish contiguity of another district. Thus, a district is contiguous if every square region in the district "touches" another square region in the district.

For the following election maps, each set of squares with the same color represent a single election district. For example, District 1 is white; District 2 is yellow; District 3 is blue; District 4 is purple, and District 5 is green, as follows:

District 1 - WHITE

District 2 - YELLOW

District 3 - BLUE

District 4 - PURPLE

District 5 - GREEN

Redistricting Map 8: Yields five districts total, with three Democratic and two Republican majority districts, all safe 56 percent or better districts for incumbents of the respective parties: R R R R R R R R R R R R R R R R D D D D D D D D D In this scheme, the Republican voters are grouped and concentrated in the smallest possible number of election districts. This results in Districts 1 and 2 being extremely safe 80 percent Republican districts. Thus, those who organized the gerrymander created even safer districts for their opposition. Another map with the same results would make District 1 the top row and District 2 the second row.

Redistricting Map 9: Yields five districts total, with five Republican and zero Democratic majority districts, all safe districts for incumbents of the Republican party: R R R R R R R R R R R R R R R R D D D D D D D D D

Redistricting Map 10: By making only slight changes from Map 9, this Map 10 yields five districts total, with three Republican and two Democratic majority districts, all safe districts for incumbents of the respective parties: R R R R R R R R R R R R R R R R D D D D D D D D D

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