I suggest drawing rectangular districts on some specified map. If the map is a cylindrical projection, then the district boundaries will be lines of latitude & longitude. But if the district boundaries are lines of latitude & longitude, it isn't necessary to use a map. A subsequent posting will state the districting formula instructions for districts bounded by latitude & longitude lines for when it's desired to define those districts without the use of a map.
When using a map:
The central meridian of that map is the meridian that's halfway inbetween the meridians that the eastmost and westmost points of the map contact.
On the map:
"North-south" means parallel to the map's central meridian.
"East-west" means perpendicular to the map's central meridian.
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A2 stands for the approximate area of the map, in square centimeters.
A1 stands for the area of the state, in square miles.
h stands for the north-south distance on the map between the northmost and southmost points of the map, in centimeters.
h1 stands for the actual north-south distance, on the ground, between the northmost and southmost points of the state, in miles. (That's approximately the latitude difference between the northmost and southmost points, in degrees, multiplied by 69).
A2 = A1(h/h1)^2
A is the average area per district, as measured on the map, in square centimeters.
N is the desired number of districts.
A = A2/N
The state map is divided into bands of east-west extent. Starting with the northmost band, make each band in turn as follows, till the entire state population is in districts in the bands:
The first band has as its upper border the top edge of the map. Each subsequent band has as its upper border the lower border of the previous band.
The initial width of the band (in the north-south direction) is sqr(A). For the 1st band (whose upper border might be irregular), that width is measured at the middle of the band. For any other band whose north border includes part of the state's border, the width measurement is made at the middle of the band.
Starting at the west end of the band, mark off rectangles in that band which contain population equal to the desired population per district. Those are districts. At the extreme east end of the band will be a district (not necessarily a rectangle, because the east border can be irregular) which has less than the desired population.
If that smaller eastmost district has more than half of the desired population, then move the lower border south till the band contains population for an integer number of districts. Repeat the procedure of dividing the band into rectangles each of which contains the desired district population.
If the smaller eastmost district has less than half of the desired population per district, then move the band's lower border north till the band contains population for an integer number of districts. Repeat the procedure of dividing the band into rectangles each of which contains the desired district population.
Mike Ossipoff
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