State borders are not as simple as they look

[November 1st, 2009]

When I was a kid, I would look at maps of the American West and be fascinated by those enormous states. Wyoming! Utah! Colorado! They looked like giant mathematical figures, crisply defined in an otherwise chaotic world. I also thought that the borders really were a dotted line on the ground, but that misconception was quickly disspelled by my parents.

<p/>

Nevertheless, I always carried in my mind the idea that western states were mostly defined by straight lines of whole latitude and longitude (parallels and meridians).

<p/>

So it was with some surprise that, while working on a <a href=”http://www.integrity-logic.com”>GIS app for the iPhone</a>, I discovered that these cherished beliefs were in fact plain wrong.

<p/>

The first misconception was that these straight lines run mostly along whole meridians and parallels. It certainly looks that way at first sight. For instance, the border between Arizona and New Mexico really does look like it’s smack on the 109th west meridian. But when you zoom in, you see that it’s actually off, and by more than a rounding error. What happened? Did the surveyors of yore goof? As a <a href=”http://www.amerisurv.com/content/view/6057/”>fascinating article in The American Surveyor</a> reveals:

<p/>

“In a case of national pride edging out common sense, the dividing line was defined as the 32nd meridian west of Washington”

<p/>

which turns out to be almost three minutes of arc west of the 109th meridian, or roughly three miles. Goodness forbid we should keep things simple. At least we can blame this one on the politicians.

<p/>

OK, so the border between Arizona and New Mexico is a bit strange. But that’s the exception, right?

<p/>

Well, no. The border between Colorado and New Mexico (and a bunch of other states) was originally defined as the 37th parallel. But man proposes, and the surveyor disposes. As it turns out, the surveyors did make a small mistake of 365 feet — not bad at all for late-19th century techniques, but enough to create a problem now. So, as described in <a href=”http://www.waymarking.com/waymarks/WM72RM_Latitude_37_North_the_border_between_six_US_States_Four_Corners”>Waymarking</a>:

<p/>

“Instead of a costly reassessment of the border line, the states affected and the US Supreme Court amended the definition of the state’s border, so now, the border between Utah, Colorado and Kansas in the North and Arizona, New Mexico and Oklahoma in the South is no longer at 37 N but at 36 59’56.34”N. “

<p/>

My belief in the cosmic order thus shaken, you can imagine how perturbed I was when I realized that, not only are these lines not exactly on whole meridians and parallels, but in fact they’re not even straight at all.

<p/>

Colorado, for instance, looks like it’s a perfect rectangle (if we ignore the spheroidal surface of the earth, of course). But if you examine closely the border with Utah, you’ll notice that there are <a href=”http://maps.google.com/?ie=UTF8&ll=38.155617,-109.008408&spn=0.478381,0.632401&z=11″>a few squiggles</a>. Small to be sure, but it is not a straight line. Same thing for the <a href=”http://maps.google.com/?ie=UTF8&ll=40.997537,-108.629014&spn=0.014349,0.019763&z=16″>border with Wyoming</a>. All these slight deviations from the geometrically ideal go back to the 19th century when surveying, although remarkably precise considering the technology available at the time, was not what it is today.

<p/>

In the case of the Colorado/Utah border, it was a <a href=”http://geology.utah.gov/surveynotes/gladasked/gladkink.htm”>surveying oopsie</a> of just over a mile. Still not bad for 1879. The border between California and Nevada was defined as a straight line from what turned out to be the middle of lake Tahoe to the intersection with the Colorado river. But wouldn’t you know it, the river moved, and the surveyor, rather than start from scratch, fudged the line. Ah, contractors!

<p/>

In fact, the closer you look, the worse it gets. All these borders are ultimately defined by monuments which were erected by surveying teams. As Ivars Peterson says in his blog, <a href=”http://mathtourist.blogspot.com/2007/08/rectangular-states-and-kinky-borders.html”>The Mathematical Tourist</a>:

<p/>

“once a border is defined on the ground and accepted by the interested parties, it becomes official, even if it doesn’t follow the written description.”

<p/>

So it’s not just that there are a few errors here and there — these seemingly-rectilinear states are actually polygons with hundreds of side!

<p/>

These vagaries can sometimes have serious consequences. A mini-war was actually fought between California and Nevada over this. Judges were thrown in jail, guns were shot in anger, and blood was shed during the <a href=”http://www.nevadacas.com/rccsawar.htm”>”Sagebrush War”</a> (more of a skirmish, really) in 1863, fought over which state Susanville belonged to. Maybe they felt like they were missing out on the Civil War and needed some excitement of their own. At one point, Susanville was the seat of two counties, one in Nevada, the other in California. They even got to elect a representative in both states.

<p/>

This all seems very quaint and amusing in this era of GPS and laser ranges, so it’s hard to believe that it’s not yet over. As recently as 1980, there was a <a href=”http://www.altlaw.org/v1/cases/384566″>lawsuit between the states of California and Nevada</a> regarding the exact location of the border. That is a long-running feud between the two states, mostly because the line has been <a href=”http://www.waymarking.com/waymarks/WM2X8Z_California_Arizona_and_Nevada”>redefined so many times</a>. And even today, in 2009, there are <a href=”http://www.nobleco.org/surveyor/index.php?q=all-in-the-meridian”>legal maneuverings between the U.S. and Canada</a> over the exact location of the Alaska border. It looks like the surveyors were a few hundred feet off (and, really, would could blame them?). A few hundred feet doesn’t sound like much until you multiply it by 647 miles. Add some oil and gas, and you can keep a sizeable herd of lawyers fat and happy for a long time.

<p/>

So this is what I’ve learned: borders are not on neat meridians and parallels, they’re not straight lines, and sometimes we’re not even completely sure where they are. Apparently, the world is not as neat and orderly as I imagined when I was a child.

When I was a kid, I would look at maps of the American West and be fascinated by those enormous states. Wyoming! Utah! Colorado! They looked like giant mathematical figures, crisply defined in an otherwise chaotic world. I also thought that the borders really were a dotted line on the ground, but that misconception was quickly dispelled by my parents.

Nevertheless, I always carried in my mind the idea that western states were mostly defined by straight lines of whole latitude and longitude (parallels and meridians).
So it was with some surprise that, while working on a GIS app for the iPhone (now defunct), I discovered that these cherished beliefs were in fact plain wrong.

The first misconception was that these straight lines run mostly along whole meridians and parallels. It certainly looks that way at first sight. For instance, the border between Arizona and New Mexico really does look like it’s smack on the 109th west meridian. But when you zoom in, you see that it’s actually off, and by more than a rounding error. What happened? Did the surveyors of yore goof? As a fascinating article in The American Surveyor reveals:

In a case of national pride edging out common sense, the dividing line was defined as the 32nd meridian west of Washington

which turns out to be almost three minutes of arc west of the 109th meridian, or roughly three miles. Goodness forbid we should keep things simple. At least we can blame this one on the politicians.

OK, so the border between Arizona and New Mexico is a bit strange. But that’s the exception, right?

Well, no. The border between Colorado and New Mexico (and a bunch of other states) was originally defined as the 37th parallel. But man proposes, and the surveyor disposes. As it turns out, the surveyors did make a small mistake of 365 feet — not bad at all for late-19th century techniques, but enough to create a problem now. So, as described in Waymarking:

Instead of a costly reassessment of the border line, the states affected and the US Supreme Court amended the definition of the state’s border, so now, the border between Utah, Colorado and Kansas in the North and Arizona, New Mexico and Oklahoma in the South is no longer at 37 N but at 36 59’56.34”N.

My belief in the cosmic order thus shaken, you can imagine how perturbed I was when I realized that, not only are these lines not exactly on whole meridians and parallels, but in fact they’re not even straight at all.

Colorado, for instance, looks like it’s a perfect rectangle (if we ignore the spheroidal surface of the earth, of course). But if you examine closely the border with Utah, you’ll notice that there are a few squiggles. Small to be sure, but it is not a straight line. Same thing for the border with Wyoming. All these slight deviations from the geometrically ideal go back to the 19th century when surveying, although remarkably precise considering the technology available at the time, was not what it is today.

In the case of the Colorado/Utah border, it was a surveying oopsie of just over a mile. Still not bad for 1879. The border between California and Nevada was defined as a straight line from what turned out to be the middle of lake Tahoe to the intersection with the Colorado river. But wouldn’t you know it, the river moved, and the surveyor, rather than start from scratch, fudged the line. That’s contractors for you.

In fact, the closer you look, the worse it gets. All these borders are ultimately defined by monuments which were erected by surveying teams. As Ivars Peterson says in his blog, The Mathematical Tourist:

once a border is defined on the ground and accepted by the interested parties, it becomes official, even if it doesn’t follow the written description.

So it’s not just that there are a few errors here and there — these seemingly-rectilinear states are actually polygons with hundreds of side!

These vagaries can sometimes have serious consequences. A mini-war was actually fought between California and Nevada over this. Judges were thrown in jail, guns were shot in anger, and blood was shed during the “Sagebrush War” (more of a skirmish, really) in 1863, fought over which state Susanville belonged to. Maybe they felt like they were missing out on the Civil War and needed some excitement of their own. At one point, Susanville was the seat of two counties, one in Nevada, the other in California. They even got to elect a representative in both states.

This all seems very quaint and amusing in this era of GPS and laser ranges, so it’s hard to believe that it’s not yet over. As recently as 1980, there was a lawsuit between the states of California and Nevada regarding the exact location of the border. That is a long-running feud between the two states, mostly because the line has been redefined so many times. And even today, in 2009, there are legal maneuverings between the U.S. and Canada over the exact location of the Alaska border. It looks like the surveyors were a few hundred feet off (and, really, would could blame them?). A few hundred feet doesn’t sound like much until you multiply it by 647 miles. Add some oil and gas, and you can keep a sizable herd of lawyers fat and happy for a long time.

So this is what I’ve learned: borders are not on neat meridians and parallels, they’re not straight lines, and sometimes we’re not even completely sure where they are. Apparently, the world is not as neat and orderly as I imagined when I was a child.