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Complex Numbers A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number. The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi. Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary. Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty. ![Merit Badge in Quill Award
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| Colorado is known for, well, several things, but mainly: mountains, weed, beer, and being a rectangle. Colorado Is Not a Rectangle—It Has 697 Sides  The Centennial State is technically a hexahectaenneacontakaiheptagon. Oh, well, at least they've still got mountains, weed, and beer. America loves its straight-line borders. The only U.S. state without one is Hawaii—for obvious reasons. There are good reasons for that, mostly involving making sure surveyors don't just give up in the middle of marking a zigzag boundary. West of the Mississippi, states are bigger, emptier, and boxier than back east. From a distance, all seem to be made up of straight lines. Can't be arsed to look it up right now, but there was a shift in the way surveying was done between the time of East Coast European settlement, and massive migrations west. That's one reason you get a lot of near-rectangular counties out West, and almost none in the East. Only when you zoom in do you see their squiggly bits: the northeast corner of Kansas, for instance. Or Montana’s western border with Idaho that looks like a human face. Never noticed that before, and now I will never not notice it. New Mexico comes tantalizingly close to having only straight-line borders. There’s that short stretch north of El Paso that would have been just 15 miles (24 kilometers) long if it were straight instead of wavy. Just guessing here, but it looks like it's wavy because it followed the Rio Grande. I use the past tense because, looking at a map, it looks like the river shifted but the boundary didn't (whether the river shift was intentional or not, I have no idea). There are great benefits in using rivers and other geological features as boundaries... at least until you remember that geological features change, and rivers in particular can rechannel themselves on human time scales. No, there are only three states whose borders are entirely made up of straight lines: Utah, which would have been a rectangle if Wyoming hadn’t bitten a chunk out of its northeastern corner; Wyoming itself; and Colorado. I've long been curious over why Wyoming bit Utah rather than the other way around, or maybe settled on a diagonal compromise. But not curious enough to delve deeper. I'm also going to quibble a bit. With a few exceptions, most boundaries are described by line segments. The exceptions include Delaware's northern boundary, but even the ones following the thalweg—that's Surveyese for the midline or channel of a river—are generally approximated by arbitrarily short line segments (putting survey monuments in a river channel tends to be cost- and labor-intensive). Whether we perceive something as a whole lot of short line segments, or a squiggle, or an arc, depends on the zoom factor. Except that they aren’t. for two distinct reasons: because the earth is round, and because those 19th-century surveyors laying out state borders made mistakes. And that's the other thing. On a Mercator and certain other map projections, latitudes appear as straight, horizontal lines. They are not. They are all circles of varying radius, centered on the poles. So when you're surveying a latitude line, you're actually describing an enormous arc (unless of course you're on the Equator). Said arc is generally approximated with line segments, as the variation from that big an arc to a line tends to be tiny. Many states and countries have latitude lines as boundaries. Perhaps the most famous is the western segment of the border between the US and Canada. And even Eastern states (theoretically) have latitude boundaries, such as most of the line between VA and NC. One begins to see why some of the US "Founding Fathers" were also surveyors. Congress defined the borders of Colorado as a geospherical rectangle, stretching from 37°N to 41°N latitude, and from 25°W to 32°W longitude. While lines of latitude run in parallel circles that don’t meet, lines of longitude converge at the poles. In contrast to latitude, longitude lines are, actually, straight (as mapped onto the ground). In theory. Those longitude numbers seem like errors. I feel like they're measured from DC and not Greenwich, because there was a time when the US tried to measure everything from a longitude line passing through Washington, DC, and things like that tend to get entrenched into surveys. I couldn't find confirmation of this, but later in the article it acknowledges that the western boundary is more like 109°02’48″W, which supports my hypothesis. This means that Colorado’s longitudinal borders are slightly farther apart in the south. So if you’d look closely enough, the state resembles an isosceles trapezoid rather than a rectangle. Consequently, the state’s northern borderline is about 22 miles (35 kilometers) shorter than its southern one. I'd love to see the flat-Earther explanation for that one. That’s not where the story ends. There’s boundary delimitation: the theoretical description of a border, as described above. But what’s more relevant is boundary demarcation: surveying and marking out the border on the ground. A friend once asked me whether the VA/NC boundary would shift with continental movements, to keep it roughly aligned with whatever latitude it's supposed to follow. No, it wouldn't; the boundary markers take precedence over the delimitation. Kind of like the NM/TX border near, but no longer on, the Rio Grande. Unfortunately, 19th-century surveyors lacked satellites and other high-precision measurement tools. I say they did remarkably well with what they had. Humans are clever when they want to be. Let’s not be too harsh: considering the size of the task and the limitation of their tools—magnetic compasses and metal chains—they did an incredible job. They had to stake straight lines irrespective of terrain, often through inhospitable land. I guess that's the polite way of saying they had to deal with mountains, Indians, and mountain Indians. Whether they should have been messing around on land that didn't really belong to them is another issue for another time. The fact remains that they did mess around. Located in a dusty, desolate corner of the desert, the Four Corners monument seems very far from the middle of anything. Yet this is the meeting point of four states: Utah, Colorado, New Mexico and Arizona. It is the only quadripoint in the United States. The monument’s exact location is at 36°59’56″N, 109°02’43″W. It's not all that desolate. And yes, I've been there. Twice. Sure, it's a tourist trap, but I'm a tourist, and spent my career working with surveyors. However, it’s not where Congress had decreed the four states to meet. That point is about 560 feet (170 meters) northwest of the quadripoint’s current location, at 37°N, 109°02’48″W. Did you drive all the way through the desert to miss the actual point by a few hundred feet? Look, it's a nice drive from any direction. The rest of the article goes into some of the more obvious deviations from straight-line surveying (though still doesn't much acknowledge that latitude lines aren't "straight.") It's worth a read if you find this sort of thing interesting. |
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