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Complex Numbers
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.
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I wanted to share this article, not just because the story isn't that widely known, but because of a tenuous personal connection to it.
The vessel floated in the shallows of the Potomac River on the leeward side of Chopawamsic Island, just off Quantico, Virginia.
It's fairly common knowledge that Maryland claims the Potomac. In contrast with standard surveying practice, which usually puts boundaries in the river's thread or main channel, Maryland gets the whole expanse, up to some defined line (I think it's the mean low tide line where it is tidal, which is all the way up past DC) on the opposite side.
This is in spite of the fact that the river is named for an Indian tribe that lived, to the best of my knowledge, exclusively in what is now Virginia. The reason for this isn't relevant to this story, though, except to note that because of this historical oddity, there just aren't that many islands in the Potomac that are part of Virginia. And this one is also claimed by the county where I (arguably) grew up, exactly (I checked) 10 miles from my childhood home.
They had to do some creative surveying to include it within Stafford County, but it's there, in a kind of pseudopod-like extrusion.
So, that's the personal connection. I told you it was tenuous.
History would be made that day, May 6, 1896, as this apparatus—a flying machine, known as Aerodrome No. 5—was started and then launched from a spring-loaded catapult.
An "aerodrome" is what we (well, mostly the British) call what's basically a small airport today. In 1896, no airports existed, so I guess the word meant something different.
The Aerodrome would take off and travel for 90 seconds some 3,300 feet in an effortless spiral trajectory and then gently land in the river.
Both time and distance exceed my best efforts at paper airplanes when I was a kid. To be fair, those experiments usually ended at a classroom wall.
The third Secretary of the Smithsonian Institution, Samuel Pierpont Langley, an astronomer who also enjoyed tinkering with his own creations, was aboard the boat. His winged invention had just made the world’s first successful flight of an unpiloted, engine-driven, heavier-than-air craft of substantial size.
Yeah... those are some specific definitions. Not quite the level of success the later Wright Brothers enjoyed, but why let North Carolina get all the credit?
As for Langley himself, yes, it was that Langley, the one the famous Air Force base is named for. He was also a Secretary at the Smithsonian (a position similar to Royal Astronomer of Britain), so of course this article is from Smithsonian.
With Langley that day, was his friend Alexander Graham Bell, the inventor of the telephone...
Another dubious claim, but that's what history insists upon, though that's really irrelevant right now.
It should be emphasized again that this flight wasn't quite as significant as the Wright one, and the article does so:
The world rightly remembers that in 1903 the Wright brothers achieved human flight at Kitty Hawk in North Carolina. “Langley’s Aerodrome No. 5 wasn’t practical and it wasn’t a working prototype for any real flying machine,” says Peter Jakab, senior curator at the museum. But the largely forgotten unpiloted flight that took place seven years before Kitty Hawk did move motorized flight from the drawing board into reality.
Langley was also apparently one of the last great polymaths:
“Langley’s real accomplishments in research were in astronomy,” says Jakab. “He had done a great deal of significant work in sun spots and solar research, some of that while at the Smithsonian.”
He had a bit of mad scientist about him, apparently:
“This was still a period when people didn’t think flight was possible,” Jakab says. “If you were a young person in the 1890s contemplating a career in engineering, flight was not exactly an area you would go into. It wasn’t taken seriously by a lot of people. The fact that someone like Langley was starting to study flight gave the field credibility.”
The article goes on to describe his fundamental errors in design, and how he lost out to those brothers from Ohio a few years later.
“Those two catastrophic failures in 1903 ended Langley’s aeronautical work,” Jakab says. “He was a broken man because he took a lot of ridicule. He spent a lot of money and did not achieve a great deal in this field.”
Perhaps not, but we also learn from failures. And by all accounts, he accomplished great things... just not so much in aviation. Still, he did what few others even dared to try, and isn't that worth remembering? |
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