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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.
March 16, 2020 at 12:52am March 16, 2020 at 12:52am
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Past the halfway point, entry 5 of 8 for
It's been a while since I listened to The Planets suite -- high school, probably.
The Great Oracle
So I consulted Wikipedia  . Apparently, and I don't know if I knew this or not, the music was inspired not by planetary science but by astrology.
An Aside
Ever been in a cathedral? Or a major mosque, or some other religious edifice, built with the intent to awe?
Humans have, in our short time on the planet, shown a great capacity for telling ourselves stories based on dubious ideas. Sometimes these ideas take root and grow, and people do stuff based on them, and sometimes the stuff we do has real, lasting emotional effects. Such is the case with cathedrals, and such is the case with certain kinds of music.
These constructs are often meant to reinforce said beliefs, and for some, that's the case. A believer might find herself, in a cathedral, closer to God; take that same believer to a Shinto shrine and all they see is the artistry. Or possibly vice-versa. Point being that we don't always align with the motivations of the creators of these structures, but we can usually appreciate the end result. There aren't any authentic followers of the ancient Egyptian religion around anymore (though some try), and yet many of us are fascinated by the Pyramids.
When it comes to art -- painting, architecture, sculpture, music, writing, whatever -- I can appreciate the end result even if I'm not on board with the motivations behind it. When Notre Dame de Paris burned, for example, I felt a great loss, even though I'm not French, or Catholic, or even religious in any conventional way. People built that thing, and it was art by any definition of the word, and its loss transcended any particular nationality, philosophy or religion. We can be inspired to do great things by made-up fairy tales, which to me only illustrates the human capacity for doing great things.
Consequently, even though I believe I have used the term "bullshit" to describe astrology in here before, I can appreciate The Planets as art, regardless of the compositions' provenance, just like I appreciate the sweeping arches and vaulted ceilings of a Gothic cathedral.
That Said...
Sometimes I have trouble interpreting art. I've gotten better at it, I think, but a lot of its subtlety is entirely lost on me. Taking the music linked above, for example, imagine listening to it for the first time without any background. What is it supposed to do? What feelings does it evoke? Do these feelings change when you know the title of the composition? What about when it's placed in the rest of the orchestral suite? How about knowing the bit about it being inspired by astrology?
I'm going to take a minute to compare this with beer, so you'll have to bear with me here. Or not; your choice, but I am doing this to make a point.
I think I'm fairly knowledgeable about beer. Not how to make it; that's too much work, and I lack the artistry to brew beer properly (or to paint, or build a cathedral, or compose a piece of music). But I have a decent working knowledge of the various styles and categories of beer.
Periodically, someone will arrange a blind taste-test, just sampling different beers without any prior knowledge of brewer, style, relative quantities of each ingredient, any adjuncts included, etc. I don't do many of these, myself, because beer isn't just about aroma and flavor; it's about the entire experience. My enjoyment (or lack thereof) of a particular brew does include what I'll call metaknowledge: what brewery it came from, what the label looks like, its name, whether I'm alone or with friends, the style it's trying to be, which hops are used, the ethanol content, and dozens of other data points. A blind tasting limits this to: color, head, aroma, flavor. And it's fine if you're the kind of person who wants a more objective experience; hell, I'd be the worst kind of hypocrite if I support scientific endeavors without allowing for a scientific beer-tasting experience.
Beer is art, much like music or architecture, and some background of the art is essential, for me, to its appreciation.
I should point out that my views on this have evolved. I have an aunt who was married to a professional artist, and we've argued about this before. Her claim was that an education in art history is important to appreciating the result; I argued that pieces of art (we were discussing, in particular, painting and sculpture) should stand on their own, and be appreciated, or not, for what they are.
I've gradually come around -- mostly -- to her point of view. Don't tell her; she's old now and her heart probably couldn't take the strain of having me finally agreeing with her on anything.
You Said You Had A Point
Yeah, I did, didn't I? I'm not sure I was entirely truthful there.
Back up there, I linked the wiki page for The Planets. In it, the composer, Holst, is paraphrased thus: "Although The Planets remains Holst's most popular work, the composer himself did not count it among his best creations and later in life complained that its popularity had completely surpassed his other works."
Which reminded me of Radiohead, who are also from England. They grew tired of their song Creep and its popularity, and for a long time refused to perform it. Which was too bad, because their other songs kinda suck. I don't have any opinion about Holst's other work, because I'm not aware of it. Could be I've heard it. Probably not. Sometimes you're the victim of your own popularity, and you can't really control which of your stuff people are going to like.
Once you put art out there, it's largely out of your control. Oh, you can defend copyrights and whatnot, but there's no accounting for taste. Art exists, independent of inspiration or source, and people are going to make of it what they will.
As for me, I think I'll crack open a cold one and listen to Jupiter again. |
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