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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.
November 3, 2023 at 11:29am November 3, 2023 at 11:29am
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No booze-related observances today, so let's dig into the slush pile. From MIT Press Reader:
"Surprisingly?" Not for people who read.
To feel small, all we have to do is look up.
Actually, it makes me feel large, because I live amongst a species that has managed to begin to comprehend that vastness.
The sun, the Moon, the stars, the planets, and the Milky Way are evidence enough that Earth is not all that is.
It wasn't very compelling evidence to some people. They honestly believed that it was all part of, connected to, and/or made for the sole benefit of the Earth.
And for as long as humans have had words, we have been sharing stories about the presumed builders and occupiers of those vaulted heavens: the gods, spirits, angels, and demons who were, in a sense, the first extraterrestrials.
Speculation about extraterrestrials has long seemed to me to be the tech-age version of speculation about gods and angels.
According to a Cherokee story, for example, the Milky Way is a great web spun across the sky by Grandmother Spider, who used it to reach the other side of the world and bring back the sun.
Presumably, they exiled arachnophobes.
In one grisly Aztec myth, the war god Huitzilopochtli sprang from his mother Coatlicue’s womb fully grown and fully armored. He beheaded his sister, Coyolxauhqui, who had been plotting to kill Coatlicue, and cast her head into the sky, creating the Moon.
The Aztec were hardcore.
Materialist interpretations of the cosmos eventually began to take the place of mythological ones. But the idea that there might be other beings in the sky has stayed with us, and it found its first protoscientific roots in Greece in the sixth century BCE.
I suppose that the article is making a distinction between supposed supernatural beings that reside in some "up" place (heaven, e.g.) and speculation about some sort of natural or created beings on other worlds.
Anaximander, a philosopher who lived in Miletus in modern-day Turkey, contributed one key idea. He was the first to propose that Earth is a body floating in an infinite void, held up by nothing.
I'm not able to find, definitively, what idea he replaced with this one, but earlier, another Greek dude, Thales, figured Earth was a giant island floating in an infinite ocean of water. Before that? I don't know. Elephants and turtles holding it up or something.
The philosopher Karl Popper called it “one of the boldest, most revolutionary, and most portentous ideas in the whole history of human thought.”
People can be right purely by accident, and this seems to be one of those cases. As the article notes, Anaximander was also wrong about a lot of things. How do we know? Science.
In the fifth century BCE, the Thracian philosopher Leucippus and his pupil Democritus invented atomism: the belief that the visible universe consists of tiny, indivisible, indestructible atoms, churning in the void without purpose or cause. In this picture, worlds aren’t divinely created; they simply form when enough atoms collide and stick together.
And scientists were premature in calling atoms "atoms," as it turned out that they weren't indivisible or indestructible, as has been repeatedly and terrifyingly demonstrated.
Both Plato (428–348 BCE) and Aristotle (384–322 BCE) lambasted Democritus’s idea of a plurality of worlds on theological grounds.
And so we come to the philosophical tension that has dominated the world ever since: some natural philosopher or scientist proposes something based on observation, and religious folk react with horror and torches.
As Christianity swept across the decaying Roman Empire in the third and fourth centuries CE, the Church Fathers ridiculed and suppressed the Epicureans and their ideas and allowed their writings to burn or crumble. Atomism, the pursuit of pleasure, the plurality-of-worlds idea — all of it slipped into darkness, where, as Wiker observes, “it stayed for nearly a thousand years.”
Such as that.
But after those dark ages—which weren't as dark as all that, but still held back scientific inquiry in the West—the article goes into the start of the Renaissance and humanist philosophy.
Copernicus is central to the story of extraterrestrials not because he believed in them — the question didn’t seem to interest him — but because he was the first person to propose, based on observation and calculation, that Earth was not the center of the visible universe.
Oddly enough, it turns out that Earth is the center of the visible universe... if you're standing on Earth. If you were standing on a planet orbiting some unnamed star in the outer reaches of the Andromeda galaxy, that would be the center of the universe.
This premise — that there’s nothing particularly special about Earth and that we aren’t in a privileged, central position to observe the universe — would come to be known as the Copernican principle, and it’s at the core of the modern-day case for doing research related to the search for extraterrestrial life (SETI).
I've harped on this in here multiple times, but just as it's arrogant to think we're the only sentient life in the universe, it's also arrogant to think that life on other worlds necessarily produces sentience, or, furthermore, looks anything like Earth life.
Copernicus knew his theory would provoke religious objections, which may be why he declined to publish it during his lifetime. His follower Giordano Bruno was not so cautious.
And as far as I know, the Catholic Church never did issue a proper apology for murdering him.
From Democritus to Galileo, thinkers treated the idea that other worlds might be home to alien beings — the word alien comes from the Latin term alius, “other” — with great seriousness. After all, believing in aliens could get you banished or burned at the stake.
It is certainly not on religious grounds that I'm a skeptic when it comes to sentient aliens (see, well, a whole bunch of earlier blog entries). I haven't encountered any positive evidence for them. I enjoy Star Trek as much as anyone and more than most, but I doubt we'll find a universe populated by tech-using aliens who play out our own foibles in metaphor.
But in 1686 a Frenchman named Bernard le Bovier de Fontenelle became the first writer to exploit the subject’s humorous possibilities.
Just to be clear, his writing was humor, fantasy, and probably satire, but not science fiction.
The people of Venus, Fontenelle mused, are “sunburnt, full of verve and fire, always amorous, loving verses, loving music, inventing celebrations, dances, and tournaments every day.” The inhabitants of Saturn, by contrast, are “quite phlegmatic. … These are people who don’t know what it is to laugh, who always take a day to answer the slightest question asked them.”
Just wild-ass guessing here, but I'm pretty sure Venus there was France, and Saturn was England.
Whewell pointed out that humans, according to the geological record then being unearthed, had been present on this planet for only an “atom of time.” If Earth had been, in effect, uninhabited through most of its history, then it wouldn’t be surprising if other distant planets were also empty. In any case, he pointed out, no planets around other stars had yet been observed, and many nebulae, star clusters, and multiple-star systems would be unsuitable places for them.
I hate, I mean absolutely hate, the fact that I generally agree with that guy, who was one of those who objected to the idea of sentient alien life on theological grounds.
Copernicus was correct to revoke Earth’s privileges as the pivot point of the universe, but that insight by itself says nothing about what else might exist in the universe.
Exactly. "We're not special" doesn't necessarily mean "We're not alone."
Anyway, I've banged on long enough. The article is relatively short, considering that it covers the highlights of over 2000 years of speculation about other worlds and alien life, and it's definitely worth a read, wherever you fall on the "aliens do(n't) exist" spectrum.
Me? I'm still waiting for real evidence. But that's not going to stop me from enjoying science fiction and fantasy. |
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