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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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October 31, 2021 at 12:03am
October 31, 2021 at 12:03am
#1020458
This will probably be the last one of these for a while, as I'm going back to doing "30-Day Blogging Challenge ON HIATUSOpen in new Window. [13+] next month. Which is tomorrow. Yikes.

How Cats Became Divine Symbols in Ancient Egypt  Open in new Window.
Felines served a useful purpose in ancient Egyptian households and were eventually associated with deities.


There's an old joke going around that goes something like:

Dogs: "These humans give us food and water and take care of us. They must be gods!"
Cats: "These humans give us food and water and take care of us. We must be gods!"

Alternatively, there's the quote I've heard attributed to Winston Churchill, which goes something like, "Dogs look up at us, cats look down on us, pigs treat us as equals."

“Though it is hard to say the Egyptians thought one thing or another, since so much change happened across their 3,000+ years of history, the ancient Egyptians, in general, did not worship animals,” says Julia Troche, an Egyptologist, assistant professor of history at Missouri State University, and author of Death, Power, and Apotheosis in Ancient Egypt: The Old and Middle Kingdoms. “Rather, [they] saw animals as representations of divine aspects of their gods.”

The distinction can be subtle, I suppose. If an archaeologist 4000 years from now dug up a typical home, they'd probably conclude that we worship the television, which is silly because the reality is that we worship the celebrities that appear on it.

Whether or not they were worshiped as deities, cats were an integral part of ancient Egyptian life. And, based on mummified cats discovered in tombs alongside humans, they carried an important role in the afterlife, as well.

Just yesterday, I saw that archaeologists uncovered a grave in Cyprus (which is kind of but not really near Egypt) from something like 10,000 years ago where some dude was buried with a bunch of his favorite toys... and his cat. The idea being that they sent the cat with him to whatever afterlife they imagined. Poor kitty was only eight months old, though. The cat didn't deserve that.

For most of the civilization’s history, ancient Egyptians saw cats as mutually beneficial companions, according to Troche. “Cats might come inside when it was hot, and they in turn would chase away dangerous animals, such as snakes—many of which were venomous—and scorpions,” she explains.

Most articles I've seen about cats emphasize their usefulness. As if that has to be a factor. How useful is the Mona Lisa? Cats are living art. Art that pukes on the couch sometimes, but still art.

The presence of cats in tombs wasn’t limited to paintings—sometimes cats were mummified and placed inside their human companion’s tomb, according to the Carnegie Museum of Natural History. One reason this was done is that when cats were mummified, they could then be used as funerary goods.

Again, the implication is that they killed the poor kitty just because its human died.

“Ancient Egyptians held cats in such high regard because of the practices and preferences of their gods, but also because their kings, the pharaohs, kept giant cats,” Skidmore says. “Members of the Egyptian royal class dressed their cats in gold and let them eat from their plates.”

And cats have never forgotten this.

In addition to appreciating their ability to keep rodents, snakes and other pests out of their homes, the ancient Egyptians understood that cats of all sizes are smart, quick and powerful.

Ah, no wonder the cats allowed that civilization to prosper for as long as it did.

Through their ubiquitous presence in the art, fashion and home ornamentation of ancient Egypt, cats served as an everyday reminder of the power of the gods.

Sure, if gods had the tendency to get underfoot and pee in your laundry basket.

Come to think of it, though, cats are more useful than gods anyway. At least you can laugh at their antics. Try that with a god and you get smited.


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