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Saturday, Dec. 21, 2024

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Complex Numbers

by Waltz Invictus

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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.

http://www.ddewey.net/mandelbrot/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Annette

about 10 hours ago
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

a day ago
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

a day ago
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

🌖 HuntersMoon

Dec 19, 2024 at 8:53am
In response to "Five Seconds"

2 Like

Aiva Raine

Dec 17, 2024 at 7:39pm
In response to "Need for Speed"

Even with the actual explanation- it still left me going- sooooo what's the difference between the special expensive "speed tape" they use for "temporary fixes" and duct tape? LOL- sure sounds like duct tape to me and they are using it for exactly the same reasons you would use duct tape. I'm guessing that the only difference between it and duct tape is the "speed tape" is slightly stronger and handles temperature differences better. It still comes down holding the plane together with tape which doesn't exactly fill me with confidence. Especially if it's a Boeing plane.

1 Like

tj-Merry Mischief Maker

Dec 17, 2024 at 6:14pm
In response to "Need for Speed"

I suppose they call it speed tape to keep people calm. If they called it what it is, temporary repair tape, they may lose passengers.

2 Like

Waltz Invictus

Dec 17, 2024 at 9:43am
In response to "Need for Speed"

Ichabod Crane-- Mince Pie

, I've heard that speed tape is extremely expensive. I can only assume the maintenance guys at Delta got a five-finger discount. But if you're going to steal from work, might as well go big and not just lift a box of paper clips.

1 Like

June 19, 2024 at 11:40am

#1072895

Matters of the Heart

Way back before the solar eclipse, I saved this Atlas Obscura article to my queue. But the topic itself is timeless, so there wasn't any rush, and the Random Number Gods saw fit to wait until today to bring it forth.

‘Total Eclipse of the Heart’ Was Almost a Meat Loaf Song
Bonnie Tyler’s classic rock epic nearly got the ‘Bat Out of Hell’ treatment.

Hardly the usual AO fare; they usually concentrate on place oddities (hence the name) or food. They were doing special "eclipse" articles, and this was one of them. Everything here tracks with stuff I already knew, but I didn't fact-check it further.

When Bonnie Tyler’s husky, powerful “Total Eclipse of the Heart” hit the radio in 1983, rock ballad fans around the world fell in love.

Well, some did. Some hated it. Still do. I can always tell if someone is worth hanging out with based on their feelings about that song alone.

But for one musician, the history of the tune, which was written by composer Jim Steinman, is as torn as the dark love ballad’s lyrics. Meat Loaf was the one who was supposed to have sung that song.

Confession time: I possess, and have read, a copy of Meat Loaf's autobiography (unsigned, unfortunately). But I can't remember whether I knew the above from that or through other sources over the years. But the way I see it, ML was famous anyway, still one of the best-selling performers in history; and it boosted Bonnie Tyler, so no real losers here. As the article notes later, ML didn't seem to hold any of this against Tyler.

Meat Loaf had long worked with songwriter Jim Steinman, including on his hit 1977 album Bat Out of Hell.

What I did get from the book, I think, was that that album was supposed to be a collaboration between singer and songwriter, not purely Meat Loaf with the writing credit added as an afterthought. (Many of the session musicians on it were also members of the E Street Band, adding to its awesomeness.)

Meat Loaf’s next planned album, the never-made Renegade Angel was poised for recording, and was supposed to include the track “Total Eclipse of the Heart.”

There are at least two things in music history that almost happened that I wish had happened, and that's one of them. The other also involves Jim Steinman—he was meant to collaborate with Andrew Lloyd Webber on a stage production. Just as well that the latter didn't happen; Broadway would have been sucked into a black hole of melodrama.

Amid mysterious and abrupt problems with his voice and a psychologically damaging aversion to fame, Meat Loaf’s success and partnership with Steinman began to slowly unravel, forever changing the sound and trajectory of the ballad “Total Eclipse of the Heart.”

It changed a lot of trajectories, really. And that sentence glosses over a lot of drama. The article dips into this a bit, then:

Meat Loaf, under contractual obligation, went ahead with his next record, Midnight at the Lost and Found, a collection of songs written by various songwriters, which Meat Loaf later came to regret. (Steinman meanwhile released songs that he initially thought up for Meat Loaf on his own solo album, Bad for Good.)

I bought that Steinman album on LP when it came out. I also acquired the CD several years later. I like it. It's not for everyone. Steinman's voice just didn't work in the same way Meat Loaf's did. Most of the tracks on it later showed up on different Meat Loaf albums, after they reconciled, and the difference is plain if you hear both versions of a song.

But that's kind of a sidebar; the point is, Steinman was a prolific songwriter, but lacked the performance talents needed to turn his ideas into popular music. For that, he needed other talent. Preferably people who could pull off melodrama, like Meat Loaf... or Bonnie Tyler.

This Rolling Stone article

has more on that.

So back to the actual topic of the article (and this entry):

The song’s music video, which featured prep school boys with glowing eyes, gymnasts, dancing men in leather jackets, and lots of floating fabric, also spawned a Literal Video spoof in the early 2000s that highlights the super weirdness of the visuals.

I've posted that Literal Video in here before, so I don't want to do it again. It's still up on YouTube, attesting to not only the power of the original song and video, but to the wonder of parody. Yes, I can like an original and also appreciate the parody.

Despite the song's title, though, it wasn't really part of my eclipse soundtrack. No, that was mostly Pink Floyd.

1 Like • Comment • View 2 comments

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