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Thursday, Oct. 17, 2024

11:23pm EDT

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Complex Numbers

by Spirit HalloWaltz

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

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

Show Recent Comments

10 Most Recent Blog Comments:

Wordsmitty ✍️

a day ago
In response to "Can We?"

"corrugated ridges add strength " - and this is so you can't use your heel to crush it like you do a beer can.

1 Like

The Ghost of Jayne's Lost Sock

a day ago
In response to "Can We?"

The top and bottom seams fail first. *As reported by a friend who used to play "can bowling" with their coworkers (overnight shelf stockers at a discount grocery chain)

2 Like

Annette

a day ago
In response to "Can We?"

By the way:

"A significant new market for the can had become apparent after 1935 when, following the development of suitable internal coatings, the first beer cans were test marketed."

Evo lution of the tin can

1 Like

Annette

a day ago
In response to "Can We?"

As much as all the modern people decry canned foods, there seems to be a fierce competition on who gets to take credit for inventing tin cans.

In a small museum in the country side in France, I learned that French scientists finally achieved to retro-engineer the exact alloy for tin cans sometime in the 18th century. They claimed to have cracked a German secret.

This article that you quote credits an English person.

Funniest part is that I was convinced it was an AmeriCAN invention.

1 Like

Mummified_Mike

Oct 15, 2024 at 10:45am
In response to "Pushing the Button"

I wonder how many of the NASA guys involved in making the Apollo 11 decision had already read Crichton's The Andromeda Strain - published about 2 months before the Moon mission - and allowed it to influence their decision-making process?

1 Like

🌕 HuntersMoon

Oct 14, 2024 at 10:13am
In response to "DeTECHtion"

I had a B.S. detector but, with the election season here, it was more like a smoke detector with a weak battery - It just went on and on and on...

2 Like

Kåre Enga in Montana

Oct 13, 2024 at 6:18pm
In response to "Spit in the Ocean"

Worthy of a long comment. Yes, I could; but, I won't.

As for your list... I'm a bag of mixed nuts.

1 Like

WakeUpAndLive️🦇Elf'sSpell

Oct 13, 2024 at 11:48am
In response to "Revisited: "I Wood Knot""

1 Like

Elisa: Stik or Treat

Oct 12, 2024 at 10:46am
In response to "A Few Observations"

Oddly enough, I've encountered similar unisex style restrooms in the US. If you wander back up to Minnesota and go to the old Schell's brewery in New Ulm, they have them. Mi amor and I joke that they're likely the most progressive bathrooms in the entire (rural) county.

1 Like

Spirit HalloWaltz

Oct 12, 2024 at 10:13am
In response to "A Few Observations"

Elisa: Stik or Treat

, oh, yeah, I'd intended to mention those things but forgot.

On accessibility: Most of the places I visited were old; in some cases, older than the US. Spiral stairs, tiny passageways, multiple levels, and those blasted cobblestones. Difficult, if not impossible, to retrofit enlightened disability access features. The few places that were more modern did it better. That's one downside to the older stuff, I think.

Restrooms: tiny, few stalls, mostly unisex. The unisex thing doesn't bother me, and I expect it better accommodates trans people... but not disabilities. I saw pay-to-poo restrooms, but I managed to avoid having to use them.

Now, some people unfamiliar with Europe might wonder how the unisex bathrooms work. There's usually one or two urinals tucked behind a corner or something, but I imagine it might be shocking for some folks from the US to peer around a corner and see some dude standing at one. But the view is from the back, and it's not like you see anything important and/or disgusting. Also, the stalls don't have those flimsy walls with big gaps and cracks; each toilet stall is usually (for the ones I used) completely enclosed, with a locking door, and private. The hand-washing facilities are shared, and sometimes not even within the restroom itself. It is entirely possible that I missed some etiquette thing, but in general, one merely needs to acknowledge that, shockingly, we're all humans and we deal with waste in much the same ways. Basically, it's just different if you're used to the US (and mostly UK) segregated style, but it's not privacy-killing.

What was more shocking to me was the near-complete absence of bidets, even in hotel rooms.

October 17, 2024 at 10:19am

#1078457

Pros and Contrails

In a sane world, this PopSci article would put an end to one of the stupidest conspiracy "theories" immediately.

Why airplanes leave white streaks in the sky
Contrails are a problem, but not how you might think

Of course, in a sane world, the "theory" would never have gotten traction in the first place.

If you’ve ever looked up at a mostly blue sky and seen straight white lines criss-crossing the horizon, or watched a plane puff out a plume as it passed above, you may have wondered, “what causes that?”

Yeah, back when I was a kid. This was before the internet, so I used the version of Wikipedia we had back then: I asked my dad, who launched into a long-winded explanation of what he called vapor trails.

I think that was his way of keeping me from asking too many questions: answer so thoroughly that I'd get bored and stop asking questions.

And no, they’re also definitely not “chemtrails.”

I'd like to think that, even without the influence of my father's scientific mind, I still wouldn't have bought into that idiocy. But I can never be sure.

What they are, instead, is frozen water vapor crystallized on soot particles, both of which are standard byproducts of a jet’s combustion engine.

Well, there you go: the answer.

And though conspiracy theories about airplanes distributing mind control chemicals aren’t correct, contrails are having a negative planetary effect.

Because of course they are. Nothing we do is allowed to actually be good for the environment.

Atmospheric conditions have to be just right to enable the jet vapor to crystallize: moist enough that the water doesn’t evaporate, and cool enough that it freezes.

But as everyone knows, thanks to the internet, nothing that's described as "moist" can ever be cool.

A growing body of scientific research, including studies conducted by both Barrett and Stettler, indicate that contrails cumulatively have a warming effect, contributing to human-caused climate change.

These studies are important, because the leading hypothesis that I'd heard was that, being basically cirrus clouds, they'd have the effect of reflecting some sunlight, reducing the "heat in" side of the climate warming equation.

Perhaps they do, but other things about them overshadow (pun intended) that.

Since contrails, like other clouds, are bright white and insulating, they reflect light and heat, and also trap it. On a sunny day, a contrail does two things simultaneously. It “acts like a blanket,” preventing heat radiating from Earth’s surface from escaping to space, says Barrett–this is the warming effect. Simultaneously, it also reflects sunlight from space away from Earth’s surface, in a cooling effect. Unfortunately, even when the sun is shining, generally the blanket effect outweighs the reflector effect, says Barrett.

At least according to this guy.

Anyway, the article goes into what can be done about them, and, surprisingly, the answer isn't "just stop flying" or "stop using plastic straws." In other words, this is one contributor to climate change that can be mitigated fairly easily, and it's not even about us doing penance for how sinful humans are (just corporations).

And yet, some of us are just going to blithely keep on believing that they're mind-control chemicals.

1 Like • Comment

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