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Friday, Oct. 18, 2024

1:19am 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

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:

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 16, 2024 at 8:12am

#1078398

Can We?

From The Takeout, answers to questions you never asked.

Here's Why Canned Food Always Comes In A Ribbed Tin

No, it's not "for her pleasure."

You might think the ribbed design on canned food is just a quirky touch meant to make jellied cranberry sauce look more interesting, but it actually serves a crucial purpose.

I might be alone in this, but I like the jellied cranberry sauce way more than the other canned kind, and way, way more than homemade. The uniformity of it appeals to me. I like to punch out both ends of the can and use one of them as a plunger, pushing out precisely 1/4" of the stuff at a time, then slicing it off at the can top. It's a far superior experience and taste.

Not that I'd ever tell that to someone who sweated over a hot stove to prepare homemade.

This is because the corrugated ridges add strength to the tin by creating a series of tiny arches along the surface. These in turn more evenly distribute pressure, similar to how the arches of a bridge help it carry heavy loads.

Always amusing to see someone unfamiliar with a subject (in this case, structural engineering) attempt to describe it to others.

Wait... is that how I sound all the time?

But the ridges have another job that goes beyond just strength.

Thus once again demonstrating the concept that things don't have to have just one purpose.

In addition to making the can stronger, the ridges on a can help it withstand heat. During the canning process, the contents are sealed and then brought to high temperatures to kill off any bacteria, helping to ensure that the food inside stays safe to eat.

In theory.

These ridges also help the can remain intact after it leaves the factory, when it might have to endure extreme temperatures during transport and storage.

I have a severe addiction to Coke Zero, and I buy it by the case. Obvously, Coke cans aren't ribbed, though their bottoms and tops are reinforced (because of how cylinders perform with pressure inside). Sometimes, on rare occasions, the reinforcement fails, and the bottom goes from concave to convex. On other occasions, it's the top that blows out, making it a whole lot harder to open the damn can. Unlike with canned food, this doesn't indicate that the contents have gone bad.

I suppose it's possible that some cans fail along the cylinder, too, and those are just culled before sale. But it seems to me that the sides of the can are not the failure points.

In the early 19th century, food was stored in glass jars that were heated to high temperatures to kill bacteria and then sealed to keep the contents safe.

Home "canners" still use glass jars. Duh.

The next big leap came in 1810 when English inventor Peter Durand developed tin-plated iron cans... Before the invention of the can opener nearly 50 years later, people had to use tools like hammers and chisels to access the food inside.

Fifty years? Man, humanity really dropped the ball on that one. I guess they were too busy opening cans to develop a better way of opening cans.

4 Like • Comment • View 4 comments

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