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Tuesday, Apr. 30, 2024

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Complex Numbers

by Robert Waltz

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:

GeminiGem of House Lannister

a day ago
In response to "Supplementary"

Working in health care for a zillion years, I've heard it all. The whole thing about something being natural, therefore it would be good for a person is very, very, very...very prevalent. An internist I worked with had a fantastic comeback to that line of thinking. She would say, "Arsenic is natural, are you going to start taking that?"

1 Like

Lazy Writer est 4/24/2008

Apr 28, 2024 at 2:14pm
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

Apr 28, 2024 at 10:43am
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

September 25, 2021 at 12:01am

#1018010

Happy Mead-ium

I have had a complicated relationship with mead.

The Quest to Recreate a Lost and ‘Terrifying’ Medieval Mead
Bochet vanished for centuries, but meadmakers are bringing it back—at least in spirit.

Any sugar can be converted into alcohol. Feed it the right yeast, and magic happens. Well, actually it's science, but it might as well be magic.

For beer, first you have to convert the starch of a grain, usually barley, into sugar. That's easy enough, chemically. Saliva does that all the time, but of course these days other processes are involved. With fruits, though, such as apples for cider or grapes for wine, it's a more direct process.

Honey is basically just sugar, and it's used to make mead. But this mead is different than the kind you occasionally find in the stores.

It starts with a cauldron, an open flame, and a good measure of raw honey. Then—double, double, toil and trouble—stir constantly until the honey spits black steam at you. Add water and stand back as it erupts, volcano-like. Throw in some yeast and spices and, after it ages a bit, behold: bochet, a mysterious and lost style of mead.

Most mead is pretty straightforward. Not that I've made it myself, but I had a friend who did. I've had a few friends who created mead, actually, hence my complicated relationship with it: unless you do it right, it tastes more like ass than like honey. As for commercial mead, it also varies in quality. Oddly enough, in all of my travels to different breweries, I've only encountered one or two that have messed around with bee vomit to make mead. There's one in particular in a suburb of Minneapolis that I would totally revisit... but I digress.

This stuff, though? Bochet? I'd never heard of it.

“Caramelizing honey is kind of terrifying,” says Ontario microbiologist Bryan Heit, the brains behind popular homebrewing reference site Sui Generis Brewing. While Heit experiments mostly with beer, particularly traditional styles, he’s been intrigued by bochet for years.

Fortunately, I don't need to know how to make it when there are other people more than willing to do it for me.

“It truly is a lost style. It’s not a historical style that has survived into the modern era,” says Heit. “It’s literally something that disappeared.”

Why revisit something that's obviously died out? Well, why do anything?

Bochet is mentioned, briefly, in French texts as early as 1292, according to research by independent scholar Susan Verberg, published in 2020 in ExArc Journal. The first and only complete recipe for the drink—and the primary source for modern recreations—turns up in 1393, in what might be considered a manual of mansplaining.

The history of fermentation isn't well-documented, but fragments exist. I think I've linked the oldest known beer recipe in here before; it was from ancient Mesopotamia, and it wouldn't have resulted in anything much resembling what we know as "beer" in all its glorious styles. So in terms of history, 1292 C.E. is quite late in the game, but from a modern perspective, it's still old. (Mead itself is probably about as old as beer.)

I should note here that as accomplished as the French are at creating wine, their beer tends to suck on ice. While I have had decent French beer, it was all from the east, near places that actually know how to brew beer, such as Belgium and Germany.

So it's not surprising that they once made mead, although the fermentation of honey is more associated with the Norse. What would be surprising would be if it was any good.

A preserved recipe for an extinct beverage is a rarity, and its unique method tantalized historic reenactors and homebrewers alike, including myself. While I’ve made small-batch meads in the past, they have all been based on modern ratios, and none involved a cauldron. Commercial meadmakers have been particularly intrigued by the idea of caramelization: The process offers new possibilities, unlocking “all those roasty, toasty, nutty flavors,” says Jen Otis of KVLT Mead in Tacoma.

Caramelization isn't unique to honey; it's used on sugarcane to make molasses, which leads to rum -- but rum is a distilled spirit; if there are fermented beverages available from sugarcane, I'm unaware of them (this doesn't, of course, means that they don't exist). Likely they did, prior to the relatively modern process of distillation.

There are distilled spirits made from honey, too, but as far as I know, I've never tried them - just the mead, which as I've noted, can be of greatly varying quality.

The rest of the article, which is fairly long, goes into the author's re-creation attempts, and is quite fascinating, but there's no need to rehash it here. There's also the modern version of the recipe that she used. I'm not tempted to do it myself -- too much like work, and I don't really have the setup for it -- but if I ever see bochet available, you damn right I'm going to try it.

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