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Tuesday, May. 7, 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:

🌕 HuntersMoon

May 5, 2024 at 8:50am
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

May 4, 2024 at 5:42pm
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

Simply Me

May 3, 2024 at 11:58am
In response to "It's a Crime"

I get it. The fear spreading jerks wants us to have an "us vs. them" attitude. The division gives them leverage. (they think)

1 Like

🌕 HuntersMoon

May 2, 2024 at 11:37am
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

2 Like

Elisa the Bunny Stik

May 1, 2024 at 10:53am
In response to "Mad as Hell"

"But if you want to really see how anger is useful, you need to consider how it's used as a tool for manipulation."

This is a line that must be regarded with care. After all, there are people that get angry about the way they are being treated on a large scale and harness that anger to change their circumstances. There is some level of manipulation in this angry but still legitimate flavor of activism. We do have to consider the possibility that this angel's usefulness can have net benefits (such as environmental cleanup and protection; see the Live Canal as an example).

GeminiGem survived GoT

Apr 29, 2024 at 1:53pm
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

October 21, 2022 at 12:08am

#1039496

It's not in Seine

One of these days, y'all're gonna be mad at me because I'mma do a whole blog entry in French.

Today is not that day, but it could have been.

‘They said it was impossible’: how medieval carpenters are rebuilding Notre Dame
Project leaders at Guédelon Castle tell how their woodwork savoir faire is proving a godsend for mission to restore Paris cathedral roof

Ils ont dit que c'était impossible.

And that's as far as I'm going to go today.

I do remember when Notre Dame burned. It was in the Before Time, so I still had a gym membership, and almost every single screen that hung from the ceiling showed the iconic spire engulfed in flame and smoke. I'm no fan of organized religion, but that shit was art, regardless of what inspired it (pun intended), and I was just as gut-punched as when I watched the Twin Towers (which may or may not have been art) burning.

Anyway. The article.

At Guédelon Castle the year is 1253 and the minor nobleman, Gilbert Courtenay, has ridden off to fight in the Crusades, leaving his wife in charge of workers building the family’s new home: a modest chateau that befits his social position as a humble knight in the service of King Louis IX.

I don't know how a château can be considered "modest" except maybe in comparison to, I dunno, Versailles. Or compared to a typical present-day American subdivision house.

The Guédelon project was dreamed up as an exercise in “experimental archaeology” 25 years ago. Instead of digging down it has been built upward, using only the tools and methods available in the Middle Ages and, wherever possible, locally sourced materials.

I'm not convinced that "locally sourced" is historically accurate, especially when it comes to upper-class housing. Hell, even Stonehenge has been shown to have been constructed out of material from many miles away. And even more kilometers.

Now, in an unforeseen twist of fate, Guédelon is playing a vital role in restoring the structure and soul of Notre Dame cathedral.

I also remember seeing some proposals for what to replace the spire with. To overwork what was once (arguably) a clever pun, none of them were very inspiring. In their defense, it's not like they always used 12th-13th century techniques in its construction; it's been modified before, with then-current materials and methods.

Paris’s imposing 13th-century cathedral, a world heritage site, was consumed by fire in April 2019, destroying its complex roof structure, known as La Forêt because of the large number of trees used in its construction.

Funny what gets translated and what doesn't. We talk about the Eiffel Tower (translated from la tour Eiffel), but Notre Dame de Paris isn't translated to Our Lady of Paris (though most English speakers don't pronounce it the same way). L'avenue des Champs-Élysées also stays French instead of becoming Elysian Fields Avenue. But le musée du Louvre is just "the Louvre." Point is, why not just call it "the forest?"

And don't get me started on La Seine, which, despite what you may have heard, isn't "the boob." After all, there's only one of it.

“After the fire, there were a lot of people saying it would take thousands of trees, and we didn’t have enough of the right ones, and the wood would have to be dried for years, and nobody even knew anything about how to produce beams like they did in the Middle Ages. They said it was impossible."

If there's one thing I know about architects, it's that if you tell them something's impossible, they'll try to find a way to prove you wrong. They're contrary that way. Sometimes, they even turn out to be right.

A number of the companies bidding for the Notre Dame work have already engaged carpenters trained at Guédelon, and more are expected to beat a path to the Burgundy clearing 200km down the autoroute du Soleil from Paris.

See, there we go again. Is "Highway of the Sun" too fucking hard to type? Or go full French and say l'autoroute du Soleil.

Stéphane Boudy is one of a small team of carpenters at the medieval site, where he has worked since 1999. Boudy, 51, trained as a baker, then an electrician, until discovering his vocation at Guédelon. He explains how hand-hewing each beam – a single piece from a single tree – respects the “heart” of the green wood that gives it its strength and resistance.

Sometimes, it pays to find a profession and stick with it. But more often, one can achieve greatness by dabbling, and finding ways to connect materials and techniques from various disciplines (I have another article in the queue that touches on this; one of these days, it'll come up).

“We have 25 years’ experience of cutting, squaring and hewing wood by hand,” he says. “It’s what we [have done] every day for 25 years. There are people outside of here who can do it now, but I tell you they all came here to learn how. If this place didn’t exist, perhaps the experts would have said: no it’s not possible to reproduce the roof of Notre Dame. We [have shown that] it is.

I mean, to me, that's the very core of ennui: doing the same thing every day, over and over. Especially if it's (ugh) work. After Day 3 I find a way to automate the process. But I can't deny it builds skill.

“This isn’t just nostalgia. If Notre Dame’s roof lasted 800 years, it is because of this. There’s no heart in sawmill wood,” he says.

Still burns just as bright.

*runs away from angry French mob*

Florian Renucci, the Guédelon site manager and a philosopher turned master mason, has already been asked to oversee training of artisans expected to work on Notre Dame.

And there we go again with the career shift. In monsieur Renucci's (or is it signore Renucci's? *googles* nope, French with a vaguely Italian name, fairly common near that border) defense, master mason almost certainly pays better than philosophy. But then again, cat litter box scraper pays better than philosophy.

Épaud is on the scientific committee at Guédelon and the committee overseeing the reconstruction of Notre Dame, as well as a member of the Centre National de la Recherche Scientifique (CNRS), France’s national research body. He says that going back to build the future is not just nostalgia.

One of those times when you don't really need a translation, just the knowledge that French usually puts the adjective after the noun. See also: Système Internationale d'Unités.

“I have studied the 13th-century technique for many years and, if we respect the internal form of the tree, the beams will last for 800 years. Guédelon is the only place in France, and I believe in Europe, where they build this kind of roof structure in wood. All those who didn’t think it was possible didn’t know about Guédelon.”

To be clear, those beams would have lasted way more than 800 years if it weren't for l'incendie.

He adds: “But it shouldn’t be rushed. Macron’s insistence that the cathedral be open by 2024 is idiotic. We are talking about a cathedral, we’re not in a hurry and we have the money to do it the right way. If we rush it, there’s a risk it [will] be done badly and something is missed. Sadly, I fear Macron doesn’t understand that.”

Politicians gonna politic. One downside of democracy (there are a few) is that politicians tend to think short-term. I think the only time in history that a politician's timeline was actually realized was Kennedy's moonshot promise, and I sometimes wonder if it would have happened if he hadn't been assassinated.

Yes, cathedrals have been known to take hundreds of years to complete, way beyond any elected leader's lifespan, let alone time in office.

The big question is: when do we replace with new materials, and when do we go back to origins? I think there's an argument to be made either way.

But "ce n'est pas possible" isn't an argument.

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