Complex Numbers (Book)

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Complex Numbers

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/

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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 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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