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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 10 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 12 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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').

1 Like

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.

June 4, 2020 at 12:40am

#984955

Building Around

I just can't resist a good architectural takedown. It's even more fun than art criticism, because architecture, unlike art, is supposed to serve a purpose. Oh, yeah, I went there.

https://www.cracked.com/article_27598_why-your-city-looks-ugly-af.html

Why Your City Looks Ugly AF

Ugly "AF?" Is this where journalism is going now: tweenspeak? Oh, wait, it's not journalism; it's Cracked. Carry on.

In case you couldn't tell by my jaded cynicism towards the film industry and insufferable snobbery, I live in Los Angeles. Recently, between a meditation session with the Mindfulness Maharishi of my spirituality-focused initiative and my appointment for a medicinal gong-ringing, I was able to take the subway (yes, LA has a subway) to Little Tokyo for a big ol' bowl of ramen -- the good shit, not the twenty-five cent brick of noodle-shaped styrofoam and sodium you're thinking of. And as I was crossing the street, I noticed something.

That all sounds more like San Francisco than Los Angeles to me. But then, I visit SF almost every year while I haven't been to LA in over 20.

Why does one look like Minas Tirith and the other look like a DMV for the Borg? Or, for those of us who didn't have sex in high school, why does one look like a beautiful altar where blood sacrifices were made to a forgotten Canaanite god and the other looks like a jail for Cubists convicted of tax fraud -- the most boring crime?

This is about where I started to lose my shit. Meaning I actually LOLed, something I rarely do when reading stuff.

In classic Cracked form, this is a numbered list. At this point I'm just used to it.

5. You Hate Modern Architecture

And modern art, but you already knew that.

The word "modern" has a problem. It can mean both "current and up-to-date" and "belonging to a particular school of art that's not current." The Modern era ended before WWII (I think). The modern era is whatever we're in now.

I had an ongoing battle with a particular landscape architect I worked with. Whenever we'd make major changes to a plan, we'd save the old one in case we had to go back to it for some reason, like when you save a video game before you go fight the Big Boss, so if you die you can go back to your old save point and nope on out of there. He always saved the old file as "filename" and the new one as "filename_new" or something along those lines. I always saved the old file as something like "filename_date" and the new one as "filename." As you can imagine, this caused some confusion. But his way, if you had to do this more than once, you'd end up with "filename_new," "filename_new_new," "filename_new_new_new..." you get the idea. My clearly superior way kept a record of when we made these save points, keeping the current file with always the same name. Should we ever go back to an older version, we could simply switch filenames.

Our boss, who kinda-sorta knew what a computer was, resolved this by saying something like "But it is the new file, so just call it new."

So I started my own company, and you're damn right I adopted the obviously better naming conventions. That particular landscape architect had the chutzpah to call me looking for a job. My short answer was "No." I'll leave the long answer to your imagination. I hear he's still designing bushes somewhere.

Anyway. That's the problem with "modern." Anything Modern is older than I am, and it's confusing as hell.

The first building I'd really consider to be "modern architecture" is the Rose Center for Earth and Space, all the way down at 33. Now before you rush down to the comments to yell at me for using terms like "Modern," "Post-Modern," "Brutalist," and "Contemporary" interchangeably, I'd like to offer a brief rebuttal: come the fuck on, you know what I mean.

No. No, I really don't. The only Rose Center I know is in NYC, and it's modern. I mean new. I mean it's a massive glass box with a sphere inside for the Hayden Planetarium. It looks cool, and I'm not sure if it was built before or after the glass box at the corner of Central Park that's actually an Apple store, but both of those buildings ought to be paying royalties to my uncle, who pioneered the whole "transparent box" art movement.

But I digress.

And really, you have to click on the article to get the full effect. There are pictures. With captions.

The point here is that most modern architecture strikes my uneducated plebeian eye as being an ugly, discordant collision of metallic angles. And statistically speaking, I'm in the majority here.

I've said this before about art, a term that I assert includes architecture, but at some point, with any art, you start to get artists who care more about winning the adoration of other artists than about being accessible to the public. This is why I hate the Oscars, and why I refuse to read lit-snob writing.

4. Common People Aren't Deciding What Buildings Look Like

Well, okay, but given that "common people" tend to vote for ocean vessel names such as "Boaty McBoatFace," this is a good thing. As the linked article points out later, we'd end up with a bunch of buildings that look like penises. But what the hell, right? We already have a bunch of buildings that look like penises. To be fair, it's hard to design a skyscraper that looks like a vagina.

When architects saw that last [I think he meant "list"] of people's favorite buildings, they said "This isn't necessarily the design professional's view of the best buildings, but the emotional connection to where people live and work and play." In other words, "Yeah, idiots! Stop liking buildings just because they make you feel good! You morons. You intractable dumbasses. What's next? Liking food that tastes good? Maybe I'll respect your opinions when you get a Gunther Ford design airbrushed onto the side of your van instead of a huge-breasted woman riding a tiger, you hookworm-ridden carny."

3. Modern Architecture isn't Just Ugly, It's Also Hard to Live In

Weirdly, architecture that's ostensibly predicated around being as brutally functional as possible actually kind of sucks ass at being a building. Not only is it ugly, it's bad at its job -- and I gotta say I don't appreciate these buildings stealing my schtick.

Got news for you, bud. Every comedy writer on the internet is ugly and bad at their job. We don't owe you royalties.

2. More Than Practical Considerations

Because what I'm getting at here is that in the arena of urban development, the idea of "comfort" seems to have been largely demoted from Literal Necessity to Frivolous Luxury. But comfort is important. It's why I exclusively wipe my ass with county fair quilts and get thirteen hours of sleep a night. Beauty is subjective, but surely an attempt to beautify our cities would at least relieve some of the crushing misery or urban life? At the very least, fewer buildings in the Concrete Mistake school of architecture could only help.

Again, we need buildings that we can live, work, and shop in, not ones that win awards from other architects. Imagine a fancy architect trying to design a Wal-Mart. A Wal-Mart has but one function: get a lot of people in there and make them want to buy stuff. That's why they all look like giant postage stamps from the air. Not a single Wal-Mart will ever win a design competition at the Columbia School of Architecture, and yet people flock to them. That's because they sell cheap-ass shit, and that's what people want, not elegant architectural sorcery.

1. Art vs. Function, Democracy vs. Authoritarianism

Now we're in a bit of pickle, because what can be done about the proliferation of ugly architecture? The Trump administration has, unsurprisingly, offered a solution that actually makes things worse: they've drafted an executive order that would put disallow modern architectural designs on federal buildings and make Neoclassical the default style. Which is not only un-democratic, it's also, frankly, kinda fashy -- since the word facism is itself an invocation of ancient Rome.

Of course, legislating art out of existence has always ended well.

But even with the ever present danger of vote brigading turning our cities into dickjungles, I think it could be worth giving the public greater say in what their cities look like. Here in LA in 2016 we voted to pass a bill that increased sales tax to fund an expansion of our Metro system, or as New Yorkers call it, "da unnagroun' piss-circus -- ayy, I'm walkin' 'ere!" I'm sorry, I shouldn't make hacky jokes about New York. (They have it hard enough what with their pizza being a Kraft single cheese byproduct melted on a greaselogged shingle and all.)

Okay, dude, I get your points about architecture, but you know nothing about pizza. NOTHING. Tofu is not an acceptable topping for pizza, or anything else, you avocado-munching California hipster.

I understand, I think, where the modern architects are coming from. Life is ugly, and art should reflect that. I don't want every movie to be a saccharine Disney lovefest, because that's not an accurate reflection of the whole of the human condition. But I also don't want every damn movie I watch to be A Serbian Film. But maybe making terrible buildings isn't just reflecting the misery of life; maybe it's actively contributing to it. I mean, living in a city is already killing us and driving us insane.

All of that, and not one mention of the architectural abortion that is Hudson Yards. Oh well, that can be forgiven since he lives 3000 miles away from that monstrous failure, and besides, a few months ago, I linked a post from the Guardian that heeled that turd down the shower drain. I can't be arsed to find it now, but if you're interested, search my blog for Hudson Yards.

Anyway, like I said, I highly recommend actually reading the article because the photos really drive home his points.

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