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Saturday, Nov. 23, 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:

Robert Waltz

Nov 20, 2024 at 2:54pm
In response to "Powers of Three"

🌕 HuntersMoon

- that's pretty much what quantum computing is, from what I've read.

2 Like

Brandiwyn🎶

Nov 20, 2024 at 2:32pm
In response to "Losing Weight Is Easy"

Wordsmitty ✍️

1 Like

🌕 HuntersMoon

Nov 20, 2024 at 1:20pm
In response to "Powers of Three"

With binary, it's either on or off. Wouldn't ternary introduce Schodenger's Cat? On, off, or both?

2 Like

Turkey DrumStik

Nov 19, 2024 at 7:43pm
In response to "Paramount Purpose"

I suspect old web filter attempts

would have combusted when confronted with this article.

1 Like

tj-turkey-jobble-jobble-hard-J

Nov 19, 2024 at 4:04pm
In response to "Losing Weight Is Easy"

Some of them break out in hives if you ask them to add 10% to something.

Yesterday I purchased an item that cost $6.42 after tax was affixed by the register. I handed the cashier a ten dollar bill and watched as she placed it in the cash register, which like most now days, showed her how much change to return. She had to try three times to count out the proper amount, and as I was leaving commented to the bagger, "Why do people have to pay with cash, don't they know how hard it makes my job?"

I didn't say anything, but was trying desperately not to break out in hysterical laughter.

1 Like

🌕 HuntersMoon

Nov 19, 2024 at 9:28am
In response to "Paramount Purpose"

At your age, Bob, take heart in the fact that it can be restored...

2 Like

Wordsmitty ✍️

Nov 18, 2024 at 12:34pm
In response to "Losing Weight Is Easy"

This brings up the old question of whether your weight changes when you inhale a helium balloon to make your voice higher.

Maybe this has to do with the difference of a mug of beer vs a tankard of ale.

The universe has so many weighty/massive questions.

3 Like

Brandiwyn🎶

Nov 18, 2024 at 11:27am
In response to "That's the Spirit"

Maybe you’d go with “Thunder” because that one’s so terrible that only divine intervention can explain its success. Wow. Missed opportunity, here. How could you let this go without a shoutout to mighty Thor, son of the Allfather, wielder of Mjolnir, He with the Long Forehead? (Said in the voice of the Soup Nazi): "No Valhalla for YOU!"

Brandiwyn🎶

Nov 18, 2024 at 11:19am
In response to "Losing Weight Is Easy"

Has anyone ever told you how annoyingly pedantic you are?

Damn your pedantic ways. I was hoping to justify a vacation to Venus (which I still argue would be a much better holiday destination than fr-fr-freezing Mars, Elon. Ugh.) Now I guess I'll have to settle for the Cancun plans we already made, since my mass (and waist diameter, and thigh diameter) will be exactly the same regardless.

3 Like

tj-turkey-jobble-jobble-hard-J

Nov 17, 2024 at 4:04pm
In response to "Revisited: "A Frank Discussion""

I became disgruntled with meat sold in local grocery stores; half the weight is water, the taste is bland, and it's "use by" date is usually one day away. I found a small town butcher/locker plant that sells locally raised meats and buy in bulk so it's actually fresher, cheaper, and better tasting. Recently, I decided to try my hand at making our own sausage as well. It's a bit of work, but I think it's worth the trouble.

1 Like

November 9, 2024 at 10:50am

#1079718

It Was a Dark and Starry Night

When science meets art, are they both elevated? Or do they make each other suck worse? From CNN:

Turbulent skies of Vincent Van Gogh’s ‘The Starry Night’ align with a scientific theory, study finds

I usually groan when I see "scientific theory" in a headline, because the word "theory" means something different in science than it does in everyday speech. In this case, though, my fears proved unfounded.

Which doesn't mean I don't have some issues with the rest of the piece.

Now, a new analysis by physicists based in China and France suggests the artist had a deep, intuitive understanding of the mathematical structure of turbulent flow.

Or, and hear me out here, he was an artist and thus observed turbulent flow in, perhaps, a river or whatever, and incorporated that observation without being able to do math beyond "this paint costs 3 francs and this other one costs 4; which one is cheaper?"

As a common natural phenomenon observed in fluids — moving water, ocean currents, blood flow, billowing storm clouds and plumes of smoke — turbulent flow is chaotic, as larger swirls or eddies, form and break down into smaller ones.

"Chaotic" is another word that means something different in science than it does in everyday speech. Again, though, credit where it's due; the article uses it in the scientific sense:

It may appear random to the casual observer, but turbulence nonetheless follows a cascading pattern that can be studied and, at least partially, explained using mathematical equations.

"Partially" is doing a lot of the work in that sentence.

“The Starry Night” is an oil-on-canvas painting that, the study noted, depicts a view just before sunrise from the east-facing window of the artist’s asylum room at Saint-Rémy-de-Provence in southern France.

There are, I think, a few paintings that even the art-blind (like me) can identify at first glance. Mona Lisa. That Michelangelo thing with God and Human. The Scream. Maybe that one with the farmers. And The Starry Night. But, on the off-chance you have no idea what we're talking about, the article provides helpful illustrations.

Using a digital image of the painting, Huang and his colleagues examined the scale of its 14 main whirling shapes to understand whether they aligned with physical theories that describe the transfer of energy from large- to small-scale eddies as they collide and interact with one another.

I would love to have seen their grant proposal. "Yeah, we're going to study... art."

The atmospheric motion of the painted sky cannot be directly measured, so Huang and his colleagues precisely measured the brushstrokes and compared the size of the brushstrokes to the mathematical scales expected from turbulence theories. To gauge physical movement, they used the relative brightness or luminance of the varying paint colors.

One wonders if they already had a conclusion in mind when they picked those criteria, which would make it questionable science.

Huang and the team also found that the paint, at the smallest scale, mixes around with some background swirls and whirls in a fashion predicted by turbulence theory, following a statistical pattern known as Batchelor’s scaling. Batchelor’s scaling mathematically represents how small particles, such as drifting algae in the ocean or pieces of dust in the wind, are passively mixed around by turbulent flow.

And here's where most of the red flags appear, to me. Paint is, and you might want to sit down for this one, a fluid. Granted, that's just about the limit of my knowledge of artists' paint, but I have a high degree of confidence in my assertion, having seen paint in its fluid form. I've even seen paint mixed, and noted the swirls and eddies of turbulence. This is kind of like seeing milk added to coffee, and I don't drink coffee either.

Where I become more speculative is in thinking: well, he painted the thing with wet paint, so of course there's turbulence at the small-scale boundaries of brushstrokes.

Beattie agreed: “It’s an amazing coincidence that Van Gogh’s beautiful painting shares many of the same statistics as turbulence,” he said.

While there is, indeed, such a thing as coincidence, I don't agree that this is an example of it.

The study team performed the same analysis and detected the same phenomenon in two other images, one a painting, “Chain Pier, Brighton,” created by British artist John Constable in 1826-7, and the other a photograph of Jupiter’s Great Red Spot, taken by NASA’s Voyager 1 spacecraft on March 5, 1979.

Now, that seems a little more like what I'd expect from science. First, another painting; perhaps as a control of sorts. Hell if I know; I don't know that painting, and CNN neglected to illustrate it for us (there is, however, a link). As for Jupiter, we're pretty sure it exhibits all the hallmarks of turbulence on its visible surface, so it's a check on their modeling assumptions.

And yet, it shouldn't boggle anyone's mind that an artist noticed turbulence and tried to recreate it, or that one as brilliant as van Gogh was able to do it.

4 Like • Comment • View 2 comments

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