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Sunday, Apr. 28, 2024

1:23pm EDT

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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

about 3 hours ago
In response to "Revisited: "Exercise""

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

Elisa the Bunny Stik

a day ago
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

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

August 19, 2020 at 12:10am

#991137

Mind Map

Today's link is a bit esoteric, and I wouldn't blame anyone for giving it a miss. But I find this stuff fascinating, so I waded through it, and of course I have comments -- else I wouldn't bother posting it.

Skip to the bottom for Mini-Contest results!

http://nautil.us/blog/new-evidence-for-the-geometry-of-thought

New Evidence for the Strange Geometry of Thought

In 2014, the Swedish philosopher and cognitive scientist Peter Gärdenfors went to Krakow, Poland, for a conference on the mind.

Philosopher and cognitive scientist? That's a dangerous combination.

In his talk, “The Geometry of Thinking,” he suggested that humans are able to do things that today’s powerful computers can’t do—like learn language quickly and generalize from particulars with ease (to see, in other words, without much training, that lions and tigers are four-legged felines)—because we, unlike our computers, represent information in geometrical space.

So, if I'm reading this right, the evidence suggests that we store concepts the same way we store, for example, the layout of our homes.

He argued that the brain represents concepts in the same way that it represents space and your location, by using the same neural circuitry for the brain’s “inner GPS.”

You know, we were talking about inventions yesterday, and as I re-read this article to talk about it, I came to the conclusion that there are, in general, two broad categories of inventions: those that can be compared to other things, and those to which we compare other things. To use yesterday's example, a wheel would be one of the latter. So, apparently, is GPS -- a truly world-changing invention, though one that's built on a multitude of earlier inventions.

The hippocampus’ place and grid cells, in other words, map not only physical space but conceptual space. It appears that our representation of objects and concepts is very tightly linked with our representation of space.

This makes all kinds of sense when you think about it (not that "making sense" means it's necessarily right). At the risk of delving into the kind of evolutionary just-so stories that I despise, our remote pre-hominid ancestors, lacking the concept of "concept," would have known nothing but their environments. It shouldn't be surprising, then, that our brains use the same sorts of brain configurations, adapting them for other purposes.

But again, I don't know; that's pure speculation on my part.

The article goes into some of the studies backing up the mapping idea.

Yet the mind is not just capable of conceptual abstraction but also flexibility—it can represent a wide range of concepts. To be able to do this, the regions of the brain involved need to be able to switch between concepts without any informational cross-contamination: It wouldn’t be ideal if our concept for bird, for example, were affected by our concept for car.

It will be when I finally get the flying car I've been promised.

Scientists still need to experimentally verify the link between the hippocampus and higher-order cognitive functions in humans. fMRI studies like the ones from the group in Oxford are, as yet, only suggestive.

I'm just including this here for anyone who's still with me. This isn't settled science; just an interesting idea with some evidence for it.

One reason the concept is intriguing is the potential impact on AI development:

Gärdenfors’ theory highlights a fruitful path, not only for cognitive scientists, but for neurologists and machine-learning researchers. It is a kind of incomplete, generic sketch on a canvas that invites refinement and elaboration. Cognitive spaces are, as Gärdenfors and Bellmund put it, a “domain-general format for human thinking,” an “overarching framework” that can help unravel the causes of neurodegenerative diseases, like Alzheimer’s, and “to inform novel architectures in artificial intelligence.”

Oh, yeah, and also to help us deal with brain dysfunction. Maybe we could even figure out how to get more people to appreciate science.

Mini-Contest results!

All great responses yesterday! Though I have to admit I'm just a little disappointed that no one said "sliced bread." Let's see, we have:

The Human Genome Project -- while I wouldn't consider that an invention, it's certainly an innovation, and a major one at that. As Sumojo

noted, it has improved human life to a vast extent.

Sterilization -- Lazy Writer est 4/24/2008

pointed out that the idea that implements and people should be, you know, clean, that's a big deal. Saved a lot of lives. I don't know why they didn't think of it sooner, to be honest; it shouldn't require knowledge of microorganisms to know that dirty things make people sick.

Writing, as per Zhen

-- I think I did a blog entry a while back on that subject. Or maybe a newsletter? If not, I'll have to do one. This one's foundational, the almost-hidden basis for a multitude of other inventions.

Printing Press -- Paul

noted that this extended the above to a wider audience.

Vaccinations (and beer!) -- Aennaytte: Free & Wild in GoT

, I'm a big fan of both (though if I had to choose, it would be beer). Both have saved lives. No, I'm not joking -- as with the sterilization one above, the process of making beer required that water be boiled, which had the unintended (pre-germ theory) effect of destroying harmful germs. Beer, for much of human history, was often safer than water to drink.

And Spring in my Sox

mostly reiterated other comments, but added "autocorrect," which I'd argue is a mixed blessing, but it's certainly made it easier for people. And yes, beer.

Now, given that it's me here, I'm really tempted to go with beer (and I will, in fact, go have a beer when I'm done posting this). There are suggestions out there that civilization is the result of people doing what needed to be done to make beer. Of course, whether "civilization" is, on balance, better or worse is arguable. (I'm arguing "better" because I like not having to hunt my dinner.) But in terms of Most Important Invention or Innovation, I'mma have to agree with writing. After all, without it, what would I do when I'm not drinking beer?

Oh, right, binge-watching Star Trek. But wait -- that show had scripts. No writing, no scripts, no Star Trek, and no blog entry on this or any other day -- a bleak alternate universe indeed. Maybe even just as bleak as one without beer. So this time the Merit Badge goes to Zhen . I'll send you the MB before I start drinking, promise!

Everyone else, thanks! Great comments, and there will be another opportunity in a few days.

5 Like • Comment

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