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Tuesday, May. 21, 2024

2:49am 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:

Aiva Raine

May 17, 2024 at 6:57pm
In response to "Knead Some Dough?"

I grew up on mostly Wonder Bread despite our family having a large dry farm (dry farm being a farm that grows wheat and barley and safflower without irrigation- it relies solely upon mother nature to rain). I was never fond of it and looked forward to the times my mom would laboriously mill and grind the grain we brought back during harvest into flour and make home-made wheat bread. There was and still is nothing better than fresh whole wheat bread with a little butter and honey.

I think we did the Wonder Bread because my mom wasn't a fan of cooking and making bread is a time-consuming process.

I quit doing the white bread decades ago. There are so many kinds of healthy, no HFCS, etc whole grain breads out there that are delicious that I have no idea why any adult would choose the crappy Wonder Bread.

1 Like

Wordsmitty ✍️

May 13, 2024 at 1:18pm
In response to "The Breakfast and the Breakfurious"

And for those who rather not read the Smithsonian article, the History Channel episode that covers Pop Tarts (if you have a cable TV account):

https://www.history.com/shows/the-food-that-built-america/season-4/episode-1

Annette

May 10, 2024 at 12:05pm
In response to "Meditation Clause"

Everytime a stranger asks me to close my eyes, I keep my eyes open to see what they are up to.
Meditation is not for me.

But I think most people would have 45 minutes every day if they took that time away from social media or TV watching.

2 Like

Wordsmitty ✍️

May 9, 2024 at 1:15pm
In response to "Super Stew?"

But . . .

What's the difference between one being in the soup or in a stew? Is one thinner or thicker? How do writers use soup and stew?

1 Like

Elisa the Bunny Stik

May 9, 2024 at 10:11am
In response to "Super Stew?"

"Served in a pot? Maybe in your country."

Honestly, in the US I'd think soup would be served in a pot due to the quantity that ends up being consumed.

1 Like

Lazy Writer est 4/24/2008

May 8, 2024 at 3:02pm
In response to "Be Positive"

Mine always tell me to eat more fruits and veggies.

Annette

May 8, 2024 at 10:52am
In response to "Be Positive"

Asking doctors for advice has its own pitfalls. Because there is no money in recommending a sensible nutrition pattern, they will say things like: Look it up online.

2 Like

🌕 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

October 25, 2020 at 12:01am

#996685

Networking

I've noted this article's thesis in a previous blog entry. Good to know it's not just my drunk ass that picks up on these things.

The Strange Similarity of Neuron and Galaxy Networks
Your life’s memories could, in principle, be stored in the universe’s structure.

I'm not sure that the subheading there is strictly accurate. But the physical similarity between the mid-scale structure of the universe and the neural mapping of the brain is too similar to be ignored. Not that there's anything supernatural at play here; patterns often repeat themselves in nature for perfectly ordinary reasons, such as the way we find Fibonacci's number in all manner of structures... but that's not important right now.

We have predicted that the void-filament boundary is one of the most complex volumes of the universe, as measured by the number of bits of information it takes to describe it.

This got us to thinking: Is it more complex than the brain?

I suspect it has to do with exactly how complexity is defined. The speed of light puts a hard limit on how information can be transmitted through such megastructures, whereas the speed of information in the brain, while much slower than the speed of light, is fast enough for our purposes.

The first results from our comparison are truly surprising: Not only are the complexities of the brain and cosmic web actually similar, but so are their structures. The universe may be self-similar across scales that differ in size by a factor of a billion billion billion.

Physical similarity, however, doesn't automatically translate to similarity of utility.

The article, while fascinating, doesn't really lend itself to a lot more quoting, especially since my own gray matter is perhaps just a bit disordered right now, thanks to a generous infusion of simple molecules involving carbon, hydrogen and oxygen. But it's worth reading, if only as an example of what can happen when you get two vastly different scientific specialties together to compare notes.

It's out of such things that creativity arises, I think -- that finding of similarities in things that are, at first glance, disparate.

But I will quote one more bit, because it relates to a phenomenon I've mentioned in here fairly recently.

Is the apparent similarity just the human tendency to perceive meaningful patterns in random data (apophenia)? Remarkably enough, the answer seems to be no: Statistical analysis shows these systems do indeed present quantitative similarities.

Now, I should probably emphasize that no, this doesn't mean that there's some mystical or metaphysical conclusion to be reached here. But, materialist that I am, I'm also a writer (of sorts) and this kind of thing is one reason why science fiction exists. Could the universe itself have a kind of consciousness? My instinct is to scoff at such an idea, but really, no one really knows what consciousness is, even though one must possess it in order to even ask the question.

But soon it will be time for me to experience its opposite for a few hours. I just thought I'd leave this here for your neural networks to contemplate.

Edit: After completing this entry, as I usually do after one of these entries, I checked the website Astronomy Picture of the Day, which updates around midnight EST. In another cosmic coincidence, today's picture is of a computer simulation that closely resembles the structure of the universe on these cosmic scales, and so I present it

as another illustration of the superficial similarity between the universe and a neural net.

6 Like • Comment • View 2 comments

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