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Wednesday, Dec. 11, 2024

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Complex Numbers

by Waltz Invictus

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:

Wordsmitty ✍️

about 4 hours ago
In response to "On the Label"

"June 25, 2024 5:40 AM" = Need time zone.

"Note to self:" = self-driving taxis

Some people definitely can be labeled, but we don't say those in polite company.

2 Like

Annette

about 4 hours ago
In response to "On the Label"

Only one label matters. Cat person or dog person. If the answer is anything except "yes" the person's life choices defy all logic analysis.

I wonder what is the difference between psychotherapist, psychologist, and psychoanalysis. Or are they like po-ta-to, po-tuh-two?

3 Like

tj-Merry Mischief Maker

Dec 9, 2024 at 2:51pm
In response to "Analog Log"

As time progresses, it amazes me how we have moved from tide clocks/analog computers to people eating tide-pods.

1 Like

Aiva Raine

Dec 9, 2024 at 2:37pm
In response to "Story of the Century"

Fantastic entry. Have I mentioned how I love your snark? Thanks once again for the grins and your awesome sarcastic responses to the articles.

1 Like

Anni Pon

Dec 8, 2024 at 10:01am
In response to "Revisited: "Millennials Killed Millennials""

It's kind of funny to me that people still use "millennials" to refer to "those darn kids" when most of them are in their 30's now with kids of their own. As an elder millennial (a Xennial, if you will), I haven't really found the generational definitions too useful for myself. I'm right on the border of generations, so I have a lot in common with both Millennials and Gen X. Growing up in silicon valley with tech savvy parents, I had earlier access to computers and technology than a lot of other people my age. But in terms of cultural milestones, I lived through and am old enough to remember things like the 90's, Y2K and 9/11. The borders are sort of arbitrary, but on a population level I can see how sometimes it can be useful to group people into generations.

2 Like

Elisa: Snowman Stik

Dec 8, 2024 at 9:40am
In response to "Revisited: "Millennials Killed Millennials""

*rubs hands together* Time for me to dive right in!

2. In this system, I'm early Generation X, which hardly anyone ever talks about, opting instead to rag on "boomers" or "millennials" or "Gen-Z." Or praise them, depending on the author's beliefs and age.

Looks like I need to change my name to "hardly anyone". Also, there has been more and more chatter about Gen X in some circles, but curiously this seems to be happening in institutions that have dropped labels (which I'll touch on in a second here).

On top of which, you'd have to convince me that our 1983-born X-er has more in common with someone born in 1966 than with someone born in 1986.

Very, very, VERY few organizations that have ever in their history used generational labels have set the Gen-X border any later than 1980. While I'm still very salty that Pew stopped this practice (one of the few things they've done wrong, IMO), I still use their Gen-X demarcation of 1965-1980. That demarcation nicely strikes the median of most demarcations. SO in this case it's not a matter of you needing to be convinced. It's a matter of the author leaning on a REALLY bad source.

Much of what sets "generations" apart, in modern terms, has to do with technological advancements and societal changes...then other people had to think that the original wearer was cool enough to start a fashion trend.

And how that message has been conveyed has drastically changed among generations, to boot.

So you know what I want to see Millennials finally kill?

Generations.

As a Millennial, I can tell you that this is probably the last thing our generation will try to kill. It's something that's persisting even as we reach middle age. And that's another thing. We're aging very strangely, namely that physically we're aging more slowly than people in Gen Z. We could have killed the concept of aging a certain way, but Gen Z's circumstances have effectively torpedoed that revolution.

Also, if anyone wants to read the original article, you can certainly try throwing the URL in the Wayback Machine. It doesn't always work, but it's worth a try. There are some other methods as well, but I haven't tried them myself yet. (Speaking of which, has anyone here tried Zette? I've been meaning to sign up for that but haven't had a chance. It's one of my goals for 2025.)

2 Like

Wordsmitty ✍️

Dec 7, 2024 at 12:41pm
In response to "Story of the Century"

Maybe it helps if you can dance - Dick Van Dyke and Coldplay - although he's just 99. (I assume everyone has read about that)

4 Like

🌖 HuntersMoon

Dec 6, 2024 at 10:07am
In response to "No Space for Time"

I think you just added clarity to the Twilight Zone...

3 Like

tj-Merry Mischief Maker

Dec 5, 2024 at 4:05pm
In response to "How To Give A Fock"

How many photons does it take to make a torpedo? Who gives a Fock! Actually, wasn't Fock Spock's half brother?

3 Like

Jayne

Dec 5, 2024 at 10:23am
In response to "How To Give A Fock"

Versatile Fock States is the name of my next thrash album.

3 Like

December 3, 2024 at 8:51am

#1080769

Coasting Along

I'm not entirely sure why I saved this particular Live Science article, and it's only been like three weeks.

What's the 'coastline paradox'?
Coastlines are challenging to measure, but why?

Yeah... it's not really a paradox.

Flanked with fjords and inlets, Alaska is the state with the most coastline in the United States.

Easy to accomplish when you're a giant peninsula with many craggy islands offshore.

But what is the length of its oceanic coast?

It depends on whom you ask. According to the Congressional Research Service, the number is 6,640 miles (10,690 kilometers). But if you consult the National Oceanic and Atmospheric Administration (NOAA), the coastal edges of the state total 33,904 miles (54,563 km).

Yep, that's a big difference, all right. Not to brag, but I knew the answer. Still, I want to say that, obviously, the former number is incorrect, because Congress always lies.

The coastline paradox occurs because coasts are not straight lines, and this makes them difficult, or impossible, to measure definitively. From an aircraft, you can see that the coast has many features, including bays, inlets, rocks and islands. And the closer you look, the more nooks and crannies you'll find.

Oh, now I remember why I saved it. It's related to fractals like the Julia set or Mandelbrot set, which involve complex numbers, and, well... you know.

As a result, the length of a coastline depends on the size of the ruler you use.

This isn't just a coastline issue. Lots of survey boundaries follow the thread of a river (or, in the case of VA/MD, the low-tide shoreline of the Potomac), which has similar characteristics.

But if you used a smaller ruler, you'll capture more complexity, resulting in a longer measurement. Hence, a paradox.

Okay, I suppose, for some definitions of paradox. Regardless, that's what it's called, so I'll run with it.

According to work published in 1961, English mathematician Lewis Fry Richardson noted how different countries had different lengths for the same shared border because of differences in scales of measurement. In 1967, mathematician Benoit Mandelbrot expanded on Richardson's work, writing a classic Science paper on the length of Britain's coastline. This later led him to discover and conceptualize the shape of fractals, a curve that has increased complexity the more you zoom in.

This is also related to why no one can agree whether China or the US is the third-largest country by area: it depends how you measure some of the boundaries, including the coasts. Also, vertical differences get thrown in; this is the same fractal problem, only in two dimensions (surface), not one (boundary line).

The concept of "dimensions" also gets modified when you're dealing with fractals; you can get fractional dimensions. Which, it should come as no surprise, gave fractals their name.

I object, however, to the idea of "increased complexity the more you zoom in." I'd argue that you get the same complexity, just at different scales. I guess the sentence can be read like zooming in reveals greater complexity.

The article also features some nice Mandelbrot set zoom animations, which I always find fascinating.

This can hold true for coastlines. You could technically measure a coastline down to the grain of sand or atomic level or smaller, which would mean a coastline's length could be close to infinity, Sammler said.

Another nitpick: no such thing as "close to infinity." In the real world, as opposed to a purely mathematical construct, there's a minimum length (it's very, very small). That minimum length implies an upper bound to how long a fractal boundary can be. It can be very, very long... but that's still not infinity.

Coastlines are also shifting entities. Tides, coastal erosion and sea level rise all contribute to the fluctuating state of coastlines. So maps from the 1900s, or even satellite imagery from a few years ago, may not resemble what coastlines really are today.

And if you want to get really technical, it changes from moment to moment, as portions are eroded and others built up. Not to mention general changes in sea level. As I've noted before, you can't step into the same river once.

So how much coastline does Alaska, the United States, or our entire planet, have? We may never know the accurate number. It's a paradox, and like many things in nature, escapes our ability to define it.

Which shouldn't mean we throw up our hands and give up. I like to think of it as a metaphor for life itself: always approaching an answer, never quite getting there. But learning more and more along the way.

3 Like • Comment • View 4 comments

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