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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Elycia Lee ☮

about 5 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 21 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 23 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

March 4, 2020 at 12:20am

#977034

Laziness is the Milkman

One of the things that we tend to take for granted -- if we think about it at all, that is -- is the ability of mathematics to model the natural world.

I don't really understand advanced math or physics, but this sort of thing is endlessly fascinating to me. I chose the word "endlessly" deliberately, there; today's link involves infinite, non-periodic patterns.

http://nautil.us/issue/69/patterns/impossible-cookware-and-other-triumphs-of-the...

Impossible Cookware and Other Triumphs of the Penrose Tile
Infinite patterns that never repeat have moved from fantasy to reality.

I first found out about Penrose tiles back in the 80s. I always wanted to cover a floor with them, but I'm entirely too lazy to do the work involved.

In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. In 1982, Daniel Shechtman, an Israeli crystallographer, discovered a metallic alloy whose atoms were organized unlike anything ever observed in materials science. Penrose garnered public renown on a scale rarely seen in mathematics. Shechtman won the Nobel Prize. Both scientists defied human intuition and changed our basic understanding of nature’s design, revealing how infinite variation could emerge within a highly ordered environment.

Sometimes, scientists discover something and have to figure out the math. Other times, the pure math comes first, followed by related discoveries. The latter happens quite a lot.

Penrose turned to five-axis symmetry, the pentagon, to create his plane of non-repeating patterns, in part, he has said, because pentagons “are just nice to look at.” What was remarkable about Penrose tiles was that even though he derived his tiles from the lines and angles of pentagons, his shapes left no awkward gaps. They snugged together perfectly, twisting and turning across the plane, always coming close to repetition, but never quite getting there.

The idea that complexity can arise from simplicity is something I've been interested in for a long time. It's right there in the blog intro.

The linked article goes on to explain, in words, what Penrose tiling is. But it helps to see more examples. Here is a more visual explanation.

In fact, Penrose tiles bridged the golden ratio, the math we invent, and the math in the world around us.

As much as I appreciate Nautilus, the promise of the headline is never realized in the text I linked above. It's only mentioned as an aside:

Their irregular atomic configuration, for instance, gives them a low surface energy, which means that not much sticks to them. As a result, quasicrystal coatings have found their way into nonstick cookware. (When Penrose created his novel tiles, there was no reason to think it would have a bearing on crystallography, let alone on frying an egg.)

The point being that all of this stuff seems really esoteric, until you realize that it has real-world, practical applications -- including one of my favorites, an invention that keeps me from having to do too much work.

They say necessity is the mother of invention. I say laziness is the milkman.

4 Like • Comment • View 2 comments

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