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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November 30, 2022 at 12:01am

#1041155

A Bridge Too Far

This article is a few years old now, but that's a rounding error compared to the subject matter.

500 Years Later, MIT Proves That Leonardo Da Vinci's Bridge Design Works
If accepted at the time, the design would have likely revolutionized architecture.

More like engineering.

Researchers at MIT have proven Leonardo da Vinci correct yet again, this time involving his design for what would have been at the time a revolutionary bridge design.

Leonardo was undeniably a genius (though, as with anyone, not always right), but one limitation on genius is the mindset of the people around you.

When Sultan Bayezid II of the Ottoman Empire put out a request for proposals for a bridge connecting capital city Constantinople (now Istanbul) with its neighbor city Galata, da Vinci was eager for the chance to win the contract.

I'm just leaving this here so you can take the time to get They Might Be Giants out of your head. It is not possible to contemplate Istanbul (not Constantinople) without thinking of their song.

Da Vinci's proposal was radically different than the standard bridge at the time. As described by the MIT group, it was approximately 918 feet long (218 meters, though neither system of measurement had been developed yet) and would have consisted of a flattened arch "tall enough to allow a sailboat to pass underneath with its mast in place...but that would cross the wide span with a single enormous arch," according to an MIT press statement. It would have been the longest bridge in the world at the time by a significant measure, using an unheard of style of design.

And see, that's why I'm putting this in the realm of engineering, not architecture. I admit I may be biased on those subjects, but bridge design is solidly in the realm of civil engineering, no matter how elegant the design may be.

There is, of course, significant overlap in those disciplines. But if the focus of the construction is structure and transportation, I'd call that civil engineering.

It wasn't just length or style that set da Vinci's bridge apart. It also had safety features unheard of at the time. One of the biggest challenges facing any bridge design is that it has to exist in nature no matter the conditions, including wind.

In theory. In practice, lots of things have brought bridges down, including unexpected floods and, yes, wind loads.

Strong winds have forced many bridge, including relatively modern bridges from the 20th century, into lateral oscillations leading to collapse.

I don't think it's possible to become a civil engineer without seeing the video of the Tacoma Narrows Bridge

failure.

To be clear, though, that collapse was due to aerodynamic effects that were barely understood even in the mid-20th century, and as smart as Leonardo was, the math for it didn't exist in his time.

Since building a full-scale bridge would have been unwieldily,[sic] the team resorted to building a model. Using 126 blocks, they built the bridge at a scale of 1 to 500, making it around three feet long.

Modeling things like this properly is a challenge in itself. You run into things like the square-cube law, which has to be taken into account.

"It's the power of geometry" that makes it work, she says. "This is a strong concept. It was well thought out." Further tests showed that the bridge could have even stood its own against earthquakes to an extent far beyond other bridges at the time.

Math: it works.

There are still mysteries surrounding the project. "Was this sketch just freehanded, something he did in 50 seconds, or is it something he really sat down and thought deeply about? It's difficult to know."

It's entirely possible that the sketch built on things Leonardo would have already been thinking about. I mean, it's basically a freestanding arch, right? They figured arches out long before his time. Putting that together with other concepts, such as the wind loads mentioned above, might not have taken him very long at all (genius, remember).

It's this combination of disparate ideas that's the hallmark of true genius, and it's one reason there's no such thing as useless knowledge.

While it's difficult to know da Vinci's intentions, one thing is now relatively certain: the bridge would have worked.

And I gotta admit, it looks cool.

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