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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.
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As sometimes happens, we get two Cracked links in a row. This one deals with two of my favorite subjects: science and food.
Not that I'm a big fan of breakfast cereal. I don't use milk enough to keep any in the house, so I mostly just eat it when I'm at a cheap motel whose idea of a free breakfast is a few stale muffins and some prepackaged cereal, along with some milk kept in questionable conditions.
Nor does this article delve into the interesting history of mass-produced breakfast cereals. It's a fascinating combination of such core American values such as puritanism, processed food, too much sugar, marketing, convenience, and profits. Maybe another time.
4. Derivatives of Position and the Rice Krispies Gnomes
Gnomes? Huh. For some reason, I always thought they were elves. I guess Keebler had the monopoly on the latter. And yes, there's a difference; ask any FRPG player and they'll tell you. At length.
Derivatives of position are a complex series of metrics within physics to allow for as accurate a description of movement as possible...
And half the readers just closed their browsers in abject terror.
It's not all that complex, really. First derivative of position is velocity; second is acceleration. I've mentioned this before. Those are things we're all familiar with on a daily basis. If you're pulling away from a stoplight, your car has position, velocity, and acceleration. Where things get wonky is that acceleration can be uneven, as well, so you get the next derivative, called jerk.
It’s the fourth, fifth and sixth that start to get breakfasty — they’re called snap, crackle and pop. These measurements — strictly speaking, “derivatives of the position vector with respect to time” — owe their names to someone simply trying to make themselves laugh.
Scientists have senses of humor, too. Why else would they call the bony spiky thing at the end of certain dinosaurs' tails a thagomizer? 
3. The Complexities of Fluid Dynamics and the Cheerios Effect
Cheerios are inherently interesting because they're tiny toroids. I call them bagel seeds.
You know how, in a bowl with only a few Cheerios left, they tend to group together? Physicists have noticed this, naming it the “Cheerios effect,” and investigated its potential ramifications within the field of fluid dynamics.
Huh. I always figured that was related to surface tension and/or slight imbalances in electrostatic charge. But I never cared enough to investigate; after all, having them clump together is a feature at breakfast, not a bug.
“The Cheerios effect arises from the interaction of gravity and surface tension — the tendency of molecules on the surface of a liquid to stick together, forming a thin film across the surface. Small objects like Cheerios aren’t heavy enough to break the surface tension of milk, so they float. Their weight, however, does create a small dent in the surface film. When one Cheerio dent gets close enough to another, they fall into each other, merging their dents and eventually forming clusters on the milk's surface.”
I was half right. Yay.
2. The Tomography of Why the Big Pieces Go to the Top
This never struck me as all that puzzling. Not if you invert the assertion: that the small pieces go to the bottom. You get breakage, and smaller pieces are more likely to slip through the voids. But again, I'm not actually a scientist.
We’ve all been there — you pull the inner bag out of a new box of cereal to open it, and all the raisins (if you’re healthy) or marshmallows (if you’re not) are up at the top, like the head on a glass of beer but more infuriating.
I think some people would call that a feature, not a bug, perhaps because all they can see is the immediate effect (more marshmallows, yay!) and don't think about the future (a bleak one indeed, featuring all the boring parts of the Lucky Charms).
It's kind of like those perverts who don't shake the orange juice and get nice smooth beverage, leaving extra pulp for the next family member.
“This will allow us to better design industrial equipment to minimize size segregation thus leading to more uniform mixtures. This is critical to many industries, for instance ensuring an even distribution of active ingredients in medicinal tablets, but also in food processing, mining and construction.”
See? It's not just a theoretical consideration.
1. Compressing Cocoa Puffs to Save Future Skiers
This is really irrelevant, but I never liked Cocoa Puffs. Or anything described as "chocolatey" instead of "chocolate."
They saw vastly more complexity in how the cereal was deformed (a science word, not an offensive one) than anyone was expecting — three different types of velocity-dependent deformation and a propagating compaction band recorded visually for the first time in granular matter.
There goes another half of readers.
They also tested Cocoa Puffs and Cocoa Krispies in the compressor, to see what difference chocolate made. It takes more pressure to crush some cereal if it has a chocolate coating.
It. Is. Not. Chocolate.
Maybe one day, science will be able to tell the difference the way my taste buds can. |
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