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 |    Not for the faint of art. |
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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. ![Merit Badge in Quill Award
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| Just in case anyone still doubts the value of science, here's incontrovertible proof of it, from Physics magazine: Cooking Flawless Pasta  Scientists have pinpointed energy-efficient ways to cook al dente pasta and developed an infallible recipe for the perfect cacio e pepe sauce. I would only object to the adjective "infallible" there. No matter how refined the recipe, someone will find a way to muck it up. A bowl of steaming hot pasta covered in your favorite sauce and dusted with a healthy dose of parmesan cheese comes high on the list of ultimate comfort foods. I don't believe in the concept of "comfort food." If your food doesn't make you feel good, re-evaluate your life choices. But cooking that pasta to perfection can be more difficult than seemingly simple recipes imply. Recipes can never tell the full story. For one thing, pasta texture is dependent on things like water hardness, a term that must confuse the living hell out of non-native English speakers. There are also variables such as elevation; it's pretty well-known that water boils at a lower temperature on mountains due to decreased atmospheric pressure.So one needs to understand one's local environment and modify recipes accordingly. In one study, Phillip Toultchinski and Thomas Vilgis of the Max Planck Institute for Polymer Research, Germany, studied whether perfectly al dente spaghetti could be prepared in a more energy-efficient way. Yes, "perfectly" is also a subjective concept, but as some of these studies note, you can decide on optimal, measurable, objective parameters and test to determine what process results in a product closest to those parameters. In a second study, Matteo Ciarchi and Daniel Busiello of the Max Planck Institute for the Physics of Complex Systems, Germany, Giacomo Bartolucci of the University of Barcelona, Spain, and colleagues developed a recipe for making perfect cacio e pepe, a three-ingredient cheese sauce that is surprisingly easy to mess up. Just how many Max Planck Institutes are there, anyway? There were other physicists, you know. Also, please take note that none of these researchers did their work in Italy. If I were writing the grant proposal, I'd leave room for a lengthy research trip to the country that's known for its pasta. You know, for science. Judging solely by the names, though, at least some of these researchers might have been Italian. “It is very difficult to make this sauce,” says Busiello. “You are almost always doomed to fail.” With that attitude, though, maybe Russian. The study by Toultchinski and Vilgis was inspired by a brouhaha over a 2022 Facebook post by physics Nobel laureate Giorgio Parisi. In that post, Parisi suggested that chefs could reduce the energy needed for cooking pasta using a “heat-off-lid-on” method. Oooh, yeah, that might stir up a good bit of anger. But chefs questioned whether this method could achieve al dente pasta—pasta that is soft on the outside and crunchy at its core. Well, chefs, all you had to do was try it. The article goes on to describe how the researchers themselves tried it, and the results. Vilgis says that while these results show that pasta can be cooked in a more energy-efficient way, when it comes to taste and texture, there is no substitute for the tried-and-tested method. “If you want perfect al dente pasta, you have to cook it the traditional way,” he says. Sometimes, science shows that we've been doing something wrong all this time. It's designed to test assumptions we've made from intuition, tradition, and the self-contradictory "common sense." But, sometimes, it turns out the other way; that the traditional method is, after all, the best. And that's good, too. The second pasta study by Ciarchi, Busiello, Bartolucci, and colleagues focused on how to make a popular but tricky sauce called cacio e pepe. For this sauce, pecorino cheese is mixed with pasta water and black pepper to make a glossy emulsion that coats the pasta in cheesy goodness. Call me ignorant, but I had no idea "cheesy goodness" was a scientific term. As before, methods and results are provided; I'm skipping over a bit. “The team came up with a really practical recipe to getting the perfect sauce every time,” Fairhurst says. He adds that these kinds of studies—where physicists apply their knowledge to food problems—can really help consumers engage in science. “It’s everyday science you can do in the kitchen; you can’t do that with particle physics problems.” Well, now, I suppose that depends on your kitchen, doesn't it? |
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