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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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One thing I insisted upon during trip planning was to visit Brouwerij Bosteels.  This is an ancient brewery which creates two of my favorite Belgian beers: Tripel Karmeliet and Pauwel Kwak. (The link there goes to the English website.)The difficulty: the brewery is located in a small village called Buggenhout, kind of between Brussels and Antwerp. And I'm still in Bruges, to the northwest. Yes, Belgium is a relatively small country, but it's not that small (about the size of Maryland, though more compact), so trains were involved. This meant that I had to walk the mile (roughly 1600 meters) from the hotel back to the train station and, yes, my nemeses the cobblestones were heavily involved, though at least I wasn't rolling luggage. From there, I caught a train directly to Ghent (or Gent), then switched trains at a heavily-under-construction station to a stoptrein (which I gather is the equivalent of a "local" in the US) going through Buggenhout. All while being mostly unable to read signs or understand announcements. While this is part of the adventure of traveling, I was on a schedule; I had to be at the brewery by 1:00pm (or 13:00 in the superior Euro style). I managed. But then I had to walk again, this time about half a kilometer, to the gigantic brewery, and here, Google failed me in a way that signs in Dutch did not, giving me the wrong location of the entrance. As the brewery is, as I noted, gigantic, this meant walking around an enormous block to find the tour location and, yes, there was a preponderance of cobblestones. I found it, albeit a few minutes late, but they let me in anyway. After a quick taste of a beer, the guide (who resembled Mirror Universe Jean-Luc Picard, in that he was bald but had a goatee) took us around the brewery operations, which, it being Saturday, weren't active at the time. This is a good thing, because we didn't get in anyone's way and I could actually hear the guide because the machines weren't operating. Point is, you know the Wonka movie? Yeah, that was me gawking at this enormous brewing operation. Now, my appreciation for Bosteels wavered a bit a few years ago, when AB/InBev bought them. (AB/InBev is an enormous Belgian company, but it's best known for its first initials, which stand for Anheuser-Busch.) But they didn't change the beer in any noticeable way. I'd hoped, because of the big corporation's greater reach and distribution network, that this meant we would be getting more Kwak in the US, but I still can only rarely find it in Virginia. I see Tripel Karmeliet quite often, though, so that's okay. Interesting thing about Kwak. Each Belgian beer has its own dedicated glassware for serving; this is one of the delightful aspects of Belgian beer. Most of them are variations on a theme, though; something resembling a goblet or Holy Grail, or perhaps a pilsner glass, or, you know, another kind of glass you'd expect to find beer in. Kwak glassware is unique, resembling instead a test tube on a wooden stand. Today's photograph, below, is of a serving of Kwak, so you can see what I'm talking about. I possess two such glasses back home, because on those rare occasions that I do find the beer near me, it's imperative that I use the proper receptacle, and there are two of them because, sometimes, I share with friends. The funny thing is that a bartender here told me that the glassware, being so different, tends to get stolen from bars. So the bars here, he said, require that you put up one of your shoes as collateral to ensure they get the glass and stand back when it's empty. This would definitely be a deterrent for me because, well... cobblestones. I didn't have to get one of my shoes taken hostage at this brewery, but they did let me have a Kwak, and yes, it is much better to drink it at the source. Between the opening beer, a serving of Tripel Karmeliet, and the Kwak, I was pretty far gone by the time the tour/tasting ended (I'm no lightweight, but Belgian beer tends to be high-octane), and I had to retrace my steps, and the trains, back to the hotel. Easier said than done, because I was still kind of out of it when I got off at Bruges, and had to walk the mile back to the hotel. Along the way, I gave up and had dinner (and a beer) at one of the many sidewalk bars in the old town. All of which is to say: Mission Accomplished. I only have a few days left here, and rumor has it there's a hurricane (yes, a hurricane, though "only" of tropical storm intensity) coming, but I've now done what I most wanted to do. As promised, a picture of how Pauwel Kwak is served (bonus: Mirror Universe Picard):  |
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