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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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| For the record, the only times I've ever urged someone to smile was when I was doing photography. The smile: a history  How our toothy modern smile was invented by a confluence of French dentistry and Parisian portrait-painting in the 1780s In part, this is because people have pushed back on it. Telling strangers to smile is rude, patronizing, and entitled. But even before I realized that, I never prompted anyone to smile (outside of a photography setting) because I can't smile. The smile is the most easily recognised facial expression at a distance in human interactions. Oh, I dunno about that. Red-faced rage might be slightly easier to spot. As well as being easy to make and to recognise, the smile is also highly versatile. It may denote sensory pleasure and delight, gaiety and amusement, satisfaction, contentment, affection, seduction, relief, stress, nervousness, annoyance, anger, shame, aggression, fear and contempt. You know, the word "fuck" does all that and more, and it's much, much easier to produce than a smile. The smile comes easy to human beings. Well, I guess that proves it, then: I'm not a human being. I mean, it's not that I can't turn up the corners of my mouth; it's that people expect to see teeth (which when you really think about it is weird), and the only way I can show my teeth is if I draw my lips back in a rictus of agony—which will never be mistaken for a smile. The smile may even predate the human species. Many great apes are known to produce them, suggesting that the smile first appeared on the face of a common ancestor well before the existence of Homo sapiens. When other animals show their teeth, it means something very different, usually. And I'd be very wary of attributing the same underlying emotion to the smile on a nonhuman animal, even that of a chimp. The smile has always been with us then, and it would appear it’s always been the same. It seems only one step further to claim that the smile has no history. But this would be far from the truth. In fact, the smile has a fascinating, if much-neglected past. I'm deferring to the author who seems to be an expert on this facial expression (but why?), although it's been my understanding for a long time that smiles can mean different things in different cultures. So this article has a decidedly European bent. The ubiquity and polyvalence of the smile means that, in social circumstances, for example, it is not enough to see someone smiling. One has to know what the smile intends. The expression needs untangling, deciphering, decoding. In this, it resembles the wink. As the anthropologist Clifford Geertz pointed out in 1973, the wink is physiologically identical to the involuntary eyelid twitch we call a blink. I'm not exactly well-versed in decoding peoples' facial expressions, but what do you mean, a wink is the same as a blink? A wink is generally performed by one eye. Blinks happen in both simultaneously (excluding certain medical conditions). They feel different to execute. Try it. Maybe I'm just weird, like with being unable to grin, but to me they're vastly different despite using the same body part. In the West, we tend to acknowledge the variability of codes in terms of space and diversity: there is a sense that Western smiling culture differs from that to be found, for example, in Japanese and Chinese societies. Okay, so the author does acknowledge this. The article goes on to link the etymology of words meaning "smile" in various languages to the evolution of the meaning of the smile. It's all quite fascinating, but doesn't lend itself well to copy-pasting. Yet if the smile was alive and well in Western culture, it was not yet our own. In Western art, it differed in one highly significant respect: the smiling mouth was almost always closed. Teeth appear in facial representations extremely rarely. Likely because dentistry wasn't a thing. Or maybe I'm a genetic throwback to those people. It is tempting to ascribe this state of affairs to the unhygienic state of mouth. But, in fact, skeletal remains from late medieval cemeteries suggest that teeth were then less affected by cavities than they would become from the 18th century onwards, with the mass advent of sugar into Western diets. Okay, yeah, he owned me there. Three factors operated to minimise representation of the expression. First, there was a close association between the open mouth and the lower orders. We still call them "mouth-breathers," for reasons not entirely clear to me. Though I think the term is meant to apply to people of limited mental capacity, to the elite, that means everyone else. Or – and this was the second factor in play, in art as in life – it betrayed a loss of reason. The mouth lolling open was an accepted way of depicting the insane, but it went further than that, and encompassed the representation of individuals whose rational faculties had been placed in abeyance, by passions or base appetites. Heresy. My base appetites are completely rational. The third factor explaining the absence of positive depictions of open mouths in Western art related to what were known as ‘history paintings’ depicting scenes from ancient history or scripture. Props to this guy for distinguishing "history" from "scripture." This part gets a little complicated, and I won't copy all his arguments here. For Le Brun, it followed that, when the soul was calm and tranquil, the face was perfectly at rest. This Le Brun person obviously never heard of "resting bitch face." In late 18th-century Paris, a new phenomenon had marked its arrival in Western culture, transgressing all the norms and conventions of Western art. The modern smile was born. After which Parisians completely forgot how to smile. That's a joke. In the article, the author connects the visible-tooth smile to La Révolution. Somehow. Seriously, just read it. I'm not convinced, but it's a new take on the subject, at least for me. The virtuous and transcendent smile showcasing healthy white teeth in the novels became a model for the Parisian social elite in real life. It became not only acceptable but even desirable to manifest one’s natural feelings among one’s peers. English travellers expressed amazement at how frequently Parisians exchanged smiles in everyday encounters. The city had become the world’s smile capital. Wait, they smiled even upon spotting an English person? If the cult of sensibility gave novel readers the wish to smile in this fashionable way, Parisians were also lucky enough to have technical assistance at hand. The French capital had become a renowned centre for dental hygiene. And now we're back to dentistry. The new dentists could clean, whiten, align, fill, replace and even transplant teeth so as to produce a mouth that was cleaner, healthier and – in smiling – attractive. I feel obliged to point out that all of this is set in the late 18th, maybe early 19th, centuries—and dental anesthesia wasn't invented until the mid-19th. I guess they're right when they say "fashion is pain." (Do they say that? I say that.)The smile went into hibernation as a public gesture in the West for more than a century... As anyone with family photograph albums going back that far will discover, it was from the 1920s and ’30s that smiles appear for the first time – precisely the period when individuals began to say ‘cheese’ when confronted with a camera. Portraiture had become democratised – and smilified. Now, reviving the smile for photographic portraiture—that's a hypothesis I can get behind. Incidentally, never tell someone, even if they're French, to say "fromage" instead of "cheese." Their expression makes it look like they've just caught a whiff of some Limburger. From the turn of the 21st century, iPhone photography and social media confirmed the preferred individual expression of social identity was through smiling. Its worth it to note that, before emoji, we had to use ASCII characters to emulate expressions. I actually preferred those simpler times. But the point is, as far as I know, the first documented ASCII art that found its way into proto-internet messaging was the colon-dash-parenthesis: the "smiley face."In 2019, the long march of the modern (Western) smile received a severe jolt, with the appearance of COVID-19. Suddenly, that expression retreated behind a surgical mask. It is true that the perceptive among us may have been aware that a genuine and sincere smile causes a detectable crinkling of muscles around the eyes. Which is one reason I like masks: I don't have to look at peoples' teeth. My own preferred face mask is a parody of the smile: the lower part of a cat's face, with the mouth open wide, showing pointy teeth and raspy tongue. It suits me, and I will continue to wear it, because I get compliments on it—and I have never, ever gotten a compliment on my actual face. Positive reinforcement: it works. I am fortunate in that I've never had to work in a place where one is forced to smile. To me, that would be actual hell. |
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