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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.
October 31, 2007 at 8:04pm October 31, 2007 at 8:04pm
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I've been getting the distinct impression that some of my latest blog entries have been met with something less than enthusiasm.
That's okay. Part of my late trend toward sporadic posting was the result of some self-examination - I do that sometimes, perhaps to a fault, but another thing I figured out was that I shouldn't be making apologies to anyone for what I am. For what I do, sometimes, sure, but to quote God and Popeye, "I am what I am."
I have a tendency to write to an audience. The "audience" here is the people who actively comment. I can see parts of you, there, glittering in the darkness like so many stars. But as the Universe seems to be composed primarily of dark matter, unseen and poorly understood, so is my audience, and all I can see is the occasional twinkling point.
It was easy for me to forget that I started journaling not to please an audience, but to - simply - write. I didn't start out to be funny, or philosophical, or provide links to fun diversions offsite. I found myself censoring myself, keeping back the most meaningful thoughts in fear that I would lose what little light I had.
But to what end? I don't have to have the most popular blog on the site. And I shouldn't be craving approval - hell, that's what I have Minions for. If I can't say something outright, well, I should be good enough at the craft of writing to talk around it and at least convey something about whatever's going on in my life.
So October comes to an end today. I almost said, "at last," but this month has flown by like a stealth jet in the night.
There's a tradition, an observance, I follow every year in October. I re-read A Night in the Lonesome October by Roger Zelazny. Each time, I find something else about it to like. Zelazny was one of my favorite authors in general; he died prematurely in 1994, October having been his last completed work (the book is also credited to macabre artist Gahan Wilson, who provided copious illustrations for it).
The book is narrated by a dog. The dog, Snuff, is the companion of Jack the Ripper, whose task it is to preserve the planet by preventing the forces of evil from opening a Door to the realm of the Elder Gods. The narrative borrows from all kinds of literary sources, including Frankenstein and Dracula as well as Sherlock Holmes and, obviously, the works of Lovecraft. It's set in late 19th century London, with all the atmosphere that implies. It's about the nature of life, love, friendship, and redemption; and it's funny and macabre and sad and uplifting all at the same time.
Just what you need in October. So, yes, that's where I took my handle from this month: Waltz in the Lonesome October.
On Halloween or, if you will, Samhain, in addition to dressing up in scary or silly costumes and making your dentist happy, do try to take a moment to reflect on what has gone before. Much has been made of the observances of this season concerning evil spirits, the last harvest, and, more recently, costuming and trick-or-treating, but one of the lesser-known aspects of Samhain was its function as a time of reflection. Those who died in the previous year are remembered and, also, those who are no longer with us, who made an impression upon us. Your grandmother, perhaps. A favorite teacher. The dog you had when you were a kid. A favorite author who left pieces of himself behind when he departed. Perhaps even your younger self, that part of you that may not have known as much, but had a particular wisdom that it might be good to recapture.
The leaves fall, the wind blows them across the deserted path. You take a minute to look back, your hair streaming past your face as you consider what was, and that which has brought you to this place. You can't go back there; you can only stand and watch the leaves slowly obscure the path - but you take a moment to look, to remember... to observe. And then, resolutely, you turn and face forward, into the biting wind, perhaps pulling your coat closer around you as you shiver from more than just the cold.
Blessed Samhain. May the Ancestors grant you all wisdom, and may you find what you seek in the coming cycle.
Such times are rare, such times are fleeting, but always bright when caught,
measured, hung, and later regarded in times of adversity,
there in the kinder halls of memory, against the flapping of the flames.
-Roger Zelazny,
A Night in the Lonesome October |
October 30, 2007 at 6:29pm October 30, 2007 at 6:29pm
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I had a meeting at an architect's office to finish off my work day. Consequently, I took a different way home than usual.
A few blocks from my street, I came upon a squirrel struggling in the middle of the other lane. It was sliding around, propelled by its front legs, leaving a trail of blood. I figured someone had hit it - squirrels around here are suicidal once the first frost hits; either they're frantically trying to score a last few nuts for their stash, or they've given it up as a lost cause and started playing chicken with cars just to end it all before they starve to death.
This one was still barely alive, obviously. A white Honda had stopped before it, in the oncoming lane. I slowed down on general principles but kept going.
I've encountered badly damaged squirrels before. One of them once dragged its ass (literally) into my front yard and collapsed there, still breathing but unable to move further. In that case, I'd grabbed a shovel and Kevorkianed the poor thing with a quick strike to the neck, then used the same shovel to give it a decent burial under my dogwood tree.
Once, when I was driving home along yet another route, I passed by a kitten struggling similarly in the street, its hindquarters immobile. It was a couple miles from my house. Someone was kneeling by it, her hand on its head. As I came closer, I saw that the someone was my wife. I turned around illegally and parked. By that time a cop had come by and was asking whose cat it was. Some of the neighbors had come out and were watching, concerned. No one admitted to belonging to the cat, though one of the neighbors, who was obviously mentally challenged in some way, said he'd seen the cat around.
My wife took matters into her own hands and scooped up the cat, who in its agony tried to bite her. She'd called the vet and they agreed to stay open until she got there - about a mile and a half from the incident. Depositing the injured kitty into a crate from her trunk, she slid it into her passenger seat and drove off, one hand on the cat's head, regardless of any further bites.
But the cat didn't bite her; nor would it bite anyone ever again. She died on the way to the vet, her last sensation a human hand caressing her head.
We've all passed dead animals in the road, I know - some more recent than others. Dogs, cats, rabbits, opossums, squirrels... once I saw a human I thought was dead, but upon summoning the appropriate authorities, it turned out he was merely dead drunk. Usually, I mutter something like, "Poor kitty," hoping that its end was swift, which is as close as I get to prayer, though I know it's very often not - as with my squirrel today.
The Honda backed up, I saw in my sideview mirror, and deliberately ran over the doomed squirrel's head, performing a crude, but presumably effective, euthanasia.
Halloween - Samhain - is traditionally a time to reflect on the end part of the cycle of life, that which binds all of us living things together whether we know it or not. Some living things, notably humans, are aware of this part of life. Morbidly aware. It permeates us. Without death or, more precisely, without the knowledge of our own and our loved ones' inescapable deaths, we wouldn't be what we are. We couldn't love as deeply as we do. We couldn't hate as passionately as we do. We couldn't laugh as heartily as we do, for laughter is essentially a way to ease the pain of mortality. And we are all aware that in order for us to live, something has to die - and this is as true for vegetarians as it is for us carnivores, for the life of a carrot is no less real than the life of a squirrel, or a kitten, or a baby.
There's a book by Samuel Delany called Dhalgren. I read it when I was entirely too young to read it. It's a massive allegory about life, death and what happens in between - and about a bunch of other stuff that I didn't comprehend then and barely comprehend now. The main character, who I don't remember being called anything but "The Kid," occasionally has the thought pop into his mind: "Someday, I will die."
Someday, I will die.
In the meantime, I'm damn well going to live. |
October 30, 2007 at 3:31pm October 30, 2007 at 3:31pm
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5 Scientific Reasons a Zombie Apocalypse Could Actually Happen
We found out recently that if you try to leave a little kid in a graveyard late at night, he'll freak out. Even if you offer to leave him a gun to protect himself. Why? It's because on some instinctual level, all humans know it's just a matter of time until the zombies show up.
http://www.cracked.com/article_15643_5-scientific-reasons-zombie-apocalypse-coul... |
October 29, 2007 at 9:49pm October 29, 2007 at 9:49pm
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In Virginia, the "Fall Line" is a fairly well-defined boundary between tidal and upland rivers and streams. I grew up spent my childhood very close to the Fall Line, on the tidal side. The town of Falmouth, in the county where I (and George Washington) lived, is derived from this geographical feature: literally, the "fall mouth" of the Rappahannock. The line roughly follows U.S. Route 1 through most of the state.
The historical date of first frost in most of the non-mountainous regions of Virginia is October 15 - that's what they tell you in almanacs and such. At one point, this had real significance, as much of Virgina was given over to agriculture. Now, the only real effect it has on most people is in helping to choose clothes to wear, and preparing those of us too poor, cheap or stubborn to have a garage to scrape frost off our windshields in the morning.
Today, October 29, was the first day I had to scrape frost off my pickup's windshield. There wasn't much of it, and it came away easily, but it reminded me that we're actually in Fall. You wouldn't know it from the temperatures around here lately, which have alternated between sweltering and cool/rainy for most of the month.
In short, it hasn't felt much like October.
I was never a big fan of Fall - or Winter, for that matter; Fall being little more to me than a precursor of Winter. When I was a kid, living on a farm where the date of first frost still had some real meaning, Fall meant two things: going back to a school I didn't much like, and being cold all the damn time.
My dad, who grew up (and in his case, I use the term precisely) during the Great Depression, was a cheap son of a bitch, especially when it came to climate control. Heating was kept to a bare minimum, and consisted primarily of wood stoves in the basement supplemented by oil heat set in the low 60s F. This in and of itself wouldn't be so bad, but Dad cannily kept the thermostat in the hottest part of the house, tricking it into thinking it was a lot warmer in the house than it actually was.
And then at night he'd drop the fucker to the mid-fifties.
My chief memory of childhood winter mornings consists of him trying to wake me up ("Rise and shine!" "Pick one!") and, when that finally succeeded, of me sitting in front of the heater vent, curled up into a little ball as the first blast of mildly warm air resulting from his setting the thermostat back to a balmy 62 shook off my overnight chills.
Dad wasn't poor, mind you. Just cheap. Paid for my college education in full, however, but as a 9 year old kid huddling near a stingy register, I couldn't have appreciated that.
I hated fall. I especially hated winter, except on those occasions when the snows would cancel school and I got to stay home, alternately playing in the snow and draping my exhausted, frozen self over the wood stove in the basement.
When I was much younger, he'd wrap me in a blanket and take me out to look at the stars. He had a sailor's knowledge of the constellations and the twinkling lighthouses in the sky - mainly because he was, after all, a sailor, but partly because he loved knowledge for the sake of knowledge. Never could pronounce Betelgeuse correctly, but could find it in a heartbeat, the orange beacon in the shoulder of Orion.
Orion became, to me, the harbinger of winter. I'd see its familiar asterism shining palely over the river in the early evening, and brace myself for quality time with the heat register in the coming mornings. The Hunter's belt became, to me, the Fall Line.
It will still be a while before I'm as old as my dad was when he took me out on those cold, winter nights to view the stars that guided him and serve me as a rich source of metaphor. But somewhen between then and now, I ceased to love summer and began to appreciate the fall and winter. Partly, I know, this is because I'm not as cheap as my dad was, and we keep our house at a civilized temperature year-round. In part, though, it's because the memory of my first glimpse of the stars is covered with a skin of frost, wrapped in a blanket of warmth and mellowed by a sailor's patient lessons. |
October 29, 2007 at 12:52pm October 29, 2007 at 12:52pm October 28, 2007 at 12:20pm October 28, 2007 at 12:20pm
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I just got done putting together the Comedy newsletter for this week. It comes out on Halloween, so I chose "Putting the Trick back in Trick or Treat" as the theme. If you're not a Comedy subscriber - which, if you're reading this, I fail to see how you wouldn't be - get on the bandwagon!
As for last night:
Ah, tequila. Drink of the gods. Unpronounceable gods, but still...
I learned more about tequila - and mezcal - last night than I had in the last mumble-mumble years. For instance: it's mezcal that has the worm, not tequila. Tequila never has a worm in the bottle. And, like the naming controls for French wine, tequila can only be produced in a limited area in Mexico - outside that area, it has to be called something else; mezcal, for instance. The flavor gradations and subtle aromas in agave-sourced drinks are almost as complex as those of wine. And there's basically three types of tequila: the cheap stuff (blanco), the good stuff (reposado, aged up to a year) and the expensive stuff (anejo, aged over a year). There is also apparently a super-expensive tequila named, unimaginatively, extra-anejo - a bottle of which can go for up to $225,000. No, I'm not kidding.
If all you're used to is Cuervo, you're missing out on a whole world of interesting flavor. |
October 27, 2007 at 5:38pm October 27, 2007 at 5:38pm
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Tonight is Tequila Tasting at a local pseudo-Mexican restaurant.
I can handle the pseudo- part because they're serious about their tequila. For example, Cuervo Gold is described as "ordinary; one step above the well" (the "well" being the parlance for cheap-ass drinks of any kind; you know, the kind that they serve you if you just order a "gin and tonic" without specifying the brand of gin).
Life, I say, is too short to drink cheap liquor, or smoke cheap cigars. And I do love tequila.
Kirstin and I have started talking about visiting England sometime either next year or the following. I want to do it before England gets so pissed off at the US that we'll get flogged at customs, and before global warming makes England colder than hell. Why global warming would have a cooling effect on Merrie Ol' I'll leave as an exercise for the reader (hint: it has to do with the Gulf Stream).
We also want to go to Budapest at some point, but at least with England we'll be able to communicate - to some extent - with the natives.
On a site note, the Genreflections contest is reopening next month:
Give it a shot.
And if anyone has any ideas for the Halloween edition of the Comedy newsletter, help me out - I'm somewhat at a loss.
...three tequila, FLOOR! |
October 25, 2007 at 5:06pm October 25, 2007 at 5:06pm
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Kids grow up so fast these days: One second you’re changing their diapers, the next you’re dropping them off at Sober Living by the Sea.
http://www.esquire.com/the-side/video/lunchbreak101907
Every once in a while, I find something that reinforces my decision to remain childfree. This is one such item.
Especially the second one down.
That particular video also reinforces my belief that being the only country to ever lose a nuclear war did something to their national psyche. Something... dark. It's as if all of the fears got concentrated to critical mass and got buried deep down, only to re-emerge over the next decades as a bizarre confabulation of tentacle porn and Hello Kitty.
(Now there's an idea.)
What's the half-life of this radioactive fallout? I don't know. And as a friend of mine pointed out, I'm sure they look at our tractor pulls and say similar things over there.
Don't get me wrong, now - I'm fascinated by the culture; I drive cars they engineered; I even love sushi. I'd like to learn the language and even visit. I have a burning desire to possess a genuine katana (doesn't even have to be made by Hattori Hanzō) that will probably never be fulfilled.
But still. |
October 24, 2007 at 3:38pm October 24, 2007 at 3:38pm October 22, 2007 at 7:16pm October 22, 2007 at 7:16pm October 19, 2007 at 1:21pm October 19, 2007 at 1:21pm October 13, 2007 at 5:23pm October 13, 2007 at 5:23pm
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Last Saturday was our fifth anniversary.
I like to make jokes like it feels like fifty, but the reality is the time's really flown by.
We went to Richmond - about an hour drive from here - and spent the day in Maymont Park. Maymont was the place where some idiot kid stuck her hand into the bear enclosure a while back and got it bit, resulting in a minor wound to the kid and the end of the bears' life because no one knew which bear bit her, or whether they had rabies or not. It seems someone's replaced the bears with new ones.
Then we went down to Shockoe Bottom, one of the oldest areas of Richmond and home of many great evening activities. After dinner (Havana 59 is awesome, if you're ever in Richmond) we did a walking tour of haunted sites in Shockoe. Now that was cool - didn't see any ghosts, of course, but the stories were fascinating. It's good to have an anniversary near Halloween so we get to do spooky shit like that.
And yeah, I haven't been around as much lately. Just been busy with other stuff, and not in any kind of mood to write - at least not for public consumption. But these things come in phases, and I'm sure I'll be around more soon. |
October 5, 2007 at 11:02am October 5, 2007 at 11:02am
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As I'm sure everyone had drummed into their heads, yesterday was the 50th anniversary of the launch of the first satellite, the Russian Sputnik I. Yesterday was also Poker Night. Combining these two observances, I felt it was necessary to invent a new alcoholic beverage and call it the Sputnik.
Sputnik
1 oz Stoli vodka (of course there has to be vodka involved somehow)
1 oz Shakka Apple (it's red. Get it? Red? Russia?)
1/2 oz Aftershock cinnamon schnapps (it's also red)
Splash grenadine (guess what color that is)
4 oz Club soda
Ice
Stick three toothpicks
| Yes, I know the Sputnik had four antennae, not three, but this is ART |
Trouble is, it's not a very tasty drink - but that is evocative of the sense of bitterness we felt in America that the damn Commies beat us into space.
Maybe next year I'll try it with pomegranate liqueur instead of Shakka Apple. |
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I'm more than happy with Carrion Luggage, however.
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