More and more fog descends each night on my town. Roads become ghastly corridors of trees with yellow and red spirits of neon moving about in the distance. The mists choke out sound and contort shadows, seeming to muffle reality until I could as well be deep within the sinister boughs of Mirkwood, the untamed forests of Legaia or the wilds of the Northern Dominance and every second it seems more likely that the Headless Horseman will come a'galloping down the row. The past few days have failed to evaporate off the entirety of the sinuous haze, meaning that each nightfall adds to the mist's mass. Weather reports claim that this cycle should soon cease, but when reality itself is in doubt meteorology hardly has firm footing.

Speed Racer and double chocolate muffins were our fare of choice last night. The Wachowski brothers stayed pretty true to the original series, in that the racing was ridiculous, the hijinks wacky, and the trunks full of children and chimpanzees. Visually the movie was spectacular (the ice cave and final race were beauteously reminiscent of the high energy collisions in a particle accelerator), and while at times it did a good job of not taking itself seriously, there were a few scenes that lapsed a little too strongly into the sentimental given the film's ludicrous premise. And like most action movies these days, it was far too long; a twenty minute episode of Speed Racer is one thing, but two hours and twenty minutes is quite another. I really enjoyed the interesting editing as well.

But you didn't come here to learn about those things. You came to learn about fractal circuits. The punchline of my discussion is this: each and every level of the fractal circuit I depicted has the same resistance. I showed that the basic building block is equivalent to a single resistor, and since any level of the fractal is just those basic building blocks in the same pattern, it must be that they all have the same resistance. I find this quite interesting, and while it certainly has no useful applications (why would you replace a single resistor with 4^n resistors, even if it leaves the circuit unchanged!?), that is absolutely not a reason to avoid considering it. Here's one last building block for a more complicated fractal:

This grid contains 42 resistors, each denoted by a red squiggle. I have designed this one such that it appears a bit more complicated, but just as before if we say that each of these 42 resistors has resistance R, it turns out that the whole grid will also have resistance R (if you lack knowledge of series and parallel resistors, you will have to trust my calculation). Here is the second level of this fractal, containing 42^2 = 1764 resistors. The beauty of course, is that it also has R resistance:

I printed out a picture of one of these at school and then copied it 42 times to get the third iteration of the fractal. It now hangs in the only place it will fit: the ceiling. That monstrosity contains 74,088 resistors and yet the intricate simplicity of fractals makes the seemingly horrendous task of computing its resistance completely trivial. 

I mentioned him earlier, and perhaps in part to appease his wrath, I shall offer him up for your pleasure, he is the Headless Horseman!

The Headless Horseman is the famous phantasm from the Legend of Sleepy Hollow. In the words of his creator, Washington Irving: "The dominant spirit, however, that haunts this enchanted region, and seems to be commander-in-chief of all the powers of the air, is the apparition of a figure on horseback, without a head. It is said by some to be the ghost of a Hessian trooper, whose head had been carried away by a cannon-ball, in some nameless battle during the Revolutionary War, and who is ever and anon seen by the country folk hurrying along in the gloom of night, as if on the wings of the wind. His haunts are not confined to the valley, but extend at times to the adjacent roads, and especially to the vicinity of a church at no great distance.... the body of the trooper having been buried in the churchyard, the ghost rides forth to the scene of battle in nightly quest of his head, and that the rushing speed with which he sometimes passes along the Hollow, like a midnight blast, is owing to his being belated, and in a hurry to get back to the churchyard before daybreak."

So I finally beat Donkey Kong Country returns (the eponymous ape actually punches the moon at the end). I got all the KONG letters, but I still have three of the super-fiendish temple levels to do before I can challenge the mysterious "golden temple": I suspect the only thing inside is death, death and more death. Overall it's a really great game, and the difficulty is quite refreshing. Also, the silhouette levels are beautiful:

Returning to mathematical matters, here's the start of that fractal from last time:

Here's the second iteration (note the 4*4=16 red squiggles):

And here's the third iteration (4*4*4=64 red squiggles):

I said I was going to start talking about fractal circuits, and believe it or not, I already have. You see, each of the above images is actually a circuit diagram; the black lines represent wires and the red squiggles symbolize resistors. One of the key concepts to circuit analysis is that of the equivalent resistance between two places in a circuit. In effect, each of the arrays of resistors between the left and right ends of these circuit diagrams creates some amount of total resistance. What this means is that the resistor array can be replaced by a single equivalent resistor whose value is the calculated total resistance without affecting the circuit in any way and making matters much less complicated:

For most circuits only two equations are required to figure out the equivalent resistor. First one must determine if the resistors in question are in series or in parallel. Series means that they are all in a line, and parallel means just that, that they are all connected to the same point on one side and the other, and thus appear to be parallel to one another. From Wikipedia (an excellent source for formulae):

Adding resistors in series is very simple, just add the values together to get the total/equivalent resistance

Adding resistors in parallel is slightly more complicated, but is still just a quick computation. 

So the obvious next question is: what is the equivalent resistance of the fractal circuits I have posted above? I'll assume that each of the red resistors has a resistance of 1 and try to compute the equivalent resistance, starting with the first iteration:

Clearly the resistors at the top are in series, so their total resistance is 1+1 = 2, and the same is true of the bottom. As seen in the diagram below this leaves us with two resistors of value 2 in parallel with each other. Using our parallel calculation we find that the total resistance of the four resistor combination is just 1.

So our first iteration is simply equivalent to a single resistor whose value is the same as each of the resistors in the diagram. But now we have the daunting task of doing this calculation for the much larger second and third iterations of the fractal circuit. I'll finish this discussion next time, but if you remember what exactly a fractal is, you probably already realize the punchline of my argument. 

Onto more villainous affairs. Continuing our trend of nefarious Disney characters, today I bring you the the twisted vizier of Aladdin fame, Jafar!

Whether you are playing a videogame, reading a book or watching a movie, remember this rule: the king's most trusted chancellor, advisor, or in this case vizier is always, ALWAYS, an evil mastermind attempting to take over the kingdom. Jafar is a magician and advisor to the Sultan in Disney's Aladdin, the nefariousness of his eyebrows and goatee are unquestionable and with his loudmouthed pet bird Iago (no, not this guy) he sets out to do just that. Finding it impossible to marry the Sultan's daughter, Jafar turns to plan B and sends the movie's protagonist to fetch him a magic lamp (the location is only accessible to those who aren't ravenously malicious) so that he might simply wish himself control of all he wants. This goes fairly well for him for a while, until he foolishly wishes himself into his own doom, proving once and for all the pitfalls of greed (yay Disney themes!).

Other than the ever-pervasive homework, my recent activities have included getting to the final world of Donkey Kong, playing racquetball, dining upon asian cuisine, kicking a giant yellow ball around and watching Beauty and the Beast. I was very impressed by how great that movie still is.

My goal this week will be to talk about fractal circuits. I choose to begin with a short discussion on fractals. A fractal is an object or image that is itself composed of that same object or image. This recursive definition will make more sense after this example:

This simple fractal is called the Sierpinski Triangle. Take a look in the picture at the diagram labeled #1; you will note that is contains three black triangles. Now imagine what would happen if you took each of those three triangles and replaced it with three more black triangles. Basically, you would be replacing each component of diagram #1 with the entirety of diagram #1. The result of this process is diagram #2. Carrying out this process on diagram #2 (replacing each of its nine black triangles with three more) would create diagram #3. This iterative process is how you create the next "level" of a fractal. Note that it is easy to say how many black triangles will be in any given level of the fractal; the first has 3 the second has 3*3=9, the third has 3*3*3=27, thus the nth level will have 3^n black triangles.

So basically the Sierpinski Triangle is a fractal because it is an arrangement of triangles that are themselves composed of triangles. The astute reader will also have noticed the other defining characteristic of fractals: they look really cool.

A tree composed of trees

A spiral composed of spirals

Fractals occur naturally too: this cauliflower looks like a tree composed of trees as well

I'll post the first iteration of one more fractal:

By replacing each of the red squiggles with the entire diagram, see if you can figure out (and perhaps sketch) the next few iterations of the fractal. I'll post what it looks like next time since this particular fractal will be important to my discussion.

After watching Beauty and the Beast I would be remiss to cite its antagonist: Gaston!

I'm in that peculiar position again when a perfect description of my villain already exists, and providing anything less than that would be a disservice. Everything you need to know about Gaston is in this video. Make sure you check out the lyrics. Whoops, almost forgot the other part.

I just learned a fun fact that connects two very distinct fields. A Kobold is a fairly weak, generally stupid civilized creature in the Dungeons and Dragons world. Apparently "Kobold" means "Goblin" in German, and the element Cobalt got its name since miners referred to it as "goblin ore." Who would have thought that there was a relation?

Another bit of intriguing wordplay that came up of late is the hilarious distinction between a flagrantly technical foul and a technically flagrant foul.

I just started reading The Hero of Ages, third and final installment of the Mistborn series. My least favorite thing about this trilogy is that a character I hate continues to gain power and be portrayed in an overly positive light; it is my most sincere hope that he dies a horrible death in the next few hundred pages. There's almost no chance of this happening though. Anyway, I'll write more after I've finished this book. I expect to have all 7 Nausicaa books under my control by the end of the week as well.

Camping plans have yet to progress, except that the pilgrimage has gained followers. I'll try to call the Park and the school's equivalent of the Outing Club later this week to arrange arrangements or plan plans. 

I'm watching my friends play Donkey Kong Country Returns right now, and frankly, its a bloodbath. They've had to continue no fewer than thirty seven times in the first 12 levels. Their riddled ruins lie mangled and strewn across the level known appropriately as Tidal Terror. 

Today I honor another classic movie villain, the unforgettable Wicked Witch of the West!

The most vile creature in all the lands, the Wicked Witch of the West is the main antagonist of the most famous rendition of the Wizard of Oz. After Dorothy lands a house on the Witch's sister, the green hag swears revenge with one of the most famous lines in all of film. The Wicked Witch attempts to foil the protagonist and regain the lost Ruby Slippers with magical spells and even unleashes the most powerful force in the universe on Dorothy and her cohorts: flying monkeys! But honestly, its not like her aggression towards the company is unfounded given the Kansas girl's involvement in the death of the Witch's sister. Unfortunately for justice and for the Witch especially, her weakness is water, and a mere splash is enough to dissolve her into a powerless puddle of vile goop. 

You might have noticed the new ornamentation on the left sidebar. The Christmas decoration was getting a little out of season, so I pulled the ole' switcheroo with the utmost of extemporaneousness. Yet in time these two/too shall be replaced.

I've always liked how proclaiming that the weather is "FOG" is done with the gusto of three consecutive capitals. It's like the haze just suddenly appears and trumps all other weather patterns: "Today we will have partly cloudy skies with a 23% chance of ... NOPE NEVERMIND IT'S GUNNA BE FOG!" Besides, even if there was some other weather going on, you wouldn't be able to see it.

The elongated weekend has left me with more time than I know how to deal with. So far I've spent it relaxing, watching football and playing Donkey Kong. I'll probably look over some E&M or Quantum Mechanics at some point. Yet I'm left thinking that taking a trip would have been pretty fun. So if I can manage it, President's Day weekend will be spent camping at Yosemite National Park. Given that it will still be in the middle of February, hopefully not too many people will be there, and since President's Day immediately follows our midterms I should be able to round up some physicists to accompany me. I'll start gathering supplies sometime soon.

Random, yet beautiful, coverpage by Oda

Today I bring you another of Eiichiro Oda's colorful villains: Enel!

"Yahahahaha!" Enel is the self-proclaimed God of Skypiea. Since eating the Goro Goro no Mi (Thunder Fruit) he has possessed the power lightening and all that that entails. He can travel close to the speed of light, cannot be harmed by conventional means and can deliver massive electrical shocks at will. Even if felled, he can actually use his power to restart his own heart! Using these alarming abilities he enslaved the people of Skypiea and ruled the land actually believing himself to be a god. So sure is he of his invulnerability, that his shock and feeling of horror when his omnipotence fails him is extreme to say the least.

During a Quantum Mechanics homework extravaganza, some of my Physics friends and I found ourselves talking about our disappointment with our latest sushi excursion. Naturally this made us quite hungry and deciding quickly that seafood was tastier than iterated integrals by parts, we grabbed our stuff and headed for yet another raw-fish purveying establishment, Sushi Nobu. It was flat-out magnificent. The service was unmatched, the variety fantastic and the freshness supreme. One had but to utter the name of any among a myriad of items and the succulent creation would appear in seconds. And so beautiful was each dish that each time one of the chefs handed you one of his glistening, colorful formulations of cuisine, the eyes were sure to feast as much as the stomach. I will be sure to return.

If you have any interest in thematic fungi, I highly recommend clicking on the following image:

My time has recently been spent pretty solely on homework. In anticipation that I may have freetime this weekend (Martin Luther King day extends it), Amazon has now provided me with the last book in Sanderson's Mistborn trilogy, the first in Steven Erickson's recent epic, and the start of Hayao Miyazaki's manga creation. I might just end up sleeping, playing racquetball and watching football though. It's hard to say at this point.

I really should get out my new digital camera and take some pictures of something. Perhaps early next week. Speaking of early next week, I've been cunningly lured into a talk on information theory by the tantalizing promise of pizza.

Today's villain is possibly the most famous non-Star -Wars robot of all time: HAL9000.

Arthur C. Clarke's 2001: A Space Odyssey is one of the most well-known science fiction novels ever written and Stanley Kubrick's brilliant film adaptation ensures that the story of HAL9000 will not soon be forgotten. Two astronauts (and a sleeping crew) are aboard a spaceship headed for Jupiter, millions of miles from any other living being. Their only companion is HAL, the ship's computer. Along with Kubrick's depiction of HAL's sensors as malevolent and omniscient red eyes, its chilling monotone conveys the uncaring, systematic nature that makes the threat and tension all the more pronounced. When the men attempt to disconnect the him due to irregularities (caused by Earth's insistence that HAL lie to the crew about the mission), this threat becomes real and HAL tricks one to his vacuum-induced doom then sucks the rest of the sleeping crew into the black void of space as well, leaving only David Bowman alive. Bowman destorys HAL in an extremely emotional death scene wherein the computer acts more human than many of the movie's characters.

The origins of units came up during our TA meeting the other day, and I thought I'd quickly outline the definitions of some of the more common units we use today. I apologize for thinking that this is interesting.

-Second: At first the sun defined the second in that it was one 60th of a minute, which was one 60th of an hour, which was one 24th of a day. Today the scientific definition for the second is the amount of time it takes a caesium 133 atom to oscillate 9,192,631,770 times. This is actually how atomic clocks keep time. It's quite exact. Of course, now we have to add leap seconds every now and then to keep the sun consistent with our caesium.

-Meter: Originally defined as the length of pendulum whose period of oscillation is 2 seconds, it was later changed to one ten millionth the distance from the Earth's equator to the North Pole. Now we define the meter as the distance that light in a vacuum travels in one 299,792,485th of a second.

-Kilogram: My personal favorite. This unit was originally defined as being the amount of mass in one liter of water, but it has a much more interesting meaning now. On the outskirts of Paris lies a mysterious institute known as the International Bureau of Weights and Measures. Deep with the confines of this complex lies a fortified, climate controlled vault; three simultaneous keys must be turned to gain access to this most hallowed of places. Inside, probably gleaming brighter than the pedestal of gold on which it surely must stand, lies a treasure beyond wealth: in hushed, furtive tones the French name it "Le Grand K." It is the International Prototype Kilogram, and the amount of mass in this physical object is actually the scientific definition of the kilogram. Since any change in this quantity would actually affect all of science, the most stringent of constraints are imposed to prevent such devilry: it was constructed as a sphere made of a platinum-iridium alloy to minimize surface area, thermal conductivity, magnetic and electrical susceptibility, and prevent oxidation. And no, I did not make any of that up.

A replica of a replica of Le Grand K that is owned by Denmark

Enough about units, but more about the French. Yesterday I watched a humorous movie that originated in that country entitled A Town Called Panic, which uses some fascinating stop motion, hilarious voice acting, and a cast composed of plastic toys to tell a delightful tale of misadventure.

We went to a new place for Sushi called Jusco Sushi Buffet yesterday, and overall it was a disappointment. It was "all-you-can-eat" in the sense that you could keep ordering food without paying more, but the rate at which the cuisine was delivered precluded any such intentions. It was still quite tasty though. This week will be my first real week back. I teach twice and homework is due in every class. Hopefully I'll be able to get some stuff done today other than Donkey Kong levels.

With nothing else to say I shall cut to the chase. Today I introduce the antagonist of Kentaro Miura's spectacular manga Berserk. His name is Griffith.

Griffith is the leader of the Band of the Hawk, a troop of mercenaries that is renowned for its prowess and skill. He is a master swordsman, a brilliant strategist and his charisma is irresistible. His country views him as a hero, yet beneath his radiant exterior lies a cold, black heart filled only with ambition. After losing for the first time in his life to his friend and the protagonist Guts in a duel, Griffith loses control and is eventually captured and tortured by the ruling government. Even after being freed by his friends in the Band of the Hawk, he receives an offer to become a demon and has no problem accepting and achieving his dream from atop the corpses of his comrades; he commits the ultimate betrayal and slaughters all but two of the Band for his own desires. As a near-divine being he now conquers his way across the lands, daring to lead a new Band of the Hawk and is sought by the revenge-minded Guts.

Well, the holidays are over, and I'm back in California. Classes have begun anew, and in addition to teaching again this semester I shall endeavor to locate an advisor with whom I can do some interesting research. I've had some free time so far while my professors get their brass knuckles out, and I have mostly spent it playing Donkey Kong Country Returns (a delightfully diverse and challenging platformer) and finishing up a series of books I had been reading during my flights. More on the book series below. Fantasy novels are my preferred way to pass the usually boring time.

Did you know that if we cut a tunnel into the Earth's crust, through the mantel and then out the other side, one could (ignoring friction) hop in such a tunnel and, thanks to gravity, be on the other side of the planet in just 42 minutes? In fact, the magic of the conservation of energy mandates that ANY destination on the Earth's surface would be a 42-minute hop away. Pretty nice, huh?

Robin Hobb's Farseer Trilogy  has quite a bit less assassination that the titles of its books seem to imply (Assassin's Apprentice, Royal Assassin and Assassin's Quest). The first book is a fairly average affair about a royal bastard with a stereotypically-mysterious past growing up amongst the nobility who, as usual, are fighting for the control of the particular country he lives in. There is a bit of magic involved and a lot of intrigue, but nothing about the book really stands out except perhaps the disappointingly lackluster ending. The second book features the same main character doing virtually nothing for a long period of time, and is saved only by the author's unconventional decision to have the vast majority of the book take place in a single building and a surprising finale. I wouldn't have bothered typing so much about this series if the final book hadn't broken this mold of mediocrity. You see, the main character dies at the end of the second book, and things only get more unusual from there. Although it seemed like a lot of time was wasted in the first two books, I really felt like I understood how the main character felt during his final quest because of that enormous amount of seemingly repetitive interaction he had with the small cast of characters in the previous books. Its quite engaging, little about the end of the series is normal for fantasy, and it really makes you think about characters who aren't perfect, glitteringly heroic superheros. I didn't expect to have to think as much as I did, either.

I've read the first two book's in Brandon Sanderson's Mistborn series so far, so I'll probably get around to talking about that series when I finish it's final installment. So far I would critique that he put a lot more thought into his magic system and over-arching plot than the minutia that really make a series worthwhile.

I guess I need to take care of the Christmas ornament on the side of my blog. I'll put something else there in a bit. In fact, it will probably have changed by the time you're reading this possibly worthless sentence.

In the meantime here's a classic villain for your enjoyment: Angel Eyes.

His heart is as black as his hat. Lee Van Cleef plays "the Bad" in The Good, The Bad and the Ugly, one of the greatest Westerns (and movies) ever made. Angel Eye's introduction is a spectacularly grim scene that leaves little doubt that the movie has cast him correctly: the gunslinger murders a man for money, kills that man's family for the fun of it, and then laughs (as seen in the image above) as he eliminates his employer following the chilling line, "but you know when I'm paid, I always see the job through." No low is too low for Angel Eyes as he extorts, tortures and murders he way around the old West, vying with the other eponymous characters for a hidden stash of gold.