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."