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!).