Fractals!!!

It seems that if you add up all the terms in the series
1-(1/2)+(1/4)-(1/8)... and so on,
you get two thirds, which is quite odd because powers of two have nothing to do with 3.

Anyway, I've always liked fractals because of their endless continuity.
One especially cool fractallish pattern is the Dragon Curve, which is not only cool for its name but also for the methods it uses. Basically, you take a vertical line segment, rotate it 45 degrees counterclockwise, and then add another copy rotated 90 degrees clockwise to it. Another, more simple way to find it in nature is to take a piece of paper, fold it in half, fold it in half again, fold it half again, and et cetera. If you look down the edge you fold, it slowly becomes the Dragon Curve. A cool Youtube animation thing by a guy that calls himself Numberphile is really good.

The most basic fractal, fortunately triangular, is the Sierpinski Triangle. If you take a regular triangle, fit in it the largest regular triangle of the upside-down orientation, fit a similar triangle in the remaining space not occupied by the second, and et cetera, you get the Sierpinski Triangle.

My favorite name, however, is the Minkowski Sausage. I mean, it doesn't look anything like a sausage, which makes it so stupidly funny! However, it's really a simple fractal with a simple iteration.

The most basic type of fractal is the base-motif fractal. In each iteration, or stage of the fractal, each part of the previous stage of the fractal that fits the description of the base is replaced by a certain motif. In other words, you replace each base with a motif, producing fractals such as Sierpinski's Triangle, the Minkowski Sausage, and the Koch snowflake.

Well, that's pretty much it for hoy.

Stay coolio...
John