# 2009:12: December Conundrum!

The philosophy of yin and yang, much like Santa, divides life into two dual halves:  naughty and nice, dark and light, low and high, etc.  The problem here is to draw one straight line through the yin yang symbol to split the yin and yang each into equal areas… and prove they’re equal!  Good luck and happy holidays!

Solution. Thanks to Daniel Phelps, a regular contributor, for the following nice write-up!

This is going to be hard to show without a visual aide, but ill try my best.

First off, you can split the yin yang into 4 equal parts (im going to arbitrarily give the yin yang a radius of 1) the black circle on top, with an area of pi/4, the equal but opposite white circle on the bottom, with an equal area of pi/4, then the pi/2 remaining area is divided equally between the left white piece and the right black piece, giving pi/4 to each of them.

Now if you draw a horizontal line across the two side pieces running tangent to the circles. you bisect the side pieces into pi/8 pieces.  This horizontal line is going to turn counterclockwise until it evenly bisects the yin and the yang.

You’ll also notice that if you swivel the the line up to 90 degrees, the line will only pass through yin on the left side and yang on the right side, so this space will be the easiest to measure out equal portions.  We already know the area of the yin above the horizontal line on the left (pi/8) and we know how much area we need to measure out to bisect the yin and yang (pi/4) which means we need to take out pi/4 from that region of pure yin in the third quadrant of the yin yang.  It just so happens that pi/4 is meted out halfway through the third quadrant, which bisects the yin and the yang.

So the bisecting line exists as a diameter 45 degrees above the horizon, and with this zen knowledge in mind, Santa can meditate until next December.