Why is the movement of a Chess King aperiodic? Imagine we have a king by itself on a chess board, making random moves around the board. Although it is apparently aperiodic, wouldn't the corresponding Markov chain to the King's movements be _periodic_ since the King could only return to a square _i_ on the board at moves 2,4,6 ... etc. (great common denominator of 2) after initially being at the square? By definition, aperiodic chains have return times to _i_ with a g.c.d. of 1.

Your mathematics is correct but your understanding of the rules of chess is not. A king on any square that is not on the edge of the board can move to any of eight squares: the four that share a common edge with the one he's on and the four that share only a common vertex.