We are now approaching the end of the story, with the final tale of the Fractal Queen approaching. But for now we must deal with with the Bishop, a close cousin to the Rook.<\/p>\n
Recall that our puzzle revolves around the maximum “unguarded” positions of a chess piece on a board; in this case Bishops. In some sense, the Bishop can be thought of a Rook turned sideways, because both pieces can only move in two orthogonal directions, but the Bishop’s directions are along the 45 degree diagonals. The difference this makes though is that Bishops have a “color” while Rooks do not. Rooks can move to any square on the board while a Bishop on black squares can only attack other black squares.<\/p>\n
![\"\"](\"https:\/\/www.nilesritter.com\/wp\/wp-content\/uploads\/2021\/04\/bishop_and_rook-1024x564.png\")
<\/a>A Bishop is a sideways Rook<\/p><\/div>\n
This limited ability to only attack half of the squares that a Rook can, makes Bishops slightly “weaker”, and so it should not be surprise (as we showed in part I), that on an $n \\times n$ board, we can place $2n – 2$ Bishops, giving them a density of roughly $\\frac{2}{n}$ for large $n$, and asymptotically going to zero on the infinite board. So in some sense the friendly Bishop’s density on the board is roughly twice that of the slightly more powerful Rook.<\/p>\n
But as with the rook, this is the two-dimensional density, while the unguarded Bishop positions clearly have a fractal dimension of one on the board. And as we saw with the Rook, once we go to the unit square board $\\mathbb{B}_I$, with a continuum of “squares” at each real coordinate, the rigid shape of the solutions can loosen, so that instead of just the diagonal set with linear measure of $\\sqrt{2}$ we can have a set of “friendly” Rooks on the board with 1-dimensional content of nearly 2:<\/p>\n
![\"\"](\"https:\/\/www.nilesritter.com\/wp\/wp-content\/uploads\/2021\/04\/maximal_rooks.png\")
<\/a>![\"\"](\"https:\/\/www.nilesritter.com\/wp\/wp-content\/uploads\/2021\/04\/maximal_bishops2.png\")
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