The other answers here are great. Inspired by them, I came up with a mathematical reason why bishops are limited to half the squares.

Define a move as a vector: <+xpos, +ypos>.

Any Rook move has the following vector basis: {<1,0>, <0,1>, <-1,0>, <0,-1>}

Any Bishop has the following vector basis: {<1,1>, <-1,-1> <-1,1>, <1,-1>}

When a Bishop moves, sum(Δx, Δy) must be even. Therefore, it must move to a square of the same color. When a rook moves, sum(Δx, Δy) may be even or odd, so it can move to a square of either color.

As an extension, for a knight, sum(Δx, Δy) must be odd. Therefore, it must move to a square of a different color.

The reason that bishops can only reach half the board is because they are limited to moves over an even amount of squares, while rooks can move over an even or an odd amount of squares.

The other answers here are great. Inspired by them, I came up with a mathematical reason.

Basically, diagonal walls have holes because bishops are limited to half the squares.

Define a move as a vector: <+xpos, +ypos>.

Any Rook move has the following vector basis: {<1,0>, <0,1>, <-1,0>, <0,-1>}

Any Bishop has the following vector basis: {<1,1>, <-1,-1> <-1,1>, <1,-1>}

When a Bishop moves, sum(Δx, Δy) must be even. Therefore, it must move to a square of the same color. When a rook moves, sum(Δx, Δy) may be even or odd, so it can move to a square of either color.

As an extension, for a knight, sum(Δx, Δy) must be odd. Therefore, it must move to a square of a different color.

The reason that bishops can only reach half the board is because they are limited to moves over an even amount of squares, while rooks can move over an even or an odd amount of squares.

This means that bishops lack the same blocking capabilities as rooks. Also, they see fewer squares (7-13 vs 14), for the same reason.