The Unicorn was invented for 3D games, and I disagree with the claim that the diagonals on a hexagonal board are the Unicorn's diagonals rather than the Bishop's. In various hexagonal adaptations of Chess or Shogi, the Bishop has been understood to move along lines of spaces that are the same color.
His reasoning for this begins with how he defined standard diagonals:
Standard diagonal (SD) directions, the diagonal of the standard square-cell geometry formed by simultaneously moving equal numbers of orthogonal steps at right angles have steps of length root-2.
I would disagree with this, defining diagonals instead as lines formed by passing through the opposite corners of spaces. This fits with the etymology of diagonal, for dia- means through, and -gonal refers to the corner. Similarly, a polygon literally means many-angled. If the shapes are not squares, then diagonal movement will not reach the same spaces as you could reach by making an equal number of moves in two directions at right angles to each other. It is merely a coincidence that it works this way on a square board, but it is not the definition of diagonal.*
The problem comes in when we start thinking about how diagonals should work on 3D boards. Since the Bishop is a colorbound piece on the 2D board, the decision was made to make it a colorbound piece on the 3D board too. But to do this, it was given a move that does not pass through the corners of a cubic space. Thinking of each space in a 3D game as a cube, the level-to-level movement of a Bishop in 3D Chess goes from one cube edge to the edge that is opposite in all three dimensions. However, this can be thought of as diagonal movement if we think of each vertical slice of a 3D board as a 2D board of squares. In fact, this is not so different from how we think of movement on the same level. We easily see the usual Bishop's moves as diagonal on one level of a 3D board, because we view it as a 2D board of squares. But if we thought of each space as a cube, the movement would no longer appear diagonal. This is easier to see, because of the way 3D boards are constructed. But mathematically, the board can be divided into vertical slices just as easily as it can be divided into horizontal slices. (Note that these vertical slices come in same-rank varieties and same-file vartities.) If we view the Bishop's movement as 2D diagonal movement through a single vertical or horizontal slice of the board, it can still be understood as diagonal in a 3D game.
Therefore, I would distinguish between 2D diagonals and 3D diagonals, and not between standard and nonstandard diagonals. Bishops move along 2D diagonals, even on 3D boards, and Unicorns move along 3D diagonals, which just don't exist on 2D boards, including 2D hexagonal boards. So it is wrong to say that the diagonal movement on a hexagonal board is how a Unicorn rather than a Bishop would move. The 3D Unicorn simply has no legal moves on a 2D board of any kind.
* As an afterthought, his definition of Standard diagonal could be tweaked for spaces of other shapes than square. On a hexagonal board, you can reach spaces along the diagonal though pairs of alternating orthogonal moves that are 60 degrees apart from each other. This works, because each angle of a hexagonal space is 60 degrees, not 90 degrees like it is for a square.
The Unicorn was invented for 3D games, and I disagree with the claim that the diagonals on a hexagonal board are the Unicorn's diagonals rather than the Bishop's. In various hexagonal adaptations of Chess or Shogi, the Bishop has been understood to move along lines of spaces that are the same color.
His reasoning for this begins with how he defined standard diagonals:
I would disagree with this, defining diagonals instead as lines formed by passing through the opposite corners of spaces. This fits with the etymology of diagonal, for dia- means through, and -gonal refers to the corner. Similarly, a polygon literally means many-angled. If the shapes are not squares, then diagonal movement will not reach the same spaces as you could reach by making an equal number of moves in two directions at right angles to each other. It is merely a coincidence that it works this way on a square board, but it is not the definition of diagonal.
*The problem comes in when we start thinking about how diagonals should work on 3D boards. Since the Bishop is a colorbound piece on the 2D board, the decision was made to make it a colorbound piece on the 3D board too. But to do this, it was given a move that does not pass through the corners of a cubic space. Thinking of each space in a 3D game as a cube, the level-to-level movement of a Bishop in 3D Chess goes from one cube edge to the edge that is opposite in all three dimensions. However, this can be thought of as diagonal movement if we think of each vertical slice of a 3D board as a 2D board of squares. In fact, this is not so different from how we think of movement on the same level. We easily see the usual Bishop's moves as diagonal on one level of a 3D board, because we view it as a 2D board of squares. But if we thought of each space as a cube, the movement would no longer appear diagonal. This is easier to see, because of the way 3D boards are constructed. But mathematically, the board can be divided into vertical slices just as easily as it can be divided into horizontal slices. (Note that these vertical slices come in same-rank varieties and same-file vartities.) If we view the Bishop's movement as 2D diagonal movement through a single vertical or horizontal slice of the board, it can still be understood as diagonal in a 3D game.
Therefore, I would distinguish between 2D diagonals and 3D diagonals, and not between standard and nonstandard diagonals. Bishops move along 2D diagonals, even on 3D boards, and Unicorns move along 3D diagonals, which just don't exist on 2D boards, including 2D hexagonal boards. So it is wrong to say that the diagonal movement on a hexagonal board is how a Unicorn rather than a Bishop would move. The 3D Unicorn simply has no legal moves on a 2D board of any kind.
*As an afterthought, his definition of Standard diagonal could be tweaked for spaces of other shapes than square. On a hexagonal board, you can reach spaces along the diagonal though pairs of alternating orthogonal moves that are 60 degrees apart from each other. This works, because each angle of a hexagonal space is 60 degrees, not 90 degrees like it is for a square.