## Sunday, April 04, 2021

### [ztdyukvt] Hyperplane rook

In 3D chess, let a file be a 1D column of cubes; let a rank be a 2D plane of cubes.  In N dimensions, let a file remain 1D, a rank a hyperplane of N-1 dimensions.

We define a non-isotropic variant of a rook which we will call an "axis rook".  It can move up and down a file, its "axis", just like 2D chess.  Within a rank, however, it can make an unlimited number of orthogonal sub-moves in one move, staying within that rank.  For example, in 3D chess, a rank is 2D, so the axis rook can make an unlimited number of sub-moves like a 2D rook within its rank.  More specifically, each sub-move is a small rook or small wazir step.  (We could also consider great rook or great wazir).  In one move, it can essentially get to any part of its rank plane that is not blocked off.  Only its final sub-move may be a capture.

This type of movement preserves the 2D-chess technique of 2 rooks checkmating a king by alternatingly forcing it back one rank at a time, then checkmating on the back rank ("ladder" or "scissor lift").  (However, because the axis rook is anisotropic, this checkmate now works only against the front and back ranks, and not against the sides of the cube.)

In higher dimensions, is it too difficult to block off a portion of a rank from an axis rook on that rank?  In 4D chess, ranks are 3D.  On one hand, there is a lot of room in 3D for the roving rook on a rank to go around obstacles: it only needs one hole to pierce a surface-like obstacle.  On the other hand, 4D chess has many pieces with which to build obstacles.  Assuming a board of width 8 and starting pieces filling the first two ranks, then, in any dimension, it takes 1/16 of one's starting pieces to build a barrier across a rank.  However, it takes an exponential number of moves to assemble such a barrier, so by the time you are done, the enemy rook likely will have long since gone to another rank, perhaps going around via another rank.  Perhaps all pieces can simultaneously move during a turn.

Goal was to define rook and pawn endgames in higher dimensions.  Pawns move forward along a file.  Left unspecified: pawn capture, king movement, what a pawn can promote to.

It's easy to generalize the axis rook to a piece able to make unlimited sub-moves in any orthogonal hyperplane, not just its rank hyperplane.  Call this a hyperplane rook.  It is a very powerful piece, though perhaps its power is justified because there is a lot of space in higher dimensions.  Previously.

Perhaps all pieces can do en passant capturing (future post).  Chess becomes a battle of intersecting hyperplanes.  For example, in 3D chess, a hyperplane rook emits a plane (actually 3 planes) within which it threatens to capture.  Another rook, of opposing color, has a plane within in which it plans to move.  The first plane intersects the second plane along a line.  If the second rook crosses that line, it can be captured en passant.  First rook chooses where on the line the capture happens.

We can generalize the hyperplane rook to a hyperplane bishop.  First, select one of the obliquely oriented colorbound planes through the bishop's position.  Draw the Voronoi diagram of the cube centers in this hyperplane.  The bishop can make an unlimited number of sub-moves of steps through facets of the Voronoi diagram.  In 3D chess, the 2D Voronoi diagram of the colorbound cube centers on the colorbound oblique plane is a hexagonal tiling.  The bishop's sub-moves are moves among adjacent hexagons in the Voronoi diagram.  Equivalently, still in 3D chess, the cross section of the lattice of cubes cut by the obliquely oriented colorbound plane is the trihexagonal tiling (triangles and hexagons).  The hyperplane bishop can make unlimited sub-moves among the hexagons of that tiling, making ferz-like moves through hexagon vertices to adjacent hexagons.  As dimension increases, there are an exponential number of colorbound hyperplanes that a bishop can move through.

The hyperplane queen can be the union of a hyperplane rook and hyperplane bishop.  But we can also consider augmenting its hyperplane rook movement.  Within an orthogonal plane, let the hyperplane queen make an unlimited number of king-like sub-moves.  The number of possible king moves increases exponentially with dimension.

Finally, instead of just the orthogonal hyperplanes and colorbound hyperplanes, consider the (exponentially) many partially diagonal partially orthogonal hyperplanes in between.  In 3D, these planes are tilings of (sqrt 2)-by-1 rectangles.  These might be hyperplane analogues of the great rook.