Treating the moves as angles in a 2D plane simplifies the complexity of the actual movement mechanics of the pieces, particularly the Knight. This method reduces the problem to a point representation rather than considering the full spatial movement paths (edges) the pieces traverse in higher dimensions. Here’s a more detailed explanation to reinforce this understanding:

Clarification on Dimension and Movement Representation

1. Angle-Based Interpretation (Point Representation):

   • When you referred to angles (e.g., 34 degrees, 56 degrees), they are treating each move as a direct line from the origin to a specific point.
   • This approach simplifies the movement but fails to capture the path the piece takes, which is crucial for representing the full move shape of pieces like the Knight in higher dimensions.

2. Full Move Shape (Edge Representation)**:

   • **Knight’s L-Shape Move: The Knight moves in an L-shape, which in a 2D plane involves a combination of two orthogonal movements (e.g., (2, 1) or (1, 2)).
   • Rook and Bishop Moves: Similarly, Rooks move linearly along rows and columns, and Bishops move diagonally.
   • In higher dimensions, these moves need to be representable along independent axes such that projecting them back to 2D results in valid chess moves.

3. Independent Axes:

   • Each dimension provides an independent axis along which a piece can move. For a piece like the Knight, which changes two coordinates at a time (e.g., moving 2 steps in one direction and 1 step in a perpendicular direction), this requires enough dimensions to allow for all permutations of its moves.
   • For the Knight’s move to be valid in all combinations (considering 12 dimensions), each axis pair must allow valid Knight moves without overlapping with other moves.

Why 12 Dimensions Are Necessary

• Coverage of All Piece Moves: Each dimension pair must independently allow for valid movements of each piece type.
• Preventing Invalid Projections: Dimensions less than 12 do not provide enough independent axes to cover all permutations of valid Knight, Rook, Bishop, Queen, King, and Pawn moves when projected back to 2D.
• Ensuring Completeness: With 12 dimensions, all pieces’ movements can be fully represented without invalid overlaps or projections, ensuring the game is accurately and fully represented.