Metric spaces in chess and international chess pieces graph diameters

TL;DR

This paper investigates the graph radii and diameters of generalized chess pieces in multi-dimensional lattices, providing exact values for some pieces and bounds for others, thereby extending classical chess metrics into higher dimensions.

Contribution

It introduces a consistent generalization of planar chess pieces to k-dimensional spaces and computes their graph radii and diameters, including exact values and bounds for various pieces.

Findings

Exact graph radii and diameters for k-rook, k-king, k-bishop, 3-queen, 3-knight, 3-pawn.

Tight bounds for k-queen, k-knight, k-pawn for all k ≥ 4.

Extension of chess piece metrics to multi-dimensional lattices.

Abstract

This paper aims to study the graph radii and diameters induced by the $k$-dimensional versions of the well-known six international chess pieces on every finite $\{n \times n \times \dots \times n\} \subseteq \mathbb{Z}^k$ lattice since they originate as many interesting metric spaces for any proper pair $(n,k)$. For this purpose, we finally discuss a mathematically consistent generalization of all the planar FIDE chess pieces to an appropriate $k$-dimensional environment, finding (for any $k \in \mathbb{Z}^+$) the exact values of the graph radii and diameters of the $k$-rook, $k$-king, $k$-bishop, and the corresponding values for the $3$-queen, $3$-knight, and $3$-pawn. We also provide tight bounds for the graph radii and diameters of the $k$-queen, $k$-knight, and $k$-pawn, holding for any $k \geq 4$.