the coordinate axes, but a sphere formed using Manhattan distance is an octahedron: these are dual polyhedra, but among cubes, only the square (and 1-dimensional line segment) are self-dual polytopes. Nevertheless, it is true that in all finite-dimensional spaces the L1 and L∞ metrics are mathematically dual to each other. On a grid (such as a chessboard), the points at a Chebyshev distance of 1 of a point are the Moore neighborhood of that point. The Chebyshev distance is the limiting case of the order- Minkowski distance, when reaches infinity. Applications The Chebyshev distance is sometimes used in warehouse logistics, as it effectively measures the time an overhead crane takes to move an object (as the crane can move on the x and y axes at the same time but at the same speed along each axis). It is also widely used in electronic Computer-Aided Manufacturing (CAM) applications, in particular, in optimization algorithms for these. Many tools, such as plotting or drilling machines, photoplotter, etc. operating in the plane, are usually controlled by two motors in x and y directions, similar to the overhead cranes. See also King's graph Uniform norm Taxicab geometry References Metric geometry Mathematical chess problems Distance