in the manhattan metric, points have length one if the lengths of their coordinates sum to 1. so you get the points (1, 0), (0, 1), (-1, 0), and (-1, -1). and then you connect these four points with straight lines to get the diamond shape. this follows from the observation that if the x coordinate decreases in length by 0.1, then the y coordinate must increase in length by 0.1.
in the euclidean metric, the points of length one lie on the unit circle, since x2 + y2 = 1 is the equation defining the unit circle.
in the chebyshev metric, points have length 1 if one of the coordinates has length 1 and the other coordinates have a length smaller (or equal to) 1. and these conditions also describe the square with sides x = ± 1 and y = ± 1.
Ah right, so “diamond” (depicted as a square rotated 45 degrees) is Manhattan, circle is Euclidean, and square is Chebyshev, then?
yeah exactly. i understand it as follows: