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Cake day: July 2nd, 2023

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  • yeah exactly. i understand it as follows:

    • 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.

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    3 months ago

    i think that’s a good point and that is a nice way to remember them. i think a lot of it just comes down to personal preference.

    i like calling them the diamond/square/circle metrics because those shapes describe the sets of points that have unit length. i’ve found this wikipedia picture to be very helpful, and the diamond/square/circle terminology is my way of paying my respects to the picture.