Beyond Euclidean Distance

Real Analysis · Axiom Academy

The straight line isn't the only way to measure "how far." Put your hands on three other distances — and watch a circle change shape. The same two points can be five apart, or seven, or four Distance feels like one fixed number — the length of the straight line between two points. But "distance" is really anything that behaves like a distance should. Pick a sensible rule, and you get a new way to measure closeness, with its own answer for the very same pair of points. Watch one pair of points, P and Q , measured three ways in a row. The straight diagonal gives the familiar Euclidean distance; walking the grid like a taxi gives a longer one; and reading off only the single biggest step in either direction gives a shorter one. Same P and Q every time — the ruler is what changes. One pair of points, three honest answers. Each is a real metric — a different lens on the word "far". Move one point; watch four distances at once Drag Q anywhere on the grid. Four rulers measure the gap to the fixed point P simultaneously — and they almost never agree. Notice the discrete metric: it ignores how far apart they are and reports only 0 when the points coincide, 1 the instant they don't. Euclidean Taxicab Chebyshev Discrete d_0=0 if equal, else 1 Whenever the points differ you'll see — the longest single step never beats the diagonal, which never beats the grid walk. A "circle" is just every point at distance r — so its shape depends on the ruler

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