Different Ways to Measure

Topology · Axiom Academy

INTRO Different Ways to Measure Exploring how different metrics give different distances in the same space How far is it from one point to another? This seems like a simple question, but the answer depends on how you measure . Imagine you're in a city with a perfect grid of streets. The straight-line distance between two corners might be 5 blocks, but if you have to walk along the streets, you'll travel much farther! In mathematics, these different ways of measuring are called metrics . Each metric is a function that assigns a distance to any pair of points. The familiar "straight-line" distance from geometry. This uses the Pythagorean theorem: the distance is the hypotenuse of a right triangle. The "city block" distance, following grid lines. Sum the horizontal and vertical distances separately, like navigating city streets. Let's see how these metrics behave with different points. Click anywhere on the canvas to move the second point and watch both distances update! Click anywhere to change the position of point B Different metrics create entirely different geometries! The choice of metric determines: What shapes "circles" make (points equidistant from a center)

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