Measuring Distance

Topology · Axiom Academy

How do we capture the idea of "closeness" in mathematics? When you think about distance, you probably imagine measuring how far apart two things are. On a number line, this seems obvious—just count the units between them. But what if we're measuring distance in a plane? Or in three dimensions? Or in even stranger spaces? Click on the canvas above to place two points and see the distance between them calculated in different ways. Notice how the same two points can have different "distances" depending on how we choose to measure! The most familiar distance is Euclidean distance —the "straight-line" distance you learned in geometry. For two points in the plane, we use the Pythagorean theorem. Move the points above and watch how the Euclidean distance changes. Notice the right triangle that forms—this is why the Pythagorean theorem gives us the distance! But straight lines aren't always practical! Imagine you're in Manhattan, walking along city blocks. You can't cut diagonally through buildings—you have to follow the grid. This gives us Manhattan distance (also called taxicab distance). The visualization shows different paths from point A to point B, all following the grid. Notice that they all have the same Manhattan distance, even though the routes differ! GPS apps use Manhattan distance for routing in grid-like cities. Robots moving along aisles use this metric for efficiency. A rook's movement distance is Manhattan distance!

This is the written version of the interactive lesson above. See the full Topology course.