Math in… Maps and Metrics

When going from one place to another, it’s useful to know how far you need to travel. But the way you travel can drastically change your travel time. For example, if asked the distance from Seattle to New York City, you might say 2400 miles.

This is the distance as the crow flies, meaning the most direct path between two points, ignoring obstacles like lakes and mountains. Highways are constructed to avoid or negotiate these obstacles. Following highways, the shortest distance is closer to 2850 miles.

How you travel can change your sense of distance. In math, we call these generalized senses of distance metrics.

For a bird, the set of points it can reach by flying half a mile form a circle.

If the grid lines are north-south and east-west city streets that you must travel because buildings block other routes, the set of points you can reach by walking half a mile instead form a square, with corners pointing north, south, east, and west.

Since the set of all points equidistant from a given point is traditionally called a circle, a square like in the last image is called a circle in taxi cab geometry.

For fun, you might pick a place in your home to start from and see all of the places you can reach by walking exactly 5 steps (and no fewer!) from that point, avoiding furniture and passing through doors.

What does this circle look like in your home’s metric geometry? What if you change the starting point or the number of steps?