Taxicab Problem
Geometry & Trigonometry
In taxicab geometry, distance is measured along a grid rather than as the crow flies. Explore how this changes basic ideas like shortest paths, circles and perpendicular bisectors.
Introduction
In taxicab geometry, movement is restricted to a grid — like a driver who can only travel along city blocks, not diagonally. Distance between two points is measured as the sum of horizontal and vertical steps, not the straight-line (Euclidean) distance. This exploration investigates how far this changes familiar geometric ideas: the shortest routes between two points, the number of such routes, and what a 'circle' (all points at a fixed distance from a centre) looks like when distance is measured this way.
Guiding Questions
- In taxicab geometry you can only travel along grid lines. How is distance measured, and how does it differ from straight-line distance?
- How many shortest routes are there between two points on a grid? Find the pattern and connect it to Pascal's triangle.
- What does a 'circle' (all points at fixed taxicab distance) look like? What about a perpendicular bisector?
- Apply it to a real grid city: how different are taxicab and straight-line distances on average?