Taxi Cab Geometry

  Taxi cab geometry does not look at the distance between two points as a straight line but the total sum of the absolute values of the differences of their coordinates.

In other words, if you look at the distance from Grand Central Station to the Empire State Building, you are not looking at the direct distance based on how far it is to fly.  You are looking at the distance based on having to work around the square blocks in town. 

The distance in taxi cab geometry is usually calculated by counting the number of blocks a cab has to travel to get from point A to point B.  Due to the fact, there may be multiple routes to follow, there may be several paths with the same distance.  The quickest way to find the minimum route is to apply the Pythagorean Theorem.

A circle in this branch of geometry often looks like a square even though it is defined as being made of four congruent segments with a slope of either 1 or -1.  In addition, the radius is still the same on both axis.  Hyperbolas and ellipses exist in taxicab geometry but they do not have the same curves we are used to.  Instead of curves, they are made up of line segments and diagonals yet they meet the standard mathematical definitions in Euclidean Geometry.

It turns out taxi cab geometry is a variety of non-Euclidian geometry formulated by Hermann Minkowski in the 19th century.  The idea is that distances are measured by adding horizontal and vertical distances rather than the direct "as the crow flies" distance.

If you'd like to introduce this topic to your geometry class, Jim Wilson has created a nice introduction using a variety of files for graphing calculators to help the student learn more about circles and analytic geometry.  Fortunately, there is an activity in Geogebra ready to be used by students as an introduction to the topic. The five files explore a different facet of Taxi Cab geometry.

In addition, Desmos has 1, 2, 3 different activities dealing with this topic. The third one compares Taxi Cab with Euclidean Geometry in a very visual way.

Let me know what your think.  I'd love to hear from you.