Taxicab Geometry: History & Formula

Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson you will discover a new type of geometry based on a different way of measuring distance between points. We'll give a little background on this strange geometry and then define the distance formula with plenty of examples.

A Different Way to Measure Distance

How far is it from the Empire State Building to the United Nations Headquarters in New York, NY? Well that depends. Do you have wings? If not, then you'll have to jump in a taxi and follow the grid pattern of streets. You'll go roughly 8 blocks east and 9 blocks north, for a total distance of 17 blocks (I realize not every block is the same length in Manhattan, but let's not worry about those details here -- as they say in the local language, 'fuggedaboutit'!).

Midtown Manhattan
Midtown Manhattan

This idea of counting distance only in the north-south (or vertical) and east-west (or horizontal) directions underlies what we call taxicab geometry. It's not that diagonal lines are not allowed, but in taxicab geometry, we have to measure the distance of every line as if it is made up of only vertical and horizontal segments.

But before getting into the mathematics, let's explore the history of this topic.

History of Taxicab Geometry

A long time ago, most people thought that the only sensible way to do Geometry was to do it the way Euclid did in the 300s B.C. But starting in the 19th century, mathematicians began exploring versions of Geometry that looked a whole lot different. Now don't get me wrong, Euclidean Geometry (the Geometry of Euclid) is still very important! In most everyday applications, like designing and building skyscrapers, Euclidean Geometry is the way to go.

However in the 19th century, mathematicians and physicists were beginning to explore ways of creating non-Euclidean Geometries. Hermann Minkowski is usually credited with introducing Taxicab Geometry, along with a whole family of different geometries, based on different ways of measuring distance between points.

Hermann Minkowski (1864-1909)
Hermann Minkowski

Now you may well ask, 'Why create new geometries?' I mean, one geometry seems good enough, right? Why complicate matters further? Well it seems that our universe sees it differently. The whole structure of space-time apparently obeys the rules of a non-Euclidean Geometry (but something much more complicated than Taxicab Geometry). The ideas of Minkowski and others led Albert Einstein towards his famous Theory of Relativity.

Distance Formulas

Euclidean Geometry is based on the Euclidean metric, which is a fancy way to talk about measuring distances. The Euclidean metric is a function that takes any two points as input and tells you the (Euclidean) distance between them. In two dimensions, this is just the familiar distance formula between points in the plane.

Euclidean distance formula between two points, (x1, y1) and (x2, y2).
Euclidean Distance Formula

The Taxicab metric is really just a sum of vertical and horizontal distance. Note the absolute values in the formula; they are very important!

Taxicab distance formula between two points, (x1, y1) and (x2, y2).
Taxicab metric

Unless the two points are directly horizontal or vertical from each other, the Euclidean distance will be smaller than the taxicab distance between them. Can you see why? Consider the diagram below.

In Taxicab Geometry, the red, blue, and yellow paths all represent shortest paths between the two points.
Lines in Taxicab Geometry


You know what a circle is, right? I know you do. But technically, the shape of a circle depends on the way you measure distances. In other words, there may be a huge difference between a Euclidean circle and a non-Euclidean circle.

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