Taxicab Geometry in Action -- By Kathy Jaqua

Not all of the situations that we see around us illustrate a familiar view of mathematics.  Taxicab geometry is one of those potentially unfamiliar areas of mathematics.  In traditional Euclidean geometry, we think of a surface or plane where we measure the distance between two points along a segment directly connecting them.  From that one definition, we develop the familiar geometry of shapes, congruence, similarity, trigonometry, and a list of topics that continues to grow even today.  The Euclidean view of geometry works great in many instances, but there can be places were the traditional definition of distance doesn’t fit the reality of the situation.  


Consider the photo of a major intersection shown here.  Imani, the author of this mathematical selfie, describes a situation where a different view of distance is needed.  “Any decent driver knows that at a four way intersection there are only four (legal) directions in which the car can be driven: up, down, left and right.”  This observation leads to a definition of distance that is based on how far apart two points are if measurements can only be made along vertical and/or horizontal segments. 

Traffic Intersection Taxicab

If you are at the intersection shown here, for example, and you need to get to another point in the city, you are likely to describe the distance in terms of blocks traveled along what is essentially a grid of streets.  This way of measuring distance is the basis of Taxicab geometry.  Imani describes distance in taxicab geometry as “the length of any path directly from A to B if all movements are right-left or up-down.” So a city built on a grid would be a perfect example of where taxicab distance is the best measure of distance between two points, but are there any other examples that we can see? 

Pac-Man Taxicab

Austin likes to play Pac-Man on line, and after exploring taxicab geometry in class, when he started a game, he saw a form of taxicab geometry in action.  He noted that “Pac-Man can only move up, down, left, and right.”  That observation started him thinking about how far away the ghosts were, and how the dots could represent the distance between two points.  The walls introduce some additional parameters, but the essence of measuring distance along grid-based pathways is clear.  A ghost on the same horizontal or vertical pathway can be much more dangerous because it is closer in game terms than one that appears closer in Euclidean terms but is not on as direct a pathway.  This is a great example of when a good player must adjust the view of distance to resemble taxi-distance and does it almost intuitively.  Without that change in focus, the ghosts are likely to finish Pac-Man off before he can clear the board.

  

Even though taxicab geometry is often represented in two dimensions, its use in three dimensions is also more prevalent than you might think.  3-D printers and other similar types of automation are a growing part of manufacturing.  In the library at WCU, a student saw a 3-D printer on display and recognized that the 2D taxicab geometry we had studied could be expanded to capture 3-D examples also.  He noted that “this 3D printer has a timing belt and motor that allows it to move up and down as well as side to side.  These are the only directions it can take, like in Taxiworld.”  Adding a third dimension did not change the essence of measuring distance along a grid-like pathway, it only introduced a third direction that is perpendicular to the existing grid.  The significance of measuring distance using the taxicab metric is critical to determining how long the timing belt will last, for example, as there are only so many revolutions that the belt can make before wearing out.  It also illustrates that any point in space, and in the plane for 2-D, can be reached using only grid-like movements.  Taxicab measurement doesn’t limit the world to just the points on the grid like buildings on a city street or dots in Pac-Man.  It can also describe how to move a machine using only perpendicular pathways to create familiar objects in 2-D and 3-D.

A different way to measure distance may seem to be pretty esoteric, but when we look at many common occurrences, we can see that measuring along a grid-like path is sometimes a more real distance than along the straight-line pathway that we have all learned to call “distance.”