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.”

STEAMing away with Mathematical Selfies -- By Axelle Faughn

The academic term STEAM came along a few years ago when Arts education advocates argued that the Arts should be added to STEM education programs (an integrated approach to Science, Technology, Engineering and Mathematics education) in order to further improve performance and creativity in the upcoming American workforce. As Kathy's previous post illustrated, Mathematical Selfies are a good way to tap into students' ability to connect mathematical concepts to real world representations while providing an outlet for individual creativity and personal artistic strengths. Here I shall expand a bit more on developing such a holistic view of learning mathematics.

Sequence on the Beach
Courtesy of Emmanuelle Forgeoux

The obvious connection involves the art of composition in photography. For instance the following student, while seeking an appealing image of a sequence, decided to create this stunning picture of sand balls on the beach. It is minimal enough to not distract the viewer's attention from the main concept, and symbolic enough to expect this image will easily be stored in long-term memory as associated with the idea of sequencing.
Using this particular representation, additional discussions could expand the notion of sequencing to the idea of function.




Another type of connection to the Arts students easily make when looking for Mathematics Selfies is the one between mathematics and architecture/engineering. Building patterns and symmetries provide a wealth of representations for the mathematical eye, and architectural structures rely heavily on mathematics for solidity, mechanical features, and visual appeal. For example, as a pre-calculus student noted, the repetition and symmetry of a church arches can illustrate the notion of periodicity, while seeking maximal thrill on a roller-coaster can be modeled by the local extrema of a polynomial function. Roofs and awnings are also good illustrations of where triangle congruency plays a role in shaping our buildings.

Architectural Symmetry
and Repetitions

Max and Min on a Rollercoaster

Triangle Congruency for Building strong Structures

The Art of Geometric Constructions

The notion of geometric construction can also be approached from an architectural standpoint. Indeed, as the following geometry student realized during an assignment on geometric constructions, in order to to construct a square shape, one could possibly start with two sets of opposite parallel sides, then adjust sizes and angles as necessary.

 You may notice that in several of these submissions the students either added in text of "painted" additional symbols on the mathematical objects in order to convey their understanding. This is a skill that Mathematical Selfies help the students develop since they have to play with a variety of digital tools in order to create these "pictures and quotes" submissions. Graphic literacy is a growing field that demands a lot of creativity and versatility in using the available technology out there, one that requires both an artistic and a mathematical understanding of symbols in order to convey the attempted message meaningfully.



All these examples are clear evidence that Science and the Arts work together in communicating ideas, and should not be considered as completely separate fields in the curriculum. There are many opportunities for bringing them together in an integrated way, Mathematical Selfies being one of them.