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.

Mathematical Selfies with an Artist’s Perspective - By Kathy Jaqua

Mathematical selfies can be a great way to see what someone knows or doesn’t know about a particular topic, as we have talked about in earlier posts, and glean some information about common understandings and misunderstandings.  Selfies also offer an opportunity to see the photographer’s passion while revealing the mathematical world of the individual.  One of my students in a liberal studies mathematics class is an artist with a passion for photography.  Her excitement about completing a project to explore mathematics through photography was evident by the gleam in her eye and the far away look that followed my description of the assignment.   I’d like to share some of her work as a way to take a quick look at one student’s combination of a personal passion with a personal view of mathematics. 

As Jess moved around campus with her camera--a quick snapshot with a phone was not sufficient for her passion--she looked for geometric shapes as one category of the assignment.  Her work, however, was much more than simple photos of common shapes.  Using her artist’s eye for perspective, she was able to effectively translate between 2D and 3D to capture images of geometric objects.  Here’s some of what she saw.

Rectangular Perspective

Across the highway from our campus is a beautiful conference center that Jess visited in her mathematical selfies walk around campus.  She began her trip to NCCAT through the tunnel that connects the two campuses.  This is a place that students have decorated with a variety of comments, slogans, and other messages.  From the title of her photo, the artist saw those messages as descriptive of the object itself, but it also reveals her view of how a 2D rectangle can be seen within a 3D object in the world.  She saw a rectangle in the light at the end of the tunnel. Her ability to combine what she knows is the shape of the other end of the tunnel with a sense of perspective allows her to turn an explosion of light into a clearly articulated rectangle complete with a description that captures the essence of the definition using a combination of formal and informal language.

Trapezoidal Shape

Continuing on through the grounds of NCCAT, Jess encountered a gate on the path to the pond that lies in front of the conference center.  Before passing through the arch, she stopped to note the geometric form at the top of the passage.  This time, her definition is formal and taken from an external source, but she does demonstrate her understanding of this definition with the figure in red that she added to the photo.  I found it particularly interesting that while she had no trouble seeing the shape and photographing it, she was not comfortable describing this shape in the informal manner that she used to describe the more common rectangle.  Her ability to see a trapezoid among all the shapes that are part of the arch reveals her ability to focus on some parts of a structure, and it also reveals her appreciation of symmetry.

A Cone in the Water

Passing through the archway, Jess shows us another part of an artist’s understanding of perspective with her photo of an overflow drain.  Rather than choosing to photograph the large jet of water rising from the center of the pond, Jess found a quieter shape that shows a cone as part of its construction.  This time, her sense of 3D perspective is obvious in the red figure traced on the top of the drain.  In her definition, again a combination of formal and informal language, Jess notes the 3D nature of the cone and describes how that definition translates into this object.  Like the 2D rectangle, this 3D shape does not push her to find an external source for a definition.  Does this indicate that a trapezoid is less commonly recognized in our world?

Mathematical selfies of geometric shapes are some of the easiest for students to capture.  In the case of a student who is drawn to visual representations and photography in particular, those selfies can illustrate a much deeper sense of the geometric structure of our world.  Not only are these photographs beautiful, they illustrate her sense of perspective that seamlessly combines 2D and 3D objects.

Pride of the Mountains Marching Band Selfies -- By Axelle Faughn

Western Carolina University is famous for the quality of its band. The fact that a mathematically inclined mind is more often than not also musically inclined, means that several of our students in the mathematics department do play an instrument in band. With intensive band practice almost every day of the week, we were bound to see some creative submissions from this aspect of their lives.

Band Placement on a Grid

Imagine yourself as a freshman, first day on campus a week before regular students arrive; indeed, band practice requires you to get acquainted with campus life before classes start. Next thing you know the band director instructs you to get into formation and be ready to follow instruction: the photograph on the right is what you must picture in your head. Of course, the grid helps you position yourself, and the numbers help understand the symmetry of the band structure. Although not a traditional cartesian coordinate plane, this is not very different from the day Descartes was lying in bed and decided to add a grid to his ceiling where a fly had just landed, so he could describe its specific position.

Octagonal Percussion

Now that you know where to stand during band practice, it is time for you and your new friends to pull out your instruments. Once again you realize mathematics is omni-present. This photograph of a drum pad nicely illustrates the need for regularity and symmetry in many musical instruments. If you pay attention you will also find this regularity and symmetry in the rhythm of the music itself.

Symmetrical Band Formation

Game day is finally here! Practice pays off as the crowd cheers you on and you aptly perform the choreography long rehearsed through a series of rotations and translations, which allows band performers to keep various symmetrical arrangements while playing. Supporters await half-time with excitement and anticipation, but of course you are ready, and you don't let them down. Well done!

String Tangency

Game is over, back to the dorm. But let us not forget your other musical friends, those whose instrument is not so conducive to band practice, but still reveals plenty of mathematical connections. This photograph of a cello with the bow positioned tangential and perpendicular to the string helps us understand the proper position for a better sound on string instruments. As a cello player myself, I can assure you that for your ear's sake, you would not want the bow to be placed any other way. The student who submitted this also noted the relationship between pitch and string width, string length, and string density, which most musicians intuitively understand, and most mathematicians can explain using derivatives.


We hope you have a chance to see our Pride of the Mountains Marching Band someday. Their performance is really worth the side trip to our small mountain town, and their fame is the reason why many prospective students choose to join the student body at Western Carolina University.