Around the end of the 19th century, a Polish-German mathematician by the name of Hermann Minkowski first introduced the Taxicab metric to the world within a collection of proposed metrics (Gardner, 1997; Reynolds, 1980), although the name “Taxicab” was not used until 1952 when Karl Menger established a geometry exhibit in Chicago (Reinhardt, 2005). It is typically first taught in college geometry courses, although many times is ignored in the curriculum. Fortunately, many strides have been taken to encourage the instruction of non- Euclidean geometry in general. In fact, geometry at the university level is no longer strictly Euclidean geometry, and has transitioned to being conceived as geometric topology (Willmore, 1970).As college educators, we should emphasize geometrical discovery and the excitement that accompanies this, along with the idea of several different geometries (Willmore, 1970).
Byrkit (1971) explains that the axiomatic system associated with Euclidean geometry is studied in depth in geometry, while other axiomatic systems receive little attention. The author
continues to explain that when non-Euclidean axiomatic systems are studied, often the examples are too difficult, too limited, or too trivial to create interest. Siegel, Borasi, and Fonzi (1998) encourage the introduction to Taxicab geometry before other non-Euclidean geometries since the simpler space makes it easier for students to reason, and thus abstract concepts. One advantage to learning Taxicab geometry is that it can be used as a model for various applications, such as optimizing driving time in cities or laying pipes in a home. Caballero (2006) even explains how it can be used to model the spread of forest fires and discusses how this can be used to improve computer code for these types of simulations. Thus, learning concepts in Taxicab geometry not only can help facilitate geometrical reasoning, but can be applicable to many individuals and their future careers. Taxicab Geometry measures distance only in horizontal and vertical motions, as opposed to Euclidean geometry which measures distance as the length of the straight line between two points. For example, imagine a city where the streets form a perfect grid system. A car can only travel forwards or backwards, with the ability to make left and right turns. Thus, driving 3 blocks straight, making a left, and driving two more blocks is a total of five blocks.
In general, the Taxicab distance between two points is measured as the sum of the change in horizontal and vertical directions between the two points, where Euclidean geometry is measured using the Pythagorean theorem. For simplicity sake, in this report when an object such as Euclidean circle, Taxicab circle, etc. is being referred to, it is intended that I am referring to this object (and associated concepts) as it exists within that particular space, rather than suggesting that object has two distinct concepts (one in Euclidean geometry and one in Taxicab geometry). For example, a “Taxicab circle” is the concept of a circle and its definition within the Taxicab metric
system. Figure 2.1 shows visual examples of both metrics. Mathematically, Euclidean distance (𝑑𝐸) and Taxicab distance (𝑑𝑇) between two points 𝑃(𝑥1, 𝑦1) and 𝑄(𝑥2, 𝑦2) are defined below:
(i) 𝑑𝐸(𝑃, 𝑄) = √(𝑥2− 𝑥1)2+ (𝑦
2− 𝑦1)2
(ii) 𝑑𝑇(𝑃, 𝑄) = |𝑥2− 𝑥1| + |𝑦2− 𝑦1|
Seen in Figure 2.1, in Euclidean geometry, a distance between two points is represented visually as a straight line segment between two given points. In particular, it is calculated as the length of the hypotenuse of a right triangle constructed with legs parallel to the axes, as can also be seen in Figure 2.1. In Taxicab geometry, a distance between two points is represented visually as a path from one point to another by “walking” only over horizontal and vertical blocks. One such path would be along the legs of the right triangle mentioned in the case of Euclidean distance, but Figure 2.1 demonstrates two such paths. It is obvious that, in the case of Euclidean distance, there is a unique geometric representation of a distance between two points, while in Taxicab geometry this is not the case.
Many interesting things occur once we change how distance is measured within an axiomatic system. For example, the triangle inequality does not hold in Taxicab geometry, circles look like squares, and the congruence criteria of Side-Angle-Side, Angle-Side-Angle, and Side-
Side-Side of two triangles does not hold anymore (for more information on Taxicab triangles and trigonometry, please see Thompson & Dray (2000)). These are some examples of why learning Taxicab geometry can be interesting and important for students. By varying assumptions, students can come to the fundamental realization that we can develop new theory and results under certain conditions (Menger, 1971), and begin to abstract and generalize their understanding of particular concepts. As an example, Smith (2013) found that through exploration in Taxicab geometry his students deepened their understanding of a locus of points.
As briefly discussed previously, for this report, when concepts are discussed in each geometry by the convention of Euclidean circle, Taxicab circle, Euclidean perpendicular bisector, etc., I am referring to this object as it exists within this geometry, not that this object is defined differently within these geometries. In other words, the definition of this object is the same, but the properties of this object may be different between the geometries because of the way distance in measured. For example, the Euclidean perpendicular bisector and Taxicab perpendicular bisector of a segment are defined as the locus of points equidistant from the endpoints of this segment in Euclidean geometry and Taxicab geometry, respectively. Thus, it is noted that for the entirety of this report a perpendicular bisector of a segment is defined as the set of points that are equidistant from the endpoints of this segment. As a result of this definition, this object has different properties in both geometries. In particular, in Euclidean geometry, this results in a straight line that intersects the segment at its midpoint at a right angle. In Taxicab geometry, this locus of points is not necessarily a straight line, nor does it necessarily intersect this segment at a right angle, depending on the slope of the segment with respect to the axes. Thus, when the concept of Perpendicular bisector is discussed in relation to a segment, it is not implied this line intersects
the segment at a right angle, because this depends on in what metric space this line is being constructed.
Regarding this idea of properties varying as a result of a definition, Smith (2013) so eloquently discusses that our own assumptions can prevent us from seeing a problem in its full depth. Along these lines, Fujita and Jones (2006, 2007), Okazaki and Fujita (2007), and Turnuklu et al. (2013) talk about prototypical images in geometry and how students use them in their personal concept definitions, which affects how they define or classify figures. Identified as the “prototype phenomenon” (Hershkowitz, 1990), I believe using Taxicab geometry in the classroom can help students to move past this phenomenon and examine definitions and the underlying reasons for the appearance of figures as a result of these definitions, as explained in Berger (2015). This author provides activities and applications of Taxicab geometry, along with Krause (1973), Dreiling (2012), Smith (2013), and Chu and Tran (2017).