Written by Yuju P. (GAFL'18)
Edited by Muchang B. (KIS'19) &
Gina L. (KIS'20)
━━ August 1st, 2018 ━━
*Due to format difficulties, we were not able to provide consistent fonts for this article.
Taxicab Geometry
𝝿 is an irrational number with the first six digits 3.14159. At the same time, 𝝿 is equal to 4. How can 𝝿, a constant, have a value other than 3.14159…? Consider the definition of 𝝿, which is “the numerical value of the ratio of the circumference of a circle to its diameter”implying that the relation between the diameter and the circumference has to be somehow different; this can be achieved by changing the measure of length, or distance. Euclidean distance is what most people think up of, if not only, when they hear the term “distance”. This is because Euclidean distance is the most frequently and widely used, although there exist various metrics or ways of measuring distance in mathematics. Despite the fact it is not as widely known as hyperbolic or elliptic geometry, taxicab geometry will be the key to answering the question how 𝝿 can equal 4.
The name taxicab itself provides insight into how the distance is defined. In a typically planned city with a grid system, people and vehicles, taxicabs included, can only move in two directions most of the time: forward and backward, and sideways. For example, the distance you travel to get to a nearby market 1 meter west and 1 meter north from your position is 1 m + 1 m = 2 meters, not 1.414 meters (square root of 2). Likewise, the taxicab distance is the sum of each measure of vertical and horizontal differences in a two dimensional plane,
Now that we have become more familiar with taxicab metric in relation to Euclidean, we move on to draw a circle in order to calculate 𝝿. A circle is “locus of points at a fixed distance from a fixed point whose center is the fixed point and whose radius is the fixed distance”. Without loss of generality, we derive an equation of a taxicab circle with the center at the origin and the radius r . Suppose P (x, y) is the locus of points equidistant by r from O(0, 0). According to taxicab distance, the equation of the circle is
When plotted on Cartesian coordinates, it results in the graph of Figure 1. It is surprising how the circle resembles an equilateral square in Euclidean geometry rather than being curved like the common Euclidean circle we know. The first step in calculating 𝝿 is to calculate the diameter, s
(Note that in this article, 𝝿 is regarded as its definition, rather than a constant, to allow convenient change of its use between Euclidean and taxicab geometry.) We finally found out that 𝝿 can be a rational number, and what is more, a natural number 4 in taxicab geometry. Referring back to the circumference and substituting 4 with 𝝿,
8r = 4 × 2r = 2πr
Despite various changes in circle due to the change in metrics, the equation for the
circumference which is given as 2𝝿r is consistent between the two geometries. The result is somewhat obvious since 𝝿 is also a constant for taxicab geometry.
From a different perspective, 𝝿 is one of the special angles in radian. A geometrical representation of angle 𝝿 in Euclidean geometry is shown in Figure 2.
Geometrical expression for taxicab geometry looks similar for both 𝝿/2 and 𝝿, ie. each radian in degree angles are equal to each other. However, such a result is only true for limited situations, and the examples are special cases where radian in each geometry have equal degrees. Extending our understanding of taxicab geometry from finding 𝝿=4, we explore how radian angles in Euclidean and taxicab geometry are related to each other in general.
A radian is defined as the length along the unit circle that is subtended by an angle which is constructed by an initial side and a terminal side, where an initial side remains in its position and a terminal side rotates. We start with a specific case where an angle is acute and is standard position, i.e. the vertex is at the origin and its initial side lies on the positive x-axis. This will less complicate the problem of expressing taxicab angle in euclidean angle, and later allow us extend the relation for general cases.
Construct a ray starting at the origin forming an acute angle with respect to the positive x-axis for the terminal side, as shown in figure 3. A taxicab radian is the length along the arc of the unit circle subtended by an angle, which will be the distance between the point (1, 0) and the intersection of the ray and the unit circle. The preceding interpretation is valid since the angle considered is acute and the circle in the first quadrant only consists of a single linear line that passes through the point (1, 0) .
Compare the equation (2) to (1). As the denominators are not always equal to
each other while the numerators are the same, it can be concluded that taxicab geometry is not rotational invariant, unlike Euclidean geometry. Stated differently, the distance between the two points will most likely change under same rotation. Referring back to a circle, this must be true since it is the Euclidean distance that is conserved under rotation while each point on a taxicab circle has different Euclidean distance from the origin. One of the consequences of rotation not being invariant would be that without the information of angle of rotation provided, we cannot know whether a triangle after rotation will be congruent to the initial triangle.
From 𝝿 being equal to 4 in rotation variant, it is surprising how taxicab distance is defined in a way that is very well integrated to our real life gives counter-intuitive results. Still, Taxicab geometry comes useful when dealing with distances along a path. For example, taxicab metric is used to determine school district boundaries in a way that enables students to attend the school of shortest travel distance from home. Taxicab geometry is also one of the most accessible non-Euclidean geometry for laymen, which could serve as an introductory course to studying non-Euclidean geometry. By breaking conventions in mathematics, one can achieve more rigorous understandings of mathematical concepts.
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Bibliography:
- “Explore Math with Desmos.” Desmos.com, www.desmos.com/.
- Lawrence S. “E-Z Geometry.” Google Books, books.google.co.kr.
- PBS Infinite Series. “When Pi Is Not 3.14 | Infinite Series | PBS Digital Studios.” YouTube, YouTube, 5 Jan. 2017, www.youtube.com/watch?v=ineO1tIyPfM.
- Math.com Glossary, www.math.com/school/glossary/glossindex.html.
Pictures:
- All images of graphs from Desmos, edited by the author.
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