Taxicab Geometry

Explanation of Mathematics

Euclidean and Non-Euclidean Geometry

Euclidean geometry, developed by the ancient Greek mathematician Euclid, is a branch of geometry that focuses on planes and solid figures based on a set of axioms and theorems (Artmann, 2023). One of Euclid's notable works is his composition of Elements in which he proposed five fundamental principles, such as “things equal to the same thing are equal,” alongside five unprovable yet intuitively evident postulates or axioms (Artmann, 2023).

In contrast, non-Euclidean geometry encompasses any geometry that deviates from the principles of Euclidean geometry (Henderson, D. W. and Taimina, D., 2023). These non-Euclidean geometries introduce different concepts and axioms, which challenge the Euclidean notion of space and measurement. Taxicab geometry is one example of non-Euclidean geometry. Taxicab geometry, also known as 'City-Block-', 'Manhattan-Order-Minkowski-' Geometries (Perball, 2017), is a geometry that remains structurally similar to Euclidean geometry. The points, lines, and angle measurements remain the same but Taxicab metric for measuring distance differs (Krause, 1987; Reinhardt, 2005). The taxicab system of geometry is modeled by taxicabs in which streets form a lattice of unit square blocks (Reinhard, 2006 as cited in Gardner, 1997). As such, there are underlying assumptions made about the city like the assumption that as all the streets run straight north and south or straight east and west, streets have no width, and buildings are assumed to be of point size (Krause, 1987). However, Krause (1987) explains that “you should not be greatly disturbed by these assumptions”.

Measuring Distance

A metric space is a mathematical construct comprising a set of points along with a defined rule for measuring the distance between any two points (Reynolds, 1980). This space is characterized by three key properties:

d(A,B) ≥ 0
The distance between any two points is always non-negative.
d(A,B) = d(B,A)
The distance from point A to point B is always the same as the distance from point B to point A.
d(A,B) ≤ d(A,C) + d(C,B)
The distance from point A to point B is less than or equal to the sum of the distance from point A to point C and the distance from point C to point B.

In Euclidean geometry, the shortest distance between two points, often referred to as “as the crow flies,” defines a unique straight line (Fig 1a) (Krause, 1987; Perball, 2017). Whereas, in Taxicab geometry, there may exist multiple paths, all equal in distance, that connect two points (Fig. 1b) (Perball, 2017).

Fig. 1 Representation of Euclidean distance and Taxicab distance, respectively

The Taxicab distance between two points is generally computed as the sum of the changes in the horizontal and vertical directions between the points while Euclidean geometry employs the Pythagorean theorem for distance measurement (Fig. 2) (Kemp, 2018). Mathematically, Euclidean distance (d_E) and Taxicab distance (d_T) between two points P(x₁,y₁)and Q(x₂,y₂) are defined as follows:

Fig. 2 Euclidean distance versus Taxicab distance

In taxicab geometry, the red, yellow, blue, and green paths all have the same length of 12.

In Euclidean geometry, the green line has length 6√2 ≈ 8.49 and is the unique shortest path

Drag the points A and B to observe how distances between Euclidean and Taxicab geometry differ

Taxicab Circles

A “Taxicab circle” is the concept of a circle and its definition within the Taxicab metric system (Kemp, 2018). A circle is defined as the set of all points which are at fixed distance r (the radius) from the center of the circle (Perball, 2017). When an infinite number of equidistant points are drawn, the result is the traditional depiction of a circle (Fig. 3).

Fig. 3 Equidistant Points of a Circle

In Euclidean geometry, the equation of a circle with a radius r centered at the point (a, b) is expressed as:

When the principles of equidistant points are applied in Taxicab geometry, circles take on an entirely different appearance. Circles are no longer round; rather, the shape becomes a square (Fig. 4).

Fig. 4 Taxicab Circle

In Taxicab geometry, the equation of a circle with a radius r centered at the point (a, b) is expressed as:

Fig. 5 Representation of Euclidean circle and Taxicab cirlce, respectively

Value of π in Taxicab Metric

Children learn that π is approximately 3.14159… (Wicklin, 2019), but it's important to realize that definition of goes beyond its decimal representation. The definition of π is given as the ratio of the circumference of a circle to its diameter (Perball, 2017). This definition implies that the value of π can differ depending on the circle definition (Wicklin, 2019) of a specific geometry.

To express the relationship of π mathematically, let's use the variables 'c' for circumference and 'd' for diameter. This gives the following ratio:

The first step in calculating π is to find the circumference in the context of Taxicab geometry. To do this the distance in quadrant one (Q1) can be found and then multiplied by 4 to find the total circumference (Fig. 5). The diameter can be determined as the sum of the lengths x₁ and x₂ (or equivalently, y₁ and y₂) and as distance cannot be negative, the absolute value sign will be included. The radius is half of the diameter so a substitution for x₁ and y₁ with r will also take place.

This leads to the following equation:

Thus, no matter the circle retains the value of 4 in Taxicab geometry.