Taxicab Geometry

Section 3.1 Taxicab Geometry

Taxicab geometry uses the same points, lines, and angles as in Euclidean geometry. However, instead of using the Euclidean distance function

\begin{equation} d_E(P_1,P_2) = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\label{Edist}\tag{3.1.1} \end{equation}

to measure the distance between two points \(P_m=(x_m,y_m)\text{,}\) we instead use the taxicab distance function

\begin{equation} d_T(P_1,P_2) = |x_2-x_1|+|y_2-y_1| .\label{Tdist}\tag{3.1.2} \end{equation}

Since the points, lines, and angles in taxicab geometry are the same as in Euclidean geometry, taxicab geometry satisfies most of the postulates of Euclidean geometry, including the parallel postulate. However, one property that fails is Side-Angle-Side Congruence (SAS), which says that two triangles are congruent (have the same corresponding parts) if two sides and the included angle are the same in the two triangles.

For example, consider the triangle shown in Figure 3.1.1. This triangle contains a right angle, and is also isosceles, since two of the legs have the same length. But, as you move point \(A\) while keeping the length of the legs constant (try it!), the length of the hypotenuse changes.

Figure 3.1.1. SAS congruence fails for taxicab geometry.

Activity 3.1.1. SAS congruence in taxicab geometry.

Can you find an equilateral right triangle?

Solution