We are all familiar with the equation relating a circle’s radius to its circumference:
Rearranging, we get
and in Euclidean geometry we determine pi’s value of 3.14158… given a circle of any radius.
But in the non-Euclidean taxicab geometry, pi equals four, which will be demonstrated below.
Taxicab geometry is a two-dimensional geometry where points can only occur at the intersection of vertical and horizontal grid lines, and paths between points can only follow the grid lines. The vertical grid lines are located at integer values of x, while the horizontal grid lines are located at integer values of y. As such, it is a discrete geometry. This system derives its name from its similarity to a grid of streets which a taxicab would traverse to get from one point to another.
The image below shows two possible paths between points. In taxicab geometry, only the path from A to B exists. The path shown from C to D (valid in Euclidean geometry) is invalid in taxicab geometry:
We calculate distance between points by counting the discrete translations required to connect the points. For example, in the image below,
the path from A to B has a length of three, while the paths between C and D and E and F have a length of four.
A circle is a collection of points having a certain radius from a center point. They are defined in taxicab geometry, but take an unexpected form:
The above example shows a taxicab circle of radius four. Notice that space between each point is empty (unlike a Euclidean circle) since points cannot be defined outside of grid intersections.
If we then consider two “vertex” points A and B, and two possible paths between them,
we see that the taxicab distance between point A and B is twice the radius. We then determine that the circumference of the circle (the circuit through points A, B, C, D and then back to A) is 4 * 2 * r = 8 * r.
Finally, plugging this circumference into the equation for pi stated above yields:
Thus pi equals four in taxicab geometry.