In taxicab geometry, circles are squares and pi equals exactly 4
Measure distance the way a cab crosses Manhattan, block by block along the grid, and geometry turns strange. A circle becomes a tilted square. The ratio of its circumference to its diameter is 4, not 3.14. The idea is older than taxis, though: a Jesuit astronomer was already using it in 1757.
Ordinary distance runs in a straight line. Taxicab distance, also called Manhattan, city block, rectilinear or L1 distance, adds up how far apart two points are along each coordinate axis separately. On a street grid that is the length of the shortest route a car can actually drive. Every such route between two corners, however it zigzags, has the same taxicab length.
Roger Joseph Boscovich used this measure in 1757 to judge how well a line fits data, and statisticians still use it in regression, where one method built on it is known as LASSO. Its reading as a genuine geometry came with the non-Euclidean ideas of the late nineteenth century, and in 1910 it appeared in work by both Frigyes Riesz and Hermann Minkowski. The catchy name came from Karl Menger, in a 1952 booklet called You Will Like Geometry written for a public exhibit at Chicago's Museum of Science and Industry.
The shapes are what make it memorable. A circle is the set of points a fixed distance from a centre, and under taxicab rules that set forms a square standing on one corner. Each side measures twice the radius, so the whole boundary is eight radii long and the circle constant comes out as 4. In higher dimensions a sphere becomes a cross-polytope, the generalisation of an octahedron. On a square grid of cells, the taxicab disc is exactly the von Neumann neighbourhood used in cellular automata.
Taxicab geometry satisfies nearly all of Hilbert's axioms for Euclidean geometry. The exception is angle congruence: two triangles can share two equal sides and an equal angle between them yet fail to be congruent. Rotating the plane also changes taxicab distances, although sliding or mirroring along the axes does not.
Source: Taxicab geometry
