Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function (or metric) called the taxicab distance, Manhattan distance, or city block distance. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length.

The taxicab distance is also sometimes known as rectilinear distance or L¹ distance (see L^p space). This geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO. Its geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski.

In the two-dimensional real coordinate space ${\displaystyle \mathbb {R} ^{2}}$, the taxicab distance between two points ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$ is ${\displaystyle \left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|}$. That is, it is the sum of the absolute values of the differences in both coordinates.

The taxicab distance, ${\displaystyle d_{\text{T}}}$, between two points ${\displaystyle \mathbf {p} =(p_{1},p_{2},\dots ,p_{n}){\text{ and }}\mathbf {q} =(q_{1},q_{2},\dots ,q_{n})}$ in an n-dimensional real coordinate space with fixed Cartesian coordinate system, is the sum of the lengths of the projections of the line segment between the points onto the coordinate axes. More formally,$${\displaystyle d_{\text{T}}(\mathbf {p} ,\mathbf {q} )=\left\|\mathbf {p} -\mathbf {q} \right\|_{\text{T}}=\sum _{i=1}^{n}\left|p_{i}-q_{i}\right|}$$For example, in ${\displaystyle \mathbb {R} ^{2}}$, the taxicab distance between ${\displaystyle \mathbf {p} =(p_{1},p_{2})}$ and ${\displaystyle \mathbf {q} =(q_{1},q_{2})}$ is ${\displaystyle \left|p_{1}-q_{1}\right|+\left|p_{2}-q_{2}\right|.}$

The L¹ metric was used in regression analysis, as a measure of goodness of fit, in 1757 by Roger Joseph Boscovich. The interpretation of it as a distance between points in a geometric space dates to the late 19th century and the development of non-Euclidean geometries. Notably it appeared in 1910 in the works of both Frigyes Riesz and Hermann Minkowski. The formalization of L^p spaces, which include taxicab geometry as a special case, is credited to Riesz. In developing the geometry of numbers, Hermann Minkowski established his Minkowski inequality, stating that these spaces define normed vector spaces.

The name taxicab geometry was introduced by Karl Menger in a 1952 booklet You Will Like Geometry, accompanying a geometry exhibit intended for the general public at the Museum of Science and Industry in Chicago.

Thought of as an additional structure layered on Euclidean space, taxicab distance depends on the orientation of the coordinate system and is changed by Euclidean rotation of the space, but is unaffected by translation or axis-aligned reflections. Taxicab geometry satisfies all of Hilbert's axioms (a formalization of Euclidean geometry) except that the congruence of angles cannot be defined to precisely match the Euclidean concept, and under plausible definitions of congruent taxicab angles, the side-angle-side axiom is not satisfied as in general triangles with two taxicab-congruent sides and a taxicab-congruent angle between them are not congruent triangles.

In any metric space, a sphere is a set of points at a fixed distance, the radius, from a specific center point. Whereas a Euclidean sphere is round and rotationally symmetric, under the taxicab distance, the shape of a sphere is a cross-polytope, the n-dimensional generalization of a regular octahedron, whose points ${\displaystyle \mathbf {p} }$ satisfy the equation: