Pi = 3.14159… Need Not Be Always!

Pi (π), the 16th letter of the Greek alphabet, is used to represent the most widely known mathematical constant. By definition, pi is the ratio of the circumference of a circle to its diameter. In other words, pi equals the circumference divided by the diameter (π = c/d). Conversely, the circumference is equal to pi times the diameter (c = πd). No matter how large or small a circle is, pi will always work out to be the same number.

Pi is an irrational number, which means that it is a real number with nonrepeating decimal expansion. It cannot be represented by an integer ratio and goes on forever, otherwise known as an infinite decimal. There is no exact value, seeing as the number does not end. Many mathematicians and math fans are interested in calculating pi to as many digits as possible. The Guinness World Record for reciting the most digits of pi belongs to Lu Chao of China, who has recited pi to more than 67,000 decimal places. The Pi-Search Page website has calculated it (with the help of a computer program) to 200 million digits.

Taxicab geometry, considered by Hermann Minkowski in 19th-century Germany, is a form of geometry in which the usual distance function of metric or Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates.

The taxicab metric is also known as rectilinear distance, L1 distance or \ell _{1} norm , snake distance, city block distance, Manhattan distance or Manhattan length, with corresponding variations in the name of the geometry. The latter names allude to the grid layout of most streets on the island of Manhattan, which causes the shortest path a car could take between two intersections in the borough to have length equal to the intersections’ distance in taxicab geometry.

The taxicab distance, {\displaystyle d_{1}}d_{1}, between two vectors {\displaystyle \mathbf {p} ,\mathbf {q} }\mathbf {p} ,\mathbf {q} in an n-dimensional real vector 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,

taxicab-metric-distance-between-two-points

where (p,q) are vectors. p=(p1,p2,…,pn) and q=(q1,q2,…qn).

For example, in the plane, the taxicab distance between (p1,p2) and (q1,q2) is
|p1-q1|+ p2-q2|.

Circles

taxicabgeometrycircleA circle is a set of points with a fixed distance, called the radius, from a point called the center. In taxicab geometry, distance is determined by a different metric than in Euclidean geometry, and the shape of circles changes as well. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. The image to the right shows why this is true, by showing in red the set of all points with a fixed distance from a center, shown in blue. As the size of the city blocks diminishes, the points become more numerous and become a rotated square in a continuous taxicab geometry. While each side would have length 2r using a Euclidean metric, where r is the circle’s radius, its length in taxicab geometry is 2r. Thus, a circle’s circumference is 8r. Thus, the value of a geometric analog to \pi is 4 in this geometry.

A circle of radius 1 (using this distance) is the von Neumann neighborhood of its center.

A circle of radius r for the Chebyshev distance (L metric) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L1 and L metrics does not generalize to higher dimensions.

Whenever each pair in a collection of these circles has a nonempty intersection, there exists an intersection point for the whole collection; therefore, the Manhattan distance forms an injective metric space.