King's metric and Grid metric

Can you think of distance from the perspective of a king in chess? And can you imagine you are driving a bike, where shortcuts have the same length as the longest path? We can get more philosophical and ask the question, What is Distance?

In simple words, it's the amount of separation between two objects — that may be anything. Mathematics generalized distance as a metric in lots of sets.

Let us consider any set A, and the distance or metric is a function d defined on every pair (x, y) of elements in set A. That function d has a few properties:

1. d(x, y) > 0 for every distinct pair of x, y.

2. d(x, y) = 0 if and only if x = y.

3. d(x, y) = d(y, x), Symmetry.

4. d(x, z) ≤ d(x, y) + d(y, z), The Triangle Inequality.

What's special about it? We can define distance as any function that satisfies these conditions. Normally, the distance is the Euclidean metric — or the distance we usually measure — defined as

When I went through these definitions, I lost my path and got into the webpages about a few different metrics. And imagining these metrics is quite interesting — especially the Discrete metric, Manhattan metric, and the very special Chebyshev Metric.

There are infinite ways to define distance. Maybe there exist countably infinite ways or uncountably infinite ways to define a metric. Why are these metrics so special?

The Manhattan metric, also known as the Taxicab metric, abolishes every shortcut. The Taxicab metric was first used as a measure of goodness of fit in 1757 by Roger Joseph Boscovich. And the development of Non-Euclidean geometries in the late 19th and 20th centuries led to the increased usage of different metrics. Hermann Minkowski and Frigyes Riesz used the Taxicab metric in their works. The Taxicab metric is defined as

d(x, y) = |x₁ - y₁| + |x₂ - y₂| + ... + |xₙ - yₙ|,

where x = (x₁, x₂, ..., xₙ) and y = (y₁, y₂, ..., yₙ).

It satisfies all the properties of a metric. If you have doubts, just verify it.

Let's dive into the imagination — if our universe had the Taxicab metric as the distance, how would it look? The first thing that hits our mind is diamond-shaped tires that work well, and the different dynamics. If you walk in a park with square grid paths, and you are standing at one endpoint of a square and want to reach the opposite diagonal, what will be the fastest way? Normally, in Euclid’s universe, taking the diagonal path saves your energy, even on a rough path. But in this Taxicab universe, forget the shortcut. Even if you take the diagonal path, it is the same as following the walkway. That’s a cruel thing for a lot of humanity.

This metric seems like a lighter and easier version of the Euclidean metric. I made this comparison because it’s easier to calculate the distance between two points.

What do you think about the Discrete metric or Isolated metric? The metric makes you isolated.

The formal definition of the Discrete metric feels like the cheat code in mathematics. The Discrete metric is defined as the distance between every distinct point being 1, otherwise 0. It follows every rule of a metric. It looks good and easy.

Imagine you spawn in a universe that has the Discrete metric as its distance. How odd would that experience be? You can see every point in the universe at an equal distance. Even your atoms are separated by the same distance. You are alive only if some mechanism exists to protect your structure and make the universe think of you as a single point. If you see a star at one unit of distance and start walking, the entire universe will walk with you. And you can’t catch the stars or anything else in that universe — including your distant lover. Such a wonderful universe.

Let’s think about another universe that has the King’s metric. Imagine you are the king in the chess world. You need to save yourself to win the war. But there are restrictions to your movement — you can move only one square. Think of the chessboard as having infinitely many squares in every direction, and if you want to move from your location to any arbitrary square of the board, the smallest number of moves it takes you to reach that arbitrary square is the King’s metric. That’s the distance in that wonderful universe.

Pafnuty Lvovich Chebyshev (16 May 1821 - 08 Dec 1894)

In math, that metric is called the Maximum metric or Chebyshev metric. It is defined as the maximum of the differences between the coordinates of two points.

d(x,y)= max(|x₁ - y₁|, |x₂ - y₂|, ..., |xₙ - yₙ|)

It satisfies the properties of a metric. And people say it is the limit of the p-metric we use (the p-metric includes the Taxicab and Euclidean metrics). This usually troubles graduate students during their courses. But it’s an interesting metric with beautiful insights.

It was used by the Russian mathematician Pafnuty Chebyshev in his work Theory of Uniform Approximation of Functions by Polynomials. It’s good to understand the dynamics of this metric. I think a universe with the Chebyshev metric as distance would definitely have more good shortcuts than any other universe.

Studying metrics is really a wonderful thing. When you start exploring more in Differential Geometry and Topological structures, you’ll get sudden twists and turns everywhere.