# Changing the Distance Metric

Posted on May 19, 2020 by Jack Kelly
Tags: coronavirus, maths

Mathematics has this idea of a metric, which generalises the idea of distances that we normally think about. If we take a set X, we can talk about a metric or distance function, which is a function d : X × X → X, that means “the distance between x and y”, and it has to have the following properties for any x, y, z ∈ X:

1. The distance between x and y is 0 if and only if they are the same thing:
d(x, y) = 0 ⇔ x = y

2. Symmetry: The distance between x and y equals the distance between y and x:
d(x, y) = d(y, x)

3. Triangle Inequality: If you’re going between x and y, detouring via z cannot be shorter:
d(x, y) ≤ d(x, z) + d(z, y)

Euclidian distance, given by the Pythagorean Theorem you might remember from school, is a metric on the set of points in 2D space — 2. There are many others. One interesting example is the discrete metric, which says that the distance between two points is zero if they are the same, and one if they are not.

It is obvious that the coronavirus lockdowns have messed with our perception of distance — someone in a different household may as well be on a different world. But it occurred to me the other day that the way we see distance has shifted from something like Euclidian distance to something a lot more like the discrete metric. While people nearby feel far away, people I used to think of as distant aren’t any more remote right now — travelling to visit them is just as impossible, instead of being a more difficult undertaking that I could sit down and plan. One upside to this whole mess is that interstate and overseas friends now feel no farther than local ones, and keeping up with them is paradoxically easier that it was under normal conditions — they’re on the other side of a screen, just like everyone else.

The analogy isn’t perfect — timezone differences still make things hard — but it’s interesting to notice and to think about.