In the past couple of months, I published a number of articles on recreational math. I did my best to keep them accessible and fun, but my goal was usually to shed light at deeper mathematical truths. For example, the discussion of 0.999… = 1 served as a springboard to highlight some of the subtler properties of real numbers and the different meanings of infinity.
Today, I have no agenda. This article exists because I discovered a somewhat obscure paper that says something unexpected and cool. It feels profound, it probably isn’t… and if you keep reading, it’s going to live rent-free in your head too.
The road to a friend's house is never long
Topologists are an odd bunch: they study the continuity of geometric shapes with no regard for appearances. To them, continuous transformations — such as stretching and squeezing — are of no consequence. A donut and a drinking straw are the same because you can knead one into another without making or mending any holes.
Fundamentally, a topologist doesn’t care about the distance between two points in space: all that matters is a more narrow concept of local continuity. Because of this, the practitioners often choose to lean on stripped-down geometrical spaces in which the notion of distance — also known as a metric — is simply not defined.
That said, if you’re not a topologist, you probably enjoy being able to measure stuff. In standard Euclidean geometry with two dimensions, if we have two points that are separated by x horizontally and y vertically, the resulting straight-line distance can be calculated as:
\(d = \sqrt{x^2 + y^2}\)
One of the most common ways to construct non-Euclidean spaces is to alter the space’s metric in some way. An example that should be familiar to many software engineers is the taxicab metric, also known as the Manhattan distance. It’s named so by analogy to a cab navigating a rectangular grid of streets, charging you per mile traveled:
An illustration of the taxicab distance.
In the taxicab universe, the distance between two points is a simple sum of the absolute distances in each axis: d taxicab = |x| + |y|. If we take another look at the earlier case — d = √(x² + y²) — it’s tempting to express the cab equation in an analogous way:
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