All of these metrics are just functions of a vector. Pop in an (x, y) pair and you get a number out. d₁, or as it's usually called, ℓ¹, |x| + |y|, gives you 7 when you pop in (3, 4). ℓ² = √[|x|² + |y|²] gives you √[3² + 4²] = 5. ℓ³ gives you ∛[|3|³ + |4|³] ≈ 4.498. None of them are moving or otherwise changing with time. They don't pertain to different numbers of dimensions; all of them are defined (in this post) as functions of two dimensions. They just assign a distance metric to every point in a two-dimensional plane.

I think where you're coming from is that the ℓ¹ metric tells you how far a taxicab would have to move, changing in one dimension at a time, while the ℓ² metric tells you how far you have to go if you go in a straight line. But the ℓ³ metric doesn't correspond to anything similar, not even in three or four or five dimensions, and neither does, for example, ℓ¹·⁵. To get to them you have to go through the formulas above.

The curves drawn are "level sets", which connect points that have the same distance metric.

The point of the post is not that "we can approximate [π] best" with a particular distance metric. Rather, it says that every metric (of this family of metrics—you can invent an infinite number of other metrics) has its own ratio of the circumference of a ball to its diameter, which we could jokingly call its "π", and that ratio is lowest for ℓ².

The ratio itself is a notion that really only makes sense in two dimensions; in three dimensions, for example, a ball has a surface rather than a circumference, and dividing the surface by the diameter gives you a length, not a number.

I am completely mystified by your remark that something "has everything to do with the nature of pi, not with the math". What is the nature of π if not math?

(not to be that guy, but I'm still curious who you're confusing me with re: https://news.ycombinator.com/item?id=45227923 )