In Math one encounters so many results that leave one with the impression that Squared Euclidean is special. One such example is Singular Value Decomposition, or equivalently the Eckart-Young theorem. Arithmetic mean also minimizes the sum of Squared Euclidean from a set of points. Squared-Euclidean's properties are also the reason why the K-means algorithm (Lloyd's algorithm) is so simple.

Note that the squared part is important in that result although the squaring destroys the metric property.

A part of beauty of Euclidean metric (now without the squaring) is it's symmetry properties. It's level set, the circle (sphere) is the most symmetric object.

This symmetry is also the reason why the circle does not change if one tilts the coordinates. The orientation of the level sets of the other metrics considered in the post, depend on the coordinate axes, they are not coordinate invariant.

Euclidean metric is also invariant under translation, rotation and reflection. It has a specific relation with notion of dot-product and orthogonality -- the Cauchy-Schwarz inequality.

A generalization of that is Holder's inequality that can be generalized beyond these Lp based metrics, to homogeneous sublinear 'distances' or levels sets that have some symmetry about the origin [0].

The Cartesian coordinate system is in some sense matched with the Euclidean metric. It would be fun to explore suitable coordinates for the other metrics and level sets, although I am not quite sure what that means.

[0] Unfortunately I couldn't find a convenient url. I thought Wikipedia had a demonstration of this result. Can't seem to find it.

Most of the special properties can be traced back to its special relationship with the inner product. And inner products have somewhat more elementary properties, so in that sense it explains the special position of the euclidean norm.

This has nothing to do with the coordinates by the way. If you want a different norm you'll first have to figure out an alternative to the bilinearity that gives the inner product its special properties.

Though bilinearity is pretty special itself, given the link between the tensor space and the linear algebra equivalent of currying.

I can't edit my comment anymore so let me elaborate a bit here.

What is this sneaky connection between squared Euclidean and Cartesian coordinates that I mentioned ? Why are they such a compatible pair ?

The answer is the Pythagorean theorem.

The squared Euclidean distances decomposes nicely along orthogonal (perpendicular) directions.

    d^2 = x^2 + y^2.

The other metrics considered in the blog post decompose as, for lack of a better name, Fermat's last theorem decomposition.

    d^n = x^n + y^n

This much is true, forget about integral triples (lattice points) for integral n > 2.

The special properties extend into statistics, where you have the Gaussian distribution which feels both magical and universal, and is precisely the exponential of a (squared) Euclidean distance, i.e. exp(-(x - x0)^2).

I have the same feeling, that Cartesian coordinates and Euclidian distances are inherently connected as a natural pairing that is uniquely suited for producing the familiar reality that we inhabit and experience.

In my opinion it holds the same place in mathematics that water holds in biology and chemistry.

One of these ways (from which Cauchy-Schwarz and the other Hilbert Space results follow) is that d_2 is the only metric that satisfies the Parallelogram Law [1]:

2 d_2(x) + 2 d_2(y) = 2 d_2(x + y) + 2 d_2(x - y)

[1] https://en.wikipedia.org/wiki/Parallelogram_law