1. Why exactly n = 2 minimizes π. The article shows this graphically, but there is no formal proof (although the Adler & Tanton paper is mentioned). It would be interesting to understand why this is the case mathematically.
2. How to calculate π for n-metrics numerically. The general idea of "divide the circle into segments and calculate the length by the metric" is explained, but the exact algorithm or formulas are not shown.
3. What happens when n → 0. It mentions that "the concept of distance breaks down," but it does not explain exactly how and why this is so.
I feel like that would have been a bit in the weeds for the general pacing of this post, but you just convert each angle to a slope, then solve for y/x = that slope, and the metric from (0,0) to (x,y) equal to 1, right? Now you have a bunch of points and you just add up the distances.
Well, if that interested you, you could have downloaded the paper and read it. To me your comment sounds a shade entitled, as if the blog author is under an obligation to do all the work. Sometimes one has to do the work themselves.
The third one we can reason about: For all cases where x and y aren't 0, |x|^n goes to 1 as n goes to 0, so (|x|^n + |y|^n) goes to 2 , and 1/n goes to infinity, so lin n->0 (|x|^n + |y|^n)^(1/n) goes to infinity. If x and y are 0 it's 0, if x xor y are 0 it's 1.
To phrase this in a mathematically imprecise way, if all distances are either 0, 1, or infinite the concept of distance no longer represents how close things are together.
You can parameterise it by other concerns if you wish, and other things follow. But as a matter of fact, this is how pi depends on the distance metric.
It'd be interested in the set where Pi(d) is constant and equal to Pi.
(disclaimer: IANAM and I haven't given it much thought)
Note that the squared part is important in that result although the squaring destroys the metric property. Arithmetic mean also minimizes the sum of Squared Euclidean from a set of points.
A part of beauty of Euclidean metrics (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, 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.
Is sqrt(2) a number to you ?
isoprophlex•1h ago
At least to me it's provocative
BrandoElFollito•40m ago
Math is very cool but I think it requires a special (brilliant) mind to go through, and a lot of patience at the beginning, where things seem to go at a glacial pace with no clear goal.