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.
whyandgrowth · 2h ago
This is very interesting, but I have 3 questions:
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.
elsjaako · 1h ago
I think lcantuf has looked at the first two and decided that the answer is too complex for a post like this. He linked to the article.
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.
mistercow · 1h ago
> 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.
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.
whyandgrowth · 1h ago
Thanks
srean · 1h ago
> The article shows this graphically, but there is no formal proof (although the Adler & Tanton paper is mentioned).
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.
whyandgrowth · 1h ago
If the author had provided links to explanations or additional materials for those who want to understand the formal reasoning more deeply.
dpassens · 1h ago
And why does the linked paper not qualify as such a link?
whyandgrowth · 23m ago
Good point. I just thought that a direct link or summary of the formal reasoning would have made it easier for readers unfamiliar with the topic. But fair enough, the linked paper does cover it.
giveita · 11m ago
Spheres and hyperbolic are also interesting. On a sphere Pi can be anything from 3.14... to 4, then decreasing to zero based on how big your circle is. Not sure about hyp space but would be interesting.
kjellsbells · 23m ago
Popular-news version of this math, from the New York Times:
Is π really a number or is it a computation? For example, fibbonaci(∞) is not a number, and π looks to be conceptually similar. Unlike fibonacci(∞), π has a limit, and we can approximate it with better and better precision, but in both cases the computation will never terminate
creata · 7m ago
There are mathematical definitions of the terms "real number", "rational number", etc., but there is no mathematical definition of the word "number".
> we can approximate it with better and better precision
In one of the three common formal definitions of the real numbers, that's what a real number is: a Cauchy sequence of rational numbers, which approximate that real number with increasing precision. (Well, a real is an equivalence class of such sequences.)
(The other two common definitions are the Dedekind reals and the reals as the unique complete ordered field.)
srean · 45m ago
To answer your question you need to define what a number is to you. There are many different kinds of numbers, naturals, integers, rationals, irrationals, computable reals, reals, infinitesimals... Not even getting into complex, quaternions, octonions etc.
Is sqrt(2) a number to you ?
If you accept computable reals as numbers then \pi is definitely a number. So is the golden ratio.
Sharlin · 32m ago
If you believe that real numbers are numbers, then, yes, pi is a number. Indeed because pi is computable, it’s actually "more" real than almost all real numbers because there is only a countable infinity of computable reals.
Anyway, in modern math every real defined as the limit of a "process", namely a Cauchy sequence. Of course, for the rational subset of reals the limit is trivial.
probablypower · 35m ago
Is 2 a number?
Is 4 a number?
Is 4/2 a number?
Is 3 a number?
Is 3/2 a number?
etc...
All of these symbols represent precise points on the numberline. Pi also represents a precise point on the numberline, so is it not a number?
teiferer · 1m ago
Ironically, that response runs into the standard problem that many "limit" arguments have.
Generally speaking just because something looks like it's converging from some angle, it doesn't mean that it actually has a well-defined limit, and if it does then it also does not mean that the limit shares the properties of the items in the sequence of which it is the limit.
Examples: 1/n is strictly positive for all n. Its limit for n going to infinity, while well-defined, is not strictly positive. Another example: You can define pi as the limit of a sequence of rational numbers. But it's not rational itself.
So, no, your argument does not prove that pi is a number.
(I'm not arguing that pi is not a number. It definitely is. It's just that the argument is a different one.)
isoprophlex · 2h ago
I love these little mathematical snippets, where I (as a math noob) can't tell if the result is trivial or deeply profound
At least to me it's provocative
BrandoElFollito · 1h ago
I had the same thoughts when studying physics (I have a PhD). Math was some kind of a toolbox for my problems - I used it without too many thoughts and a deeper understanding. Some of the "tools" were wonderful and I was amazed that it worked; some were voodoo (like the general idea of renormalisation, which was used as a "Deus ex machina" to save the day when infinities started to crawl up).
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.
Yeah I noticed that too, and had to comment (now deleted) to test it was the website here, and not the original copied and pasted text.
skrebbel · 1h ago
I really suck at math, especially when continuous functions are involved (ie non-CS-y math). Usually when mathy articles are posted on HN, I quickly give up, but I just ate this article up. I'm really impressed with the clear explanation, it's quite something! Thanks for writing this!
amelius · 1h ago
But remember that "we" chose the generalization which this all depends on.
mjburgess · 1h ago
Well, it's pi parameterised by the distance metric, Pi(d)
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.
amelius · 59m ago
Yes, but we're looking at a specific set of distance metrics.
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.
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.
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.
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.
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.
https://www.nytimes.com/interactive/2025/06/09/science/math-...
> we can approximate it with better and better precision
In one of the three common formal definitions of the real numbers, that's what a real number is: a Cauchy sequence of rational numbers, which approximate that real number with increasing precision. (Well, a real is an equivalence class of such sequences.)
(The other two common definitions are the Dedekind reals and the reals as the unique complete ordered field.)
Is sqrt(2) a number to you ?
If you accept computable reals as numbers then \pi is definitely a number. So is the golden ratio.
Anyway, in modern math every real defined as the limit of a "process", namely a Cauchy sequence. Of course, for the rational subset of reals the limit is trivial.
Is 4 a number?
Is 4/2 a number?
Is 3 a number?
Is 3/2 a number?
etc...
All of these symbols represent precise points on the numberline. Pi also represents a precise point on the numberline, so is it not a number?
Generally speaking just because something looks like it's converging from some angle, it doesn't mean that it actually has a well-defined limit, and if it does then it also does not mean that the limit shares the properties of the items in the sequence of which it is the limit.
Examples: 1/n is strictly positive for all n. Its limit for n going to infinity, while well-defined, is not strictly positive. Another example: You can define pi as the limit of a sequence of rational numbers. But it's not rational itself.
So, no, your argument does not prove that pi is a number.
(I'm not arguing that pi is not a number. It definitely is. It's just that the argument is a different one.)
At least to me it's provocative
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.
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)