Folks, we have the best π(lcamtuf.substack.com) |
Folks, we have the best π(lcamtuf.substack.com) |
You can read more about the curves of Lamé plotted in this article at https://en.wikipedia.org/wiki/Superellipse. If you're in Sweden, the layout of https://en.wikipedia.org/wiki/Sergels_torg is a superellipse design by Piet Hein. Martin Gardner wrote a delightful column about this in the September 01965 Scientific American: https://www.scientificamerican.com/article/mathematical-game... "The superellipse: a curve that lies between the ellipse and the rectangle" which I don't have a copy of, except the slightly corrupted copy at https://piethein.com/superellipse/. It begins lyrically:
> Civilized man is surrounded on all sides, indoors and out, by a subtle, seldom-noticed conflict between two ancient ways of shaping things: the orthogonal and the round. Cars on circular wheels, guided by hands on circular steering wheels, move along streets that intersect like the lines of a rectangular lattice. Buildings and houses are made up mostly of right angles, relieved occasionally by circular domes and windows. At rectangular or circular tables, with rectangular napkins on our laps, we eat from circular plates and drink from glasses with circular cross sections. We light cylindrical cigarettes with matches torn from rectangular packs, and we pay the rectangular bill with rectangular bank notes and circular coins.
This column is included in one of Martin Gardner's books, which is where I read it in my childhood.
Superquadrics are a generalization of the three-dimensional case (see https://en.wikipedia.org/wiki/Superquadrics); Ed Mackey's 01987 "Superquadrics" screensaver is included in xscreensaver, which you can easily install if you're running Debian or Android with F-Droid: https://f-droid.org/en/packages/org.jwz.xscreensaver/
Viewed as level sets of vector norms (https://en.wikipedia.org/wiki/Norm_(mathematics)) these curves are called "balls": https://en.wikipedia.org/wiki/Ball_(mathematics)#In_normed_v.... Vector norms are fundamental to approximation theory, and because people often do math on measurements from the real world [citation needed] which are always imprecise [citation needed], approximation theory is pretty widely applicable. It's often convenient to use one of the alternative norms mentioned in Michał's article for your proofs.
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.
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.
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 think it does. Both decompose along orthogonal directions. See my comment here https://news.ycombinator.com/item?id=45248881
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.
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.
Cartesian coordinates are orthogonal, which is great.
Euclidean distance is great because it makes space flat and rotationally symmetric.
2 d_2(x) + 2 d_2(y) = 2 d_2(x + y) + 2 d_2(x - y)
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.
Especially if they are complex differentiable functions - then they are wholly determined by their values in any tiny (complex) neighborhood around 1 (complex) value. Basically just equivalent to power series at that point.
While even finding the number of ways to give change is extremely challenging.
I like to think of these two metrics and "rook" and "queen" distance. Manhattan distance is how far away two points are if you are traversing using a rook in chess which can only move horizontally and vertically. Chebyshev distance is how far they are if you can also move diagonally.
Can someone explain what d(3)=(|x|^3+|y|^3)^(1/3) would actually mean as the blog seems to suggest something more profound than the below?
If d=|x|+abs|y} is moving in 2 dimensions, one dimension at a time and d(2)=(x^2+y^2)^(1/2) is moving in 2 dimensions at the same time, d(3)=(|x|^3+|y|^3)^(1/3) would have to mean moving 3 dimensions at once in two dimensional space (as it is missing the 3th position z) and for all n moving n dimensions at once in two dimensional space.
Now pi comes down to the constant calculating circumference. The blog shows we can approximate it best ignoring all other dimensions but those two in two dimensional space. Seems obvious, but that has everything to do with the nature of pi, not with the math.
d=(|x|^3+|y|^3+|z|^3)^(1/3) would approximate pi better in 3 dimensional space than in any other, etc.
The area of the Euclidean unit disk (area of a circle with radius 1) is equal to π. However, the volume of the unit ball (ball with radius 1, one dimension more) is larger, namely 4/3π ≈ 4.18879. So how does the hypervolume for unit n-balls change in higher dimensions, where the 1-ball is a circular disk and the 2-ball an ordinary ball?
Surprisingly, it first increases but then converges to 0. The maximum is achieved for a unit 5-ball with a hypervolume of about 5.2638. For higher dimensions the value decreases again.
However: If we allow fractional dimensions, the 5-ball isn't at the peak volume. The n-ball with the largest volume is achieved for n≈5.256946404860577, with a volume of approximately 5.277768021113401, which are slightly larger numbers.
These were computed by GPT-5-thinking, so take it with a grain of salt. But the fractional dimension for peak volume is also reported here on page 34: http://lib.ysu.am/disciplines_bk/8d6a1692e567ede24330d574ac3...
Curiously, the paper above says that the area of the hyper surface of the n-ball (rather than its volume) peaks at n≈7.2569464048, while ChatGPT calculated it as n≈6.256946404860577, so exactly one dimension less than the paper. I assume the paper is right?
Also curiously, as you can see from these numbers, that fractional dimension with the peak hyper surface area is exactly two (according to the paper) or one (according to ChatGPT) dimension larger than the fractional dimension of the peak volume.
I haven't clicked the link, but I guess this is a well written blog post, since the place where I asked the question is precisely where they link to the paper. Nice.
Fun fact: the article uses U+03C0 GREEK SMALL LETTER PI (π), but Unicode also has several pi codepoints meant to be used specifically for math. E.g., U+1D6D1 MATHEMATICAL BOLD SMALL PI (let’s see whether HN strips it: [edit: it does, see [0]]).
My point is that when it comes to π it's just like people don't care like they used to. I hope you don't take that as a criticism.
[I dropped my physics major in college in favor of computer science, mostly because I couldn't handle the math, so I acknowledge that this could be a stupid/non-sensical question.]
Imagine the Earth is a sphere. You make circles centered in the north pole:
* If the circle is tiny, the Earth is almost flat and you get almost pi.
* If the circle is the equator, you have to walk 1/4 of length the circle from the pole to the equator, so the result is 4/2=2
* If the circle is so big that you walked almost to the south pole, the result is almost 0.
I guess my point is that Pi is only a minimum in the selected family of metrics that the article examines. There are plenty of other metrics where Pi is as small or as big as you want.
The great circle is the one that passes through those points and has the center of the sphere as it's center.
https://www.nytimes.com/interactive/2025/06/09/science/math-...
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 what a real number is, is defined as the limit of a "process", namely a Cauchy sequence. Of course, for the rational subset of reals the limit is trivial.
Surely it’s not dimensions, since all of these examples were two-dimensional (x and y). So I’m a little lost here.
So if you pick n=1 you get d(x, y) = |x| + |y|, which is the taxicab metric. You can apply this metric to a Euclidean space of whatever dimension you like, just substituting the appropriate definition of |x| and |y|. For 1-dimensional space you would use |x| = abs(x[0]), for 2-dimensional space you would use |x| = sqrt(x[0]**2 + x[1]**2), etcetera. Hope that helps.
So in 3-dimensional space, for n=3, the "length" of a vector u=(u_x, u_y, u_z) is: d(u) = (|u_x|^3 + |u_y|^3 + |u_z|^3)^(1/3)
In 2-dimensional space, for n=4 and u=(u_x, u_y) you get the following: d(u) = (|u_x|^4 + |u_y|^4)^(1/4)
If you want to use the norm as a metric to work out the "distance" between two vectors u and v then you just compute d(u-v) where u-v is ordinary vector subtraction.
0: https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_...
If weird fonts bother you, consider disabling custom fonts in your browser settings. I do it occasionally, but it also breaks fontawesome (and similar), so it's not a clear win.
As for n=0, can't you prove that pi=inf for n=0 using limits?
My conclusion therefore isn't "we have the best pi", but is rather "we have the only pi", because pi is simply not applicable, as soon as you alter the rules of there being a 2-dimensional plane and there being real-world distance, that the definition of pi depends on.
Anyway, I am not a mathematician, maybe I'm just too stuck in the boring old real world to get it!
Having defined what a "circle" is and what its "circumference" and "radius" are, "pi" is defined: it's half the ratio of the circumference to the radius.
(I don't think it was very nice of whoever downvoted you, presumably because you're wrong, given you explicitly allowed that you might not be getting it.)
I believe this is how this website works: if someone thinks you are wrong, they will downvote your comment. It's best not to think about in terms of niceness but more about getting the content most people agree with, or considered the most valuable by the majority, to the top so that more people can view and discuss it.
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)
Memory from my Analysis 4 class in college.
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?
In the article 2π(d) = the ratio of the circumference to the radius. This is dimensionless, in the sense that the circumference and the radius are both lengths (measured in meters, or whatever), so 2π(d) is really just a number.
But the (hyper)volumes you're talking about depend on dimension, which is exactly why you say "hyper". In 2 dimensions the volume is the area, πr^2, which has dimensions L^2 [measured in m^2 or whatever]. But in 3 dimensions the volume is 4/3 πr^3, which has dimensions L^3. The 5 dimensional (hyper)volume has dimensions L^5, and so on.
So, "comparing" these to find out which is bigger and which smaller is not really meaningful---just like you shouldn't ask which is the bigger mass: a meter or a second? Neither is, they aren't masses.
Anyway, it seems independently interesting that this value peaks for the 5-ball, or the ~5.2569-ball. The non-fractional difference between fractional dimension of peak hyper volume and peak hyper surface area seems also interesting. (I assume there is some trivial explanation for this though.)
sure, you already got the complaint about comparing values of different units - but observe HOW this question is actually sidestepped! We divide hyper-volume of n-sphere by hyper-volume of n-cube!
Now this raises the question: WHAT n-cube are we taking?
If you take hyper-cube with side-length of sphere's diameter, you'll have nice relation between cube and its inscribed sphere - and, predictably, as n goes up, number of cube's "corners" also goes up. So this ratio consistently goes down
But what about your numbers? Well that result happens when you take cube with side-length of sphere's RADIUS. That way you arbitrarily add a scaling factor 2^n - and there's nothing geometric about this behaviour
> 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 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.)
There are a bunch of very strange metrics, e.g. a metric for which d(x,x) = 0, d(x,y)=1, that is, all points are at a distance of 1 to each other (this satisfies all axioms).
1. F = ma
2. -dV/dx = m d2x/dt2 # force is the negative gradient of potential. Acceleration is the second time derivative of displacement.
3. Rewrite d/dx as 1/v * d/dt via the chain rule: d/dt V = d/dx V * dx/dt => d/dx V = 1/v d/dt V => d/dx = 1/v d/dt.
4. Rearrange (2). 0 = dV/dt + m v d2x/dt2.
5. Integrate both sides by t. E = V + 1/2 m v^2 # where the constant of integration is a conserved quantity (energy).
Circling back to the blog post here, it seems like the author is specifically trying to discuss things in a way that non-experts can still follow along with I took a lot at a couple of other posts on the blog after finishing this one, and the one about a "proof" that pi equals 4 had a section that felt pretty similar to me, where it cites an explanation of why it's wrong conveyed in as way that probably doesn't do much to help the people who are most likely to need it in understanding why the proof is wrong: https://lcamtuf.substack.com/p/4
Then come the theoreticians that try to find something deeper that would link or emerge the models.
In the case of energy conservation the "root" reason for the conservation are Noether principles that state some immutable entities (basically an experiment is not dependent on time and on geometric transformations such as translation or rotation). The link is not immediately visible (5 pages of math :)) but then suddenly tadaaam! and conservation of energy!
But yes, your point is very correct and this is why people wondered (and to some extend still wonder) if the masses in mhg=mv^2/2 are the same masses :)
Physics is really cool
Physics is a model after all. I don't think you are so far off. Lot of physics is predicated on it's ability to make predictions, and predictions often come in a form of calculation of something. There is even a saying "shut up and calculate" in physics. It is not without contention, but it does describe a lot of physics especially today.
It is also notable that you can have multiple different models about the same thing. Notoriously we have both lagrangian and newtonian mechanics, both which are valid but can be useful in different scenarios.
However, once you take appropriate roots of hypervolume to get same units you can safely compare. Or the otherway round take appropriate powers of length to get same units as hypervolume.
Another fair comparison is between dimension-dependent lengths is the ratio of the (hyper)volume to the surface (hyper)area V(n)/A(n). This monotonically decreases from n=1.
What I was getting at is that the x and y grid is orthogonal for all Lp norms. The interesting stuff happens if you pick a different line as x-axis. Orthogonal projection to a different axis still works though, the resulting grid just won't be orthogonal in the conventional sense.
Or, more to the point, suppose someone came with a different system of units and said: look, one of our standard lengths is 10^-6 of one of yours, but one of our standard areas is defined just like yours: 1 ha = (100m)^2 and (1 of our areas) = (100 of our lengths)^2.
In other words, their standard length = 10^-6m; their area = 10^-12 ha = 10^-8 m^2.
Is their standard area bigger or smaller than their standard length?
Notice that in proportion the lengths and areas are the same.