Folks, we have the best π
And some novel ideas for bicycle wheel design.
In the past couple of months, I published a number of articles on recreational math. I did my best to keep them accessible and fun, but my goal was usually to shed light at deeper mathematical truths. For example, the discussion of 0.999… = 1 served as a springboard to highlight some of the subtler properties of real numbers and the different meanings of infinity.
Today, I have no agenda. This article exists because I discovered a somewhat obscure paper that says something unexpected and cool. It feels profound, it probably isn’t… and if you keep reading, it’s going to live rent-free in your head too.
The road to a friend's house is never long
Topologists are an odd bunch: they study the continuity of geometric shapes with no regard for appearances. To them, continuous transformations — such as stretching and squeezing — are of no consequence. A donut and a drinking straw are the same because you can knead one into another without making or mending any holes.
Fundamentally, a topologist doesn’t care about the distance between two points in space: all that matters is a more narrow concept of local continuity. Because of this, the practitioners often choose to lean on stripped-down geometrical spaces in which the notion of distance — also known as a metric — is simply not defined.
That said, if you’re not a topologist, you probably enjoy being able to measure stuff. In standard Euclidean geometry with two dimensions, if we have two points that are separated by x horizontally and y vertically, the resulting straight-line distance can be calculated as:
One of the most common ways to construct non-Euclidean spaces is to alter the space’s metric in some way. An example that should be familiar to many software engineers is the taxicab metric, also known as the Manhattan distance. It’s named so by analogy to a cab navigating a rectangular grid of streets, charging you per mile traveled:
In the taxicab universe, the distance between two points is a simple sum of the absolute distances in each axis: dtaxicab = |x| + |y|. If we take another look at the earlier case — d = √(x² + y²) — it’s tempting to express the cab equation in an analogous way:
If we look at these two formulas — Euclidean and taxicab — it’s clear that we can generalize this to a whole family of related metric spaces:
Again, the case of n = 1 is the taxicab universe; n = 2 nets us the standard Euclidean space. There are some unnamed geometries in between, but if n approaches infinity, we get what’s called the Chebyshev distance. In this case, even the tiniest difference between x and y makes one of the exponentiated values vastly smaller than the other one, so the distance is just:
A circle by any other name
In each of these spaces — from d1 to d∞ — there is a well-defined notion of a “circle”: it’s a set of points that are equidistant under the chosen metric from some center point.
To understand how these circles might look like, we can note that if y is zero, the earlier distance formula always simplifies to just dn = |x|:
Similarly, if x is zero, the formula is just dn = |y|. This means that in any of our spaces, if we’re trying to draw an n-circle with a radius r = 1, the points at (1, 0), (0, -1), (-1, 0), and (0, 1) will be a part of the shape:
For the taxicab metric (d1), the distance from (0, 0) is a simple sum of x and y coordinates, so a circle of radius r is just a collection of points that satisfy the criteria |x| + |y| = r. In our r = 1 example, this includes (0.9, 0.1), (0.8, 0.2), (0.7, 0.3), and so on. In effect, each of the four quadrants of the circle is just a ±45° diagonal:
For higher-order n-circles, the constraint for (x, y) points that make up the “circle” is:
For a more elegant version of this equation, we can raise both sides to n:
The following illustration shows examples calculated for n = 1.5, 2, 3, and 5:
Earlier on, we also mentioned that as n approaches infinity, we get what’s known as the Chebyshev distance, d∞ = max(|x|, |y|). This ∞-circle is a collection of all points where one coordinate is equal to ±r, while the other one stays within (-r, r):
So… you promised something about π?
Right. This brings us to an interesting question: does the value of π — that is, the ratio of circumference to diameter of a circle — change for “circles” drawn in these non-standard spaces?
Let’s have another look at the taxicab scenario:
What’s the circumference of that shape? It’s tempting to say that each quadrant is just the diagonal of a 1×1 square (= √2), so the circumference is 4·√2. The diameter of the circle is 2·r = 2. Hence, the value of the taxicab π (π1) must be 4·√2/2 ≈ 2.828.
Except… that’s a bit of a category error. We’re attempting to measure the circumference of a circle drawn in a non-Euclidean space by using the Euclidean definition of length. In the taxicab space, the diagonal of a 1×1 square is not √2; it’s a simple sum of the edges of the square. So, if we’re to be consistent, the correct answer is that π1 = 4·2/2 = 4.
Curiously, the ∞-circle case nets the same result: each quadrant consists of two straight segments of length 1, so the result is π∞ = 8·1/2 = 4.
For other n-circles, exact answers require some non-trivial calculus, but we can approximate the value numerically. To do this, we solve the circle equation at some constant angular intervals and then connect the dots, yielding a shape constructed out of straight segments.
For example, for the familiar case of n = 2, we get:
We can calculate the length of these segments using the appropriate metric, then divide it by the diameter (always 2). With the crude 20-segment approximation shown above, we get π2 ≈ 3.129. A more precise model with a thousand segments nets us π2 ≈ 3.14159.
Several other computed values are shown below:
And here’s a plot of πn showing even more data points:
So, that’s the somewhat unexpected revelation: of all these metric spaces constructed through intuitive extrapolation from Euclidean geometry, our π is the “minimal” π. We live in some sort of a π-hole.
👉 Credit: in their 2000 paper, mathematicians Charles Adler and James Tanton provide a formal proof of this property, and this is where the idea for this blog post comes from.
Hol’ up… what about n < 1?
We can do that, although this results in a space where the metric no longer corresponds to the shortest possible path between two points. For example, for n = 0.5, the calculated distance between (0, 1) and (1, 0) is:
But if we instead walk along the axes of the coordinate system, the vertical distance from (0, 1) to (0, 0) is 1; this is also the horizontal distance from (0, 0) to (1, 0). In other words, we can find a path that’s shorter than what the metric implies:
This violates one of the usual axioms of metric space; still, if we ignore the fact that the meaning of distance gets wonky at n < 1, we can follow the earlier algorithm to construct concave “circles”, as shown below:
The measurements corresponding to the image are π0.8 ≈ 4.7, π0.5 ≈ 7.2, π0.3 ≈ 11.9; the trend continues until a singularity at n = 0.
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Nice. Don’t all the distance formulas miss a '+' though?
Have you read about Piet Hein's superellipse? It uses the same formula as what you are exploring (except the ellipse-scaling factor, which is presumed 1 for the supercircles). Hein's superellipses were the subject of an early-1970s Mathematical Games article by Martin Gardner.