Pi day 2026: Other values of pi

It is the last hour of Pi Day (March 14th in the US is written as 3/14¹) and I can finally post the article about \(\pi\) that I was planning since the article about the 3 possible regular 2D tilings and getting these tiles into my math AI benchmark for 2026 (where 80 different models performed well when they could cheat).

The AI articles focused on spirals on the square and hexagonal grids, but what are spirals but points moving on circles with a constantly expanding radius? And where there are circles, there are pies.

First, let’s start with the definitions. A circle is the collection of all points that are a fixed distance from a given point called the center. The fixed distance is called the radius. A disk is the collection of all points that are enclosed by a circle (that is, all points with distance to the center less than the radius threshold). From the classical Euclidean geometry, we have that the perimeter of a circle or radius \(r\) is \(P = 2\pi r\) and the area is \(A = \pi r^2\). From these, we can build two expression for \(\pi\) that should be valid in other geometries too – with the relevant squinting implied. For the perimeter:

\[ \pi_p = \frac{P(r)}{2r} \]

and for the area:

\[ \pi_a = \frac{A(r)}{r^2} \]

In both cases, we explicitly wrote \(P(r)\) and \(A(r)\) as functions of \(r\), as we follow this algorithm to determine the values of \(\pi\):

1. Determine expressions for \(P(r)\) and \(A(r)\) based on the geometry of the space
2. Use the formulas for \(\pi_p\) and \(\pi_a\), eventually taking the limit as \(r\) grows to be very large.

Would we get the same values of \(\pi\) in the area and the perimeter formulas? Let’s see.

First, let’s begin with the square grid, like the one in the following image:

We define a circle here as being all the squares that would be that same distance away from the center, moving horizontally, vertically and diagonally. This is the same that arises from the spiral problem.

Looking at the image, we quickly see that for a radius \(r\), the perimeter is \(8r\) and the area is \(4r^2 + 4r + 1\) (for this one, you need to add the perimeters of all circles of radius up to \(r\)). So, \(\pi_p = 4\) and \(\pi_a \rightarrow 4\). In this case, we got the same value of \(\pi\), but it’s no longer 3.14…

The careful reader might ask us to stop here and consider a different type of distance metric. Rather than counting diagonal moves as equivalent to horizontal/vertical, we should count them as 2 moves. That is, we should use the Manhattan distance which sums the horizontal and vertical displacements. In that case, our circles of radius 1 and 3 from the previous image would look like in the following one:

Here, the perimeter becomes \(4r\) and the area \(2r^2+2r+1\), using the same type of math as before. So, \(\pi_p = 2\) and \(\pi_a \rightarrow 2\). We still get a similar value for \(\pi\) from both area and perimeter, but it’s a new one, again.

Next, let’s look at hexagonal grids. Here, the distances are easier to compute, so the circles look like in the following image:

Convince yourself, dear reader, than the perimeter is \(6r\) and the area is \(3r^2+3r+1\), using the math as in the other grid cases. Thus, \(\pi_p = 3\) and \(\pi_a \rightarrow 3\). Again, we get similar values for \(\pi\).

In all 3 cases, we obtained similar values of \(\pi\), as summarized in this table:

One observation we can make from this table is that it seems that as we gain more degrees of freedom – more possibilities to move with equal cost –, the value of \(\pi\) grows. This is interesting. Other interesting findings from this experiment is that we got the same value from area and perimeter and that all the values of \(\pi\) are integers.

I’ll end the article here, before the end of \(\pi\) Day, even though there are more questions and extensions I’d like to pursue now. You should read this interactive short article from Steven Strogatz which also covers Manhattan-distance based grids². Or, wait a few days until Grant Sanderson will publish a video about other values of \(\pi\), based on other ways of defining geometries. He just tweeted about it!