The Golden Ratio Is Hiding in Projective Planes

ζ φ Play

Click points on either view. Drag to interpolate between ζ and φ.

I’ve been playing around with projective planes over finite fields and stumbled into something I wasn’t expecting: the golden ratio shows up in a completely natural way, and it drags Lucas numbers and Euler products along with it. Here’s the story.

The setup

Take a prime $p$ and build the projective plane $\mathrm{PG}(2, \mathbb{F}_p)$ — the set of one-dimensional subspaces of $\mathbb{F}_p^3$. It has $p^2 + p + 1$ points, the same number of lines, and every line passes through $p + 1$ points. Standard stuff.

Now ask a simple question: pick two distinct points $P$ and $Q$. They determine a unique line. If you pick a third point $R$ at random from the remaining $p^2 + p - 1$ points, what’s the probability that $R$ lands on the same line?

There are $p - 1$ other points on the line through $P$ and $Q$ (the $p + 1$ on the line, minus $P$ and $Q$), so the collinearity probability is

$\mathrm{coll}_p = \frac{p - 1}{p^2 + p - 1}.$

That denominator, $p^2 + p - 1$, is where things get interesting.

The golden ratio appears

The polynomial $x^2 + x - 1$ has roots

$x = \frac{-1 \pm \sqrt{5}}{2},$

which are $1/\varphi$ and $-\varphi$, where $\varphi = (1 + \sqrt{5})/2$ is the golden ratio. So we get an exact factorization:

$p^2 + p - 1 = \left(p - \frac{1}{\varphi}\right)(p + \varphi).$

This isn’t an approximation. It’s algebraically exact for every prime $p$. The golden ratio just… shows up, uninvited, from the combinatorics of projective planes.

Building an Euler product

With an interesting polynomial in hand, the natural move is to build an Euler product — take the product over all primes:

$\mathcal{P}_{\mathrm{PG}(2)} = \prod_p \frac{p^2 + p - 1}{p^2} = \prod_p \left(1 + \frac{1}{p} - \frac{1}{p^2}\right).$

This product diverges, but in a controlled way. Expanding the logarithm:

$\log\left(1 + \frac{1}{p} - \frac{1}{p^2}\right) = \frac{1}{p} - \frac{3}{2p^2} + O(p^{-3}),$

so the divergence comes entirely from $\sum_p 1/p$ (which diverges by Mertens’ theorem), and everything else converges to a constant $C \approx -0.5323$.

primes 6 Play

$p$	$1 + 1/p - 1/p^2$	running $\prod$
2	1.250000	1.250000
3	1.222222	1.527778
5	1.160000	1.772222
7	1.122449	1.989229
11	1.082645	2.153628
13	1.071006	2.306548

The running product grows without bound — but the log splits into a divergent piece (Σ 1/p) and a convergent remainder.

Factoring into convergent pieces

The golden ratio factorization of $p^2 + p - 1$ splits the Euler product into two pieces:

$\prod_p \frac{p^2 + p - 1}{p^2} = \prod_p \left(1 - \frac{1}{\varphi p}\right) \times \prod_p \left(1 + \frac{\varphi}{p}\right).$

Both individual products diverge (the first to $0$, the second to $+\infty$), but you can tame them. The trick is a fractional-power Mertens normalization — you compare each factor against the corresponding power of the Riemann zeta function’s Euler product. This gives two convergent products:

$\mathcal{C}_1 = \prod_p \frac{1 - 1/(\varphi p)}{(1 - 1/p)^{1/\varphi}} \approx 1.0956,$

$\mathcal{C}_2 = \prod_p \frac{1 + \varphi/p}{(1 + 1/p)^{\varphi}} \approx 0.8745.$

That second one is suspiciously close to $7/8$. Within $0.06\%$. And $7 = L_4$, the fourth Lucas number. Coincidence? Maybe.

Lucas numbers everywhere

The real punchline is the constant $C$. It has an exact series representation:

$C = \sum_{n=2}^{\infty} (-1)^{n+1} \frac{L_n}{n} P(n),$

where $L_n$ is the $n$-th Lucas number and $P(n) = \sum_p p^{-n}$ is the prime zeta function. Written out:

$C = -\frac{3}{2}P(2) + \frac{4}{3}P(3) - \frac{7}{4}P(4) + \frac{11}{5}P(5) - \frac{18}{6}P(6) + \cdots$

The Lucas numbers $3, 4, 7, 11, 18, 29, \ldots$ show up as numerators. This isn’t a coincidence either — it comes from the power series expansion of $\log(1 + t - t^2)$, and the fact that $1 + t - t^2$ has roots at $\varphi$ and $-1/\varphi$ means the Binet formula naturally produces Lucas numbers as the coefficients.

terms 4 Play

-3/2

L₂ = 3

+4/3

L₃ = 4

-7/4

L₄ = 7

+11/5

L₅ = 11

-18/6

L₆ = 18

+29/7

L₇ = 29

-47/8

L₈ = 47

+76/9

L₉ = 76

-123/10

L₁₀ = 123

+199/11

L₁₁ = 199

-322/12

L₁₂ = 322

+521/13

L₁₃ = 521

-843/14

L₁₄ = 843

+1364/15

L₁₅ = 1364

Each term's numerator is a Lucas number. The partial sums oscillate and converge to C ≈ -0.5284.

Where does regular zeta fit?

The Riemann zeta function $\zeta(s) = \prod_p (1 - p^{-s})^{-1}$ comes from the polynomial $p^2 - 1 = (p-1)(p+1)$, which counts the multiplicative group $\mathbb{F}_{p^2}^*$. The roots are $\pm 1$ — rational.

Our projective plane polynomial is $p^2 + p - 1$, with irrational roots involving $\varphi$. The difference between the two polynomials is just $p$, which represents the contribution from the line at infinity in projective geometry.

You can actually interpolate between them. Consider $f_\alpha(x) = x^2 + \alpha x - 1$ for $\alpha \in [0, 1]$. At $\alpha = 0$ the roots are $\pm 1$ (zeta territory), and at $\alpha = 1$ they’re $1/\varphi$ and $-\varphi$ (golden territory). The roots slide continuously from one to the other.

And there’s a uniqueness result: $\alpha = 1$ is the only value for which the associated constant $C(\alpha)$ converges. For any other $\alpha$, you get a leftover $(\alpha - 1)\sum_p 1/p$ term that diverges. The golden ratio isn’t just a coincidence from projective geometry — it’s analytically forced.

The Fano plane

The smallest case is $p = 2$: the Fano plane, with 7 points and 7 lines. Here $p^2 + p - 1 = 5$, a Fibonacci number, and the point count $7$ is a Lucas number. Both the point count and the collinearity denominator land in the Fibonacci-Lucas family. For larger primes this stops happening pretty quickly, which makes the Fano plane a nice minimal example where the golden ratio structure is maximally visible.

Open questions

A few things I haven’t resolved:

• Is $\mathcal{C}_2 = 7/8$ exactly, or is there a tiny correction?
• Do these geometric zeta functions have functional equations like the Riemann zeta function?
• What happens for higher-dimensional projective spaces $\mathrm{PG}(n, \mathbb{F}_p)$ with $n > 2$? What algebraic numbers show up?
• Is there an analogue of the Riemann hypothesis here?

Whether any of this leads somewhere deep or is just a nice curiosity, I’m not sure yet. But the golden ratio emerging from projective plane combinatorics and then pulling in Lucas numbers and Euler products feels like it’s pointing at something.