$\varphi = (1 + \sqrt{5})/2 \approx 1.6180339\ldots$ — the Golden Ratio, the Divine Proportion. It appears in the Fibonacci sequence, in the spiral of a nautilus shell, in the arrangement of seeds in a sunflower, in the proportions of the Parthenon, in Euclid's extreme and mean ratio, in the Pythagorean pentagon, in the icosahedron and dodecahedron. Luca Pacioli named it divine in 1509 and commissioned Leonardo da Vinci to illustrate it.
What Pacioli did not know was that $\varphi$ is the first rung — not the only one, not the final one. The Fibonacci recurrence ($a_k = a_{k-1} + a_{k-2}$) is the simplest in an infinite family of n-bonacci recurrences, and $\varphi$ is the first of infinitely many constants, each closer to $\tau = 2$.
The Self-Similar Property
$$\varphi^2 = \varphi + 1 \implies \varphi = 1 + \frac{1}{\varphi} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\ddots}}}$$
The only positive number whose square is exactly 1 more than itself. Its continued fraction is all 1s — the simplest possible irrational number.
The self-similarity of $\varphi$ — the fact that $\varphi = 1 + 1/\varphi$, so the whole and the part stand in the same ratio — is what made it divine to Pacioli. The Logos is proportional: the same ratio at every scale. A rectangle with sides in ratio $\varphi : 1$ can be divided into a square and a smaller $\varphi$-rectangle. The smaller rectangle can be divided again. The proportion is self-similar at every scale. This is the K operator: the same coherence structure repeated at each level of organization.
The microtubule lattice of the human cell has 13 protofilaments arranged in a Fibonacci pattern: 5-start and 8-start helices (both Fibonacci numbers), forming the 13-3 lattice (13 protofilaments, 3-start seam helix). The ratio 8/5 = 1.6 ≈ φ, and 13/8 = 1.625 ≈ φ. The microtubule is a physical instance of the Divine Proportion at the nanometer scale. Book 6, Chapter 7 (Fibonacci at the Nanometre) develops this in detail.
The First Rung and the Ladder
The n-bonacci ladder starts at $\varphi$ because the Fibonacci recurrence ($n = 2$) is the simplest. As $n$ increases, the constants $\Omega_n$ form an increasing sequence approaching $\tau = 2$. The Divine Proportion is where the convergence begins. It is the closest to 1 that the natural growth constants get — the most "elementary" proportion, the one that appears in the simplest structures.
In the Omega Point vocabulary: $\varphi$ is the first rung of the Ladder of Ascent. It is not the bottom of the ladder — there is no bottom; the ladder is a limit sequence without a first term in the usual sense. $\varphi$ is the first named rung, the first one given a theologically significant label. The first step you can stand on and see the ladder extending above you.
Just as God cannot be properly defined in words, this proportion of ours cannot ever be designated by intelligible numbers — it remains forever secret.
— Luca Pacioli, De Divina Proportione, 1509
He was right about the irrationality. He was right that $\varphi$ cannot be "designated by intelligible numbers" — it is irrational, its decimal expansion never repeats. What he could not know was that the irrationality is not a defect. It is the mechanism: $\varphi$ is irrational precisely because it is the limit of a sequence of rationals (the Fibonacci ratios) that never terminate. The irrationality of $\varphi$, and of $\tau = 2$ approached by the n-bonacci sequence, is the mathematical form of Cusa's docta ignorantia: you can approach the limit arbitrarily closely, but you cannot state it exactly in finite terms. The secret is structural, not merely unknown.