Four things worth knowing about ρ that the first page didn't cover: where the name comes from (it has nothing to do with the material), what makes it Pisot and why that's the property you actually want in a substrate, what its contracting plane looks like — it's a fractal, and it tiles — and where it genuinely beats Fibonacci, which is a narrower set of places than the enthusiasm suggests.
Nothing to do with polymers. Dom Hans van der Laan (1904–1991) was a Dutch Benedictine monk and architect, and he named it het plastische getal in 1928 in the sense of the plastic arts — sculpture, modelling, things shaped in three dimensions. The Greek root plassein means to mould.
His claim was a proportional one. The golden ratio governs two-dimensional relations — the division of a line, the proportions of a rectangle. Van der Laan argued that three-dimensional form needs a different constant, one that relates volumes the way φ relates lengths, and that ρ is it. He built the abbey church at Vaals to that system: every dimension in the building is ρ-related to every other.
He also made a psychophysical claim — that ρ ≈ 1.325 sits at the threshold where two sizes stop reading as the same and start reading as different. Below it, your eye files them together. That's why he chose the smallest ratio that still does perceptual work. Treat this as design theory rather than measured psychophysics; it rhymes with Weber–Fechner but he did not prove it. It is, however, exactly the property you want in a level-of-detail ladder.
A Frenchman, Gérard Cordonnier, had found the same number four years earlier, in 1924, and called it le nombre radiant — the radiant number. That name lost, which is a shame: Cordonnier had priority, and "radiant" describes what the number actually does here. It is the growth rate of a cone radiating from now. We use Cordonnier's name. Every external source you read will say "plastic"; they mean this.
Stack powers of ρ and something falls out: ρ⁷ ≈ 7.16. Van der Laan took that as one complete "order of size" — the full span over which a human can hold sizes in a single perceptual relation — and divided it into eight measures. Each rung is the smallest step that still reads as a step.
This is a ready-made LOD ladder. The conventional 2× zoom step is ρ³ ≈ 2.32 — you are already skipping three of van der Laan's rungs every time you zoom.
Take an algebraic number greater than 1 — a root of some polynomial with integer coefficients. It has conjugates: the polynomial's other roots. They're the number's shadows; the algebra can't tell them apart.
A Pisot number is one where every conjugate is strictly inside the unit circle — every shadow shrinks when you raise it to a power. And here's the consequence. The sum of a number's powers and all its conjugates' powers is always a whole number (it's a trace — the symmetric functions force it). So:
That is the whole trick. A Pisot number is irrational — its powers never land on integers — but they pass ever closer to them. It is the only way to be irrational and quantised at once. Pick a number and watch the distance from αⁿ to the nearest whole number:
The green bars are effectively zero — the number is chasing the integers. Try 1.5 or 1.3: their bars stay tall and jump around forever. They are not Pisot, and their powers never settle.
Aperiodic, so no two events collide, nothing resonates, nothing aliases. Quantised, because ticks and holons are discrete and you have to checkpoint on whole numbers. A rational gives you the second and destroys the first. A generic irrational gives you the first and destroys the second. A Pisot number is the resolution — and ρ is the smallest one that exists (Siegel, 1944). Least growth, least action: the slowest self-similar expansion the integers permit.
This is not an obscure corner. Pisot expansion factors are why quasicrystals work — aperiodic atomic order with sharp diffraction peaks, which won Shechtman the 2011 Nobel. The Penrose tiling's inflation factor is φ. Aperiodic, yet perfectly ordered. Same trick.
The cone has one expanding direction (time, growing by ρ) and a two-dimensional contracting plane (space, rotating and shrinking). So far the plane has been an abstraction. Here it is.
Padovan has a substitution rule that generates it letter by letter — a→b, b→c, c→ab. Apply it forever and you get an infinite, never-repeating word whose letter counts are Padovan numbers, and whose letter frequencies converge on powers of ρ. Now take each prefix of that word, count its letters, and project that count onto the contracting plane. Plot where it lands.
It does not run away. After 60,000 steps the points are still trapped inside a small bounded region — because the plane contracts, everything falls inward. The region they fill is the Rauzy fractal, and it splits into exactly three subtiles, one per letter, whose areas are in ratio ρ⁻¹ : ρ⁻² : ρ⁻³. It tiles the plane periodically with copies of itself.
This is a deterministic embedding: feed it the n-th event in causal order, get back a point in the plane. It never escapes the tile, it never collides, it is self-similar at every scale, and it is a pure function of n — no RNG, no force simulation, no layout pass to converge. If you want to lay holons out on a canvas such that position encodes causal history and the layout is stable, auditable, and infinitely zoomable, this is the map. It is what janus:axis:cone's space plane is, rendered.
Look hard at the two recurrences and one difference jumps out:
Padovan does not use the previous term. Whatever happened one step ago contributes nothing yet. There is a built-in gap between an action and the moment it can bear fruit.
That is a statement about lag, and it is the honest one for planning. Work committed this sprint does not add capacity this sprint — someone has to be onboarded, a review has to land, a build has to go green. Fibonacci implicitly assumes the last step is already available to compound. Padovan assumes it isn't. Real organisations, supply chains, and deliberative bodies all have a refractory period. Padovan is the smallest recurrence that admits one.
It is the same {2,3} structure from the first page, seen from the planning side rather than the causal side — which is why the number of distinct causal histories under a 2-or-3 tick delay is exactly Padovan.
Fibonacci is right for estimation. The whole point of story points is that the gaps are coarse enough that nobody argues 5 versus 6. But the same coarseness is fatal for a horizon, because each bucket is 1.618× the last and the error compounds:
| looking out… | Fibonacci bucket | Padovan bucket | |
|---|---|---|---|
| a month | 13 d | 9 d | 1.44× finer |
| a year | 144 d | 114 d | 1.26× finer |
| ten years | 1597 d | 1081 d | 1.48× finer — both swallow years |
| cells to a decade | 16 | 23 | 50% more gradations — the real win |
An earlier version of this page put the Padovan cell at ten years at 465 days against Fibonacci's 1597,
and called Fibonacci "unusable". That was wrong — an artifact of a ladder that never actually
reached ten years and silently reused its last cell. The real figure is 1081 days. Both ladders swallow
years that far out: Padovan swallows 3.0 of them, Fibonacci 4.4.
The argument that does survive is the last row. Reaching a decade takes 23 Padovan cells against
16 Fibonacci — 50% more usable gradations across the same span. And ρ is provably the floor:
the smallest growth rate any self-similar integer recurrence can have. You cannot coarsen more gently than this
and stay self-similar. That case never needed the inflated number.
Padovan degrades gracefully because ρ is the gentlest self-similar coarsening there is — and by van der Laan's argument, it is gentle right down to the threshold where two buckets would stop looking different at all. You cannot do better and still be self-similar.
The first version of this work seated events in the cone at ρ's rotation, 139.6719°, and claimed a bounded gap at any count "forever". That was wrong, and the code has been fixed.
ρ's angle, measured in turns, has continued fraction [0; 2,1,1,2,1,2…]. Those 2s make it comparatively easy to approximate with a fraction — and 183 × 139.6719° lands 0.038° from a whole turn. So the 184th event sits on top of the 1st. The golden angle's continued fraction is all 1s (the "most irrational" number), and the three-gap theorem bounds its worst gap ratio at φ² ≈ 2.618 for every count, forever. No resonance can exist.
| events in the cone | hash % 360 (old) | plastic 139.67° | golden 137.51° |
|---|
139.6719° is a fact about the Padovan operator's dynamics — it is what the causal tick genuinely does, and it is the right angle when you are modelling the tick. 137.5078° is the answer to a different question: how do you pack points around an axis without collision. Phyllotaxis converges on φ for exactly that reason, and so should a renderer. Both encode causal order equally well, so nothing is lost by seating with the one that cannot clump. ρ still governs the cone's geometry — how fast it opens — which is where it is derived rather than chosen.
1/ρ = 0.754878 is a seductive number. It feels like it should be the rate at which a pheromone trail evaporates, or money demurrages, or memory fades — the natural dual to a cone that opens at ρ. I went looking for the derivation. Two candidates failed. One survived, and it is not the one you'd expect.
The story: the cone opens at ρ per tick, so any old commitment accounts for an ever-smaller slice of the futures still live — weight it by ρ−age and you're just tracking its true share. Elegant. And false. Compute it: fix a commitment at t=6, then measure what fraction of all length-n paths still pass through it as n grows. It converges to 0.37007 and stays there — at age 34, at age 44, at age 54. A commitment's share of the future is set the moment it is made and never erodes. The cone grows, but the commitment's cone grows with it, at the same rate. Nothing decays.
The story: deposits echo through a {2,3}-latency network, so they arrive at a rate growing like ρⁿ; retention must be exactly 1/ρ to balance that, putting the trail at criticality — slower and the trail runs away (premature convergence, the classic ACO pathology), faster and it never forms. Also false. Simulate it and the trail-to-deposit ratio converges for every retention q < ρ. There is no knife-edge. Nothing is singled out. The tempting phase transition simply isn't there.
Here is where the number genuinely lives. Take the Euler–Lotka equation — the foundational equation of demography, which asks what growth rate r a life history implies. Write it for an organism that matures late and reproduces exactly twice: once at age 2, once at age 3.
So ρ is the intrinsic growth rate of a creature with a two-tick maturation delay and a two-tick reproductive window. And the standing age structure of such a population — the fraction of individuals in each age class — falls off by exactly 1/ρ per class. That is a theorem of stable population theory, not a metaphor.
And it is already on this page. Scroll back to the Rauzy fractal: its three subtiles have areas ρ⁻³ : ρ⁻⁴ : ρ⁻⁵ — successive ratios of exactly 1/ρ. The subtiles ARE the age classes. The fractal is a picture of the stable age distribution.
| if you are modelling… | use | status |
|---|---|---|
| standing age structure of a {2,3} renewal process | ρ⁻ˣ per class | derived — Euler–Lotka |
| pheromone evaporation (ACO / Physarum) | a free parameter, tuned | ρ gives you nothing here |
| demurrage on a holding | set by policy intent (Gesell), not by ρ | ρ gives you nothing here |
| human attention / relevance decay | power law, not exponential | Ebbinghaus / Wixted — better supported than any exponential |
That last row is the one worth internalising. Every exponential decay constant — ρ-derived or otherwise — assumes memorylessness. Human forgetting is measurably power-law: it has a heavy tail, and things you learned long ago decay far more slowly than an exponential predicts. If your substrate is modelling human relevance rather than a physical trail, the constant is the wrong argument to be having — the functional form is wrong first.
In the ant-colony-optimisation literature, τ ← (1−ρ)τ + Δτ — that ρ is the evaporation rate, typically 0.1–0.5, and has nothing whatever to do with 1.3247. Two different ρ's in one codebase is a bug with a long fuse.
Ordered by how much weight each one bears. The green ones follow from the algebra. The grey ones are proposals I find plausible and would want to test before shipping — they are not derived, and I would rather say so than dress them up.
Deterministic map from causal index → bounded 2D point. Stable, collision-free, self-similar at every zoom, no force simulation, no RNG. Position means causal history.
derivedIf a proposal can't be voted on the tick it's raised (people must read it — 2 ticks) and goes stale after 3, your governance channel carries exactly log₂ρ = 0.4057 bits/tick of distinguishable outcomes. Push proposals through faster than that and the body is provably saturated — it is rubber-stamping, and you can now measure that rather than argue about it.
derivedZoom in steps of ρ rather than 2×: the finest gradation that still reads as a gradation (van der Laan), and Pisot means the scale factors land near integers, so tiles snap to the pixel grid instead of drifting off it.
derivedAssign reviewers, jurors, or rotating duties by walking a fixed irrational angle around the member circle. Deterministic (no RNG to trust — anyone can recompute it), low-discrepancy (no clique ever re-forms), and non-repeating. Use the golden angle, per the correction above.
derivedAny process with a maturation delay of 2 and a renewal window of {2,3} has intrinsic growth rate exactly ρ, and its standing age classes fall off by exactly 1/ρ. Euler–Lotka gives this in two lines. It is the one place 1/ρ is genuinely forced.
derivedI proposed this, then tested it. It fails. The premise was that an option's value tracks the share of futures it keeps open, and that this share shrinks as the cone grows. It doesn't: a commitment's share of length-n paths is constant in age. There is nothing to discount. Same failure kills "ρ-demurrage".
retracted — tested, falseKeep Fibonacci for story points; model team capacity as P(n) = P(n−2) + P(n−3), so work committed now contributes only after the onboarding gap. The recurrence is the lag, stated honestly.
proposal — test firstBack off by 1.3247× instead of 2×: more attempts inside the same window, self-similar spacing, near-integer intervals. Fine for heartbeat and prefetch. Not fine for congestion control without a cap — 2× exists for a reason.
proposal — test firstρ is not a nature constant. φ has (contested, often overstated) phyllotaxis claims; ρ has essentially none — I searched. Its home is architecture, and any system with a maturation lag. Pheromone evaporation at 1/ρ and ρ-demurrage were tested and refuted, not merely unproven. Tune those like any other free parameter.
refuted