the second hundred · metaphor 195
There are truths you can only approach — never say outright, only get nearer to. And the honest way to approach one is in steps: the best simple version you can manage now, then a slightly less simple one that sits a little closer, then closer again — each the nearest you can be without more complexity, none of them ever quite it.
Think of how understanding actually arrives. You don't leap to the whole truth of a person, a grief, a craft; you hold a rough picture that is almost right, then a better one that costs a little more nuance, then a better one after that. Each picture is the closest you could be at that level of simplicity — and each is wrong, just less wrong than the one before. The gap narrows; it does not close.
Most truths reward this. Spend a little more care and you swallow most of what's left; the picture snaps into focus and the remaining error goes tiny. But a few truths resist — no simple account of them ever gets much purchase, and you inch closer forever, gaining ground so slowly it can feel like standing still. Those are the ones worth naming: the truths that punish every attempt to say them simply.
| k | pk / qk | decimal | |x − pk/qk| |
|---|
The best fraction in turn
Any number can be written as a continued fraction: its whole part, plus one over (its next whole part, plus one over (…)). Truncate that tower at each level and you get a sequence of ordinary fractions — the convergents p₀/q₀, p₁/q₁, p₂/q₂, …. They are not just a way in; they are provably the best way in. Each convergent is closer to the target than any fraction with a smaller denominator. To beat pk/qk you cannot be cleverer at the same level of simplicity — you must accept more complexity.
The convergents also bracket the truth: one lands just above, the next just below, the next above again, each overshoot smaller than the last. And the speed of approach is written in the coefficients. A large coefficient means the previous convergent was already almost perfect, so the error takes a sudden plunge — π's fourth coefficient is 292, which is exactly why 355/113 is famous for nailing π to seven places. A run of small coefficients means slow, grinding progress.
What to try
Every fraction and every error in the panel is generated live from one short loop — nothing is tabulated. Pick π and watch the error curve plunge: it drops a couple of decades at a time and reaches the machine's precision floor within a dozen steps. Then pick √2 for a steady geometric slide, or e for a patient march with no dramatic leaps. Each step down the table is a fraction that beats every simpler one before it.
Now pick φ, the golden ratio, and watch it crawl. Its curve is the shallowest, straightest line on the chart — after the same number of steps it has barely descended while π has bottomed out. That is not an accident of scale: φ = [1; 1, 1, 1, …] is, by a theorem, the hardest number in existence to approximate. Its convergents are ratios of consecutive Fibonacci numbers, and the readout q²·|φ − p/q| keeps settling near 0.447 — the exact wall no irrational can cross. Dial the custom slider onto a clean value and the opposite happens: the expansion ends, and the error drops to nothing.
The mapping
Read the target as a truth you can never write down exactly, and each convergent as your current best simple account of it — the nearest you can be while keeping the story this simple. Getting closer always costs: a longer denominator, a finer distinction, more of your attention spent. The error never reaches zero, and that is not failure but the shape of honest approach — forever nearest, never exact.
The coefficients tell you what kind of truth you're facing. A large one is a truth that suddenly yields — one extra idea and the gap collapses, the way a hard problem sometimes gives way to a single clean sentence. A truth that is all small coefficients is the cruel kind: nothing about it is ever nearly captured by anything simple, and every step forward is grudging. The golden ratio is the mathematical portrait of that truth — the one that resists every simple approximation the hardest.
Read as life lessons
A convergent is the closest you can get at that complexity. To do better you don't think harder at the same level — you pay for a finer instrument. Knowing which you need is half the wisdom.
The approach isn't one-sided. You overshoot, then undershoot, each miss smaller — above the truth, below it, above it. Being wrong in alternating directions is how you home in, not a sign you're lost.
A single large coefficient means a sudden leap — one added idea closes most of the gap. Other truths never offer that gift; they only ever reward slow, patient denominators.
In the wild
A year is ≈365.2422 days. Its continued fraction gives 365 + 1/4 (Julian), then the closer 365 + 8/33, then 365 + 97/400 — the Gregorian leap-year rule, a convergent chosen for being simple and accurate.
You can't cut a fractional tooth. Clockmakers and Huygens' planetarium picked gear-tooth ratios that were convergents of the true orbital periods — the best whole-number ratio a modest number of teeth could hold.
Twelve semitones per octave comes from log₂(3/2) ≈ 0.585 ≈ 7/12 — a convergent. Push further and you get 53-tone tuning, the next fraction that lands even nearer the perfect fifth.
The mapping, exactly
| Mathematics | Life |
|---|---|
| the target irrational x | A truth you can never state exactly — only get nearer to. |
| convergent pk/qk | Your current best simple account of it — the nearest you can be at this level of detail. |
| best at its denominator | The closest you can get without adding complexity; to do better you must pay for more. |
| the error never reaches 0 | Forever nearest, never exact — the honest shape of approaching a truth. |
| convergents alternate above/below | Approaching by over- then under-shooting, each correction smaller than the last. |
| a large coefficient ak | A truth that suddenly yields — one added idea closes most of the gap. |
| φ = [1;1,1,…], all ones | The truth that resists every simple approximation hardest of all. |
The honest model
Hand the panel a number x. It runs one loop: take the whole-number part a₀ = ⌊x⌋, subtract it, and flip what's left — x₁ = 1/(x − a₀) — then repeat on x₁. That stream of whole numbers a₀, a₁, a₂, … is the continued-fraction expansion. Fold them back with a two-line recurrence — pk = akpk−1 + pk−2, qk = akqk−1 + qk−2, started from p₋₁=1, p₋₂=0, q₋₁=0, q₋₂=1 — and each pk/qk is a convergent. Nothing is stored in advance; the fractions you see are built from those two recurrences, and the error is simply |x − pk/qk| evaluated each step.
Why is the golden ratio worst? A large coefficient ak+1 means the last flip barely left a fraction to flip — the convergent you just formed was already nearly perfect, so the error takes a huge step down. φ is [1; 1, 1, 1, …]: every coefficient the smallest a whole number can be. No term is ever large, so no convergent ever gets a shortcut, and the error shrinks by the slowest constant factor there is (about φ² ≈ 2.6 per step). Its convergents are consecutive Fibonacci numbers, and qk²·|φ − pk/qk| settles near 1/√5 ≈ 0.447 — the exact bound Hurwitz proved no irrational can beat. φ lives pressed against that wall.
One honesty note, printed in the panel where it bites: a double-precision number is really a nearby rational, so after roughly a dozen steps the leftover fraction is pure rounding and the tail of the true expansion becomes fiction. The panel caps the depth and stops once the error hits the floating-point floor (~10⁻¹⁶); every fraction it shows above that line is exact arithmetic. Dial the slider to a clean value and you'll see the expansion legitimately terminate — the last convergent equals x with error zero.
Where the metaphor tears
Convergents march monotonically inward because x never moves and the error only ever shrinks. Real approach to a truth is nothing like that: new evidence shifts the target, a better account can turn out worse, and you can end a year further from understanding than you began. The metaphor models a fixed truth patiently narrowed — not a live one that fights back.
Convergents are provably best only against a clean, agreed measure of simplicity — the size of the denominator. Human understanding has no such metric. What counts as the "simpler" account of a person or a grief is contested, and two honest people's next-best approximation won't even be the same fraction. Strip out the tidy definition of simple and "best simple approximation" loses its teeth.
The whole picture — forever nearest, never exact — is the irrational case. A rational number's expansion terminates: you reach it exactly, in finitely many steps, and you're done. Not every truth is unreachable. Some yield completely to a simple description and ask nothing more of you, and mistaking those for the infinite kind is its own quiet error.