Concept

Self-similarity — where it appears

The property of looking the same after a change of scale. A logarithmic spiral has it exactly, and so does the ladder of parastichy pairs, whose rungs sit a fixed factor apart in the rise.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A logarithmic spiral growing by 3.20× per turn. Fitting log r against angle on the drawn points returns 3.2000, and the model's worst residual is 9e-16 in log r.

Growth as a rule

A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit.

shells · Spiral

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21. Consecutive transitions are 0.382, 0.382, 0.383, 0.382 of the previous rise — 1/φ² is 0.3820.

The Fibonacci ladder

Lower the rise on a cylinder and the parastichy pair climbs — 1 and 2, then 2 and 3, then 3 and 5 — each rung the sum of the two before it. Nothing in the arithmetic mentions Fibonacci, the transitions sit at computable rises, and consecutive ones stand in the ratio 1/φ².

cylinder · Ladder

A spiral with no clock

The growth factor a shell's curve gives up is a rate per turn of the shell's own coiling, and a turn is not a unit of time. Four clocks — the aperture advancing at a constant angular rate, adding a constant length, a constant area, a constant volume — trace the identical curve: every mark one of them leaves lies on r = W to the power theta over two pi exactly, so a fit through any of them returns the same factor. What differs is where the marks are, and the difference is enormous: at 3.2 per turn the outermost of three whorls holds 33.4, 71.0, 90.3 and 97.0 per cent of the record. It also breaks the instrument. The routine that recovers a growth factor unwraps the angle by assuming successive points advance less than half a turn, and five of the twenty readings here leave gaps past that — returning 3.33 where the curve was built at 3.20, and 8.19 where it was built at 6.85.

shells · Spiral

A pair of dividers opened to 1/50 of a shell's outer radius, walked over the outer 3 of its six turns. A nautilus spiral of six turns growing 3.2 times a turn. The section is taken to preserve only its outer 3 turns; the lost inner whorls are drawn faint. Dividers opened to 1/50 of the outer radius walk from the innermost preserved whorl to the rim in 48 steps; the right panel magnifies the first steps and the whorls inside them. In radii of the whorl the walk starts on the opening is 0.66, against the safe 0.789, so the walk reads the factor exactly. A walk at this opening is exact over any span shorter than 3.16 turns.

The rim sets the opening

A pair of dividers reads a nautilus's growth factor exactly when its opening is under 0.789 of the radius of the whorl it starts on — a quarter of a millimetre at the true centre of a real shell, which no hand can set. A real section starts where its whorls can be read, and a person sets the dividers against the shell in front of them. Measured that way, the rule becomes a span: dividers opened to a share f of the outer radius are exact over the last log((√k − 1)/f)/log k turns of any shell — 3.76 turns at a hundredth, 3.16 at a fiftieth — and the change-over falls exactly there at every opening tried. Held against the rim, their error grows with the span rather than falling, so a section with its centre broken away is read more exactly, not less; and the whole budget still refuses the golden spiral at every span from three quarters of a turn to six for any opening up to a fiftieth.

shells · Nautilus

Named alongside it

The objects these essays reach for when they reach for this one.

Growth factor Logarithmic spiral Whorl Honest limits Model scope Claim testing Continued fraction Convergents Dominant pair Error propagation Fibonacci Gnomonic growth

All concepts