Phyllotaxis
the spirals are counted, not admired
Concept

Measurement sensitivity — where it appears

How much a reading moves when one of the conditions it was taken under moves, as against how much it scatters when nothing about them changes. A growth factor fitted to a nautilus section is sensitive to its assumed centre roughly as the inverse square of the span of arc, so one offset costs a half-turn fragment sixty times what it costs three turns.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

A golden spiral and a nautilus spiral over 2.5 turns, from the same start. After 2.5 turns the golden curve is 7× larger. The growth factors are 6.85 and 3.2, a factor of 2.14 apart.

The nautilus question

A golden spiral grows by 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14 — not a rounding error, not an artefact of where the centre is assumed to be, and not close.

shells · Nautilus The nautilus error budget with its three span-dependent entries priced at one span, against the gap the golden claim asks it to clear. The budget with the dividers set aside, but with its centre, span and oblique entries replaced by the worst joint error of a quarter-radius centre offset and a ten-degree tilt at one span of arc. 0.5 of a turn: 210.4%; 0.75 of a turn: 111.5%; one turn: 61.0%; 1.5 turns: 31.9%; 2 turns: 29.3%; 3 turns: 25.1%. The upper level line is the gap, 114.2%; the lower is the budget as it was priced, 54.4%, with its three coupled entries at 32.2% together. The budget clears the gap from 0.75 of a turn and not below it.

Three entries and one span

The error budget for a nautilus added seven ways a growth factor read off a section can be wrong, and asked whether they were independent. Three of them are not: the displaced centre, the span of arc and the oblique view are one error priced three ways, at two turns, over a turn and more, and at half a turn — 32.2 per cent together. A section has one span. Read together at one span, a quarter-radius centre and a ten-degree tilt come to 7.1 per cent at two turns and 188 at half a turn, and in their worst orientation they always add to more than their sum. So the budget refuses the golden spiral from three quarters of a turn of shell upward, and below that it cannot.

shells · Nautilus A pair of dividers opened to 1 of the innermost radius, walked along two and a half turns and along four. A nautilus spiral growing 3.2 times a turn, stepped at a fixed opening of 1 of its innermost radius from the inner end outward. Over 2.5 turns the walk takes 17 steps against a floor of 22 and reads the factor +18.2% too high; Over 4 turns the walk takes 104 steps against a floor of 132 and reads the factor +4.2% too high. The opening at which every span reads exactly is the square root of 3.2 less one, 0.789 of the innermost radius: under it the first step never carries the angle past half a turn, whatever the span.

The dividers belong to the opening

The nautilus error budget's largest entry, dividers walked along the shell at 278 per cent, was set aside as the historical method. Priced at one span it turns out not to belong to the span at all. A person sets a pair of dividers to an opening and walks until the curve runs out, so the step count grows with the arc exactly as fast as the floor on it does: opened to less than the square root of the growth factor less one — 0.789 of the innermost radius for a nautilus — they read the factor exactly over every span from half a turn to six, and opened wider they read it too high over every span, least over the longest. The 278 per cent was nine steps along five turns, an opening of 37 innermost radii. With the dividers opened to anything up to five radii, the whole budget refuses the golden spiral from three quarters of a turn upward.

shells · Nautilus How much a twist of the rim changes the divergence between neighbouring organs, across a 900-organ head, for six shapes of twist. A twist of 0.25 radians at the rim, shaped as a·(r/R)^p, changes the angle between one organ and the next by a·(p/2)·ρ^(p−2)/(N − 1) at a distance ρ from the centre: the six lines are exponents 0.5, 1, 1.5, 2, 3, 4. At two the change is the same everywhere — 0.0159° — which is a different divergence angle; below two it is largest at the centre, above two at the rim. The shaded ring is the outer annulus, and the band between dotted lines is the flag's resolution on an untwisted 900-organ head: grown 0.0125° off the golden angle it counts consecutive Fibonacci numbers, 0.015° off it does not. At this twist the outer annulus passes it for exponents 2, 3, 4.

The flag reads the angle, not the twist

A twisted seed head is caught first by its counts: at half a radian some band stops returning two consecutive Fibonacci numbers, which an untwisted golden head never does. Twist the head in proportion to the square of the radius instead and the organs land exactly where a head grown at the golden angle plus a/(N − 1) puts them — to a billionth of a spacing — so the two read the same pairs in every band, and the flag fires on both. The flag detects a divergence that is not golden, and it has a resolution: 0.015° on 900 organs, 0.01° on 2,400, 0.003° on 9,000. Every twist shape from a half to four is flagged exactly when its change to the divergence in the outer annulus passes that resolution. So a flagged head is not golden, and nothing in its counts says whether it was twisted or grown that way.

lattices · Recovery The divergence read from the positions ring by ring on a 900-organ head twisted half a radian at the rim, for six shapes of twist. A 900-organ golden head twisted by 0.5 radians at the rim as a·(r/R)^p, displaced by 0.1 of a spacing, read in six rings from 0.35 to 0.95 of the radius by the organs' phase sums within the counts' interval. Above the golden angle, ring by ring: p 0.5: 0.0371°, 0.0212°, 0.0179°, 0.0126°, 0.0114°, 0.0091°; p 1: 0.0451°, 0.0306°, 0.0271°, 0.0214°, 0.0192°, 0.0176°; p 1.5: 0.0426°, 0.0324°, 0.0320°, 0.0277°, 0.0264°, 0.0253°; p 2: 0.0374°, 0.0318°, 0.0334°, 0.0310°, 0.0316°, 0.0321°; p 3: 0.0241°, 0.0237°, 0.0306°, 0.0334°, 0.0385°, 0.0437°; p 4: 0.0148°, 0.0156°, 0.0249°, 0.0313°, 0.0414°, 0.0530°. The square law reads flat, 0.0310° to 0.0374°, as a head grown at another angle does; every other shape falls or rises across the head.

The positions read the twist's shape

A skipped Fibonacci count says a seed head's divergence is not golden, and cannot say whether it was twisted or grew that way; only the radial profile of the angle between neighbours can, and the counts are too coarse to see it. The organs' positions are not. The birth order is not in them — consecutive organs at the rim of a 900-organ head are a hundred and sixth of a spacing apart in radius, so a hundredth of a spacing of displacement scrambles a quarter of them — but each counted family is, and the sum of its phase angles across the organs peaks at the local divergence with no organ named. Displaced by a tenth of a spacing, a 900-organ head's outer annulus reads its divergence to 0.0001°, a hundred times finer than the flag resolves it, and the two annuli tell every twist shape from a changed angle at a quarter of a radian or less, no later than the flag fires. The square-law twist stays inside the untwisted heads' spread at every size and displacement: it is a changed angle, organ for organ.

lattices · Recovery A seed head with half its rim missing, and three places its centre could be put. A 900-organ golden head, every organ displaced by 0.1 of a spacing, with the organs beyond seven tenths of the radius removed over half the head. The centroid of what is left lies 3.099 spacings from the centre the head grew about; the centre about which the inner annulus's two counted families are most coherent lies 0.0132 spacings from it. The inset magnifies half a spacing around the true centre.

The centre the spirals give

Reading a seed head's divergence from its organs' positions needs the head's centre, and a photograph does not give it. A misplaced centre turns every organ's angle by an amount that varies round the head and grows toward the middle, so it could read as a twist. It does not, until it is large: a 900-organ head tolerates a centre a sixth of a spacing off, a 2,400-organ head a quarter. The centroid of the organs finds a whole head's centre to a hundredth of a spacing, and read about it the positions separate every twist shape exactly as well as with the centre given. The radius law, which looks like the natural fit, does not find it at all. And when half the rim is missing the centroid moves three to five spacings and turns an untwisted head into a twisted one, while the centre about which the counted spirals are most coherent still lands within a hundredth of a spacing.

lattices · Recovery A golden seed head photographed off its axis, with the squash its second moments find. A 900-organ golden head, every organ displaced by a tenth of a spacing, photographed 20° off its axis: every coordinate along a bearing of 7° shortened by the cosine of the tilt. The head's second moments about its centroid put the short axis at 7.7° and imply a tilt of 19.75°. Read square on, the inner annulus gives a divergence of 137.5080° and the outer 137.5080°; stretched back along the short axis first, 137.5078° and 137.5077°, against the golden angle's 137.5078°. The bands count 34 and 55 spirals inside and 55 and 89 outside.

A head photographed from the side

A seed head photographed off its axis is squashed by the cosine of the tilt across one bearing, and the squash turns every organ's angle by an amount that goes round the head twice. Read square on, a 900-organ golden head tolerates twelve to fourteen degrees of tilt before its two annuli disagree by more than any untwisted head's do, and the spiral counts never notice at all. Past that, the reading is not a smooth drift but a scatter of heads misread one at a time. Stretching the head back along the short axis its own second moments find restores the square-on reading exactly to thirty degrees, even though the moments cannot see a tilt under three or four degrees. And the square-law twist stays hidden: it is a changed angle organ for organ, so no camera, at any angle, sees it differently.

lattices · Recovery A golden seed head photographed twenty degrees off its axis by a camera close enough to see it in perspective. A 900-organ golden head, every organ displaced by a tenth of a spacing, photographed by a pinhole camera 5 head radii from its centre and tilted 20° across a bearing of 7°: the near side of the head larger in the picture than the far side. Read square on, its inner annulus gives 137.5064° and its outer 137.5090°; stretched back along its second moments' short axis, 137.5069° and 137.5013°; un-projected through a camera fitted from the organs — tilt 19.9°, bearing 10°, distance 5.15 radii — 137.5083° and 137.5078°, against the golden angle's 137.5078°. Of the ten heads photographed this way, 1 are read as untwisted square on, 0 are read as untwisted by the ellipse, 9 are read as untwisted through the fitted camera.

A head seen in perspective needs a camera, not an ellipse

A seed head photographed off its axis from far away is squashed, and stretching it back along the short axis of its own second moments restores its reading exactly. Photographed from close up, the near side of the head is larger than the far side, and the stretch stops working: at twenty degrees off the axis it reads every head right from ten head radii and none from five. The organs' centroid drifts towards the near side as the camera comes in, but that is not why — the same stretch about the head's true centre fails at the same distances. What fails is the taper. A camera fitted from the organs alone, its tilt, bearing, distance and the head's centre chosen so that the head it un-projects is round, centred and evenly packed, reads seven to ten heads in ten right at every distance down to two radii.

lattices · Recovery A golden seed head laid on a dome and photographed twenty degrees off its axis from infinitely far. A 900-organ golden head, every organ displaced by a tenth of a spacing, laid evenly on a paraboloid dome 0.2 head radii high and photographed by a camera infinitely far away, tilted 20° across a bearing of 7°. The dome lifts the centre of the head towards the camera, and seen from the side every organ moves along the bearing by its height times the sine of the tilt, more at the centre than at the rim. Of ten such heads, 0 are read as untwisted after the moments' ellipse and 0 through the camera fitted for a flat head, which puts the camera 7.4 head radii away at a tilt of 20.2° (median).

A dome is read as a nearer camera

A camera fitted from a photographed seed head's organs alone restored the reading of a flat head seen in perspective from as close as two radii. Lay the same heads on domes and the camera is fooled in a particular way: it finds the tilt, its median within about a degree, and puts itself nearer than it is, by an amount set by the dome's height, so that a dome a fifth of the head's radius high photographed from infinitely far is read as a flat head photographed from seven radii. The reading it hands on is wrong. Off the axis, a domed head is misread once its height times the sine of the tilt passes about four hundredths of the radius — the same number at which a flat head's perspective broke the old correction — and only a photograph taken square on is safe.

lattices · Recovery

Named alongside it

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

Honest limitsDivergence angleGolden angleIdentifiabilityParastichy pairRound tripVogel's modelDisplacementClaim testingGolden spiralGrowth factorLogarithmic spiral

All concepts