Phyllotaxis
the spirals are counted, not admired
Concept

Joint distribution — where it appears

Two quantities measured together on the same items and drawn as a cloud, rather than one of them summarised within classes of the other. A cell's area against its side count is one, and it shows how much of the area the side count accounts for where a line through class means cannot.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Cell area against side count, at 0% disorder. The dashed line is Lewis's law, (n−2)/4. The fitted slope here is 0.014 against his 0.25, and the side count accounts for 20% of the variation in area.

Lewis's law wants disorder

Cell area rises linearly with side count — measured on cucumber epidermis in 1928 and quoted ever since as a property of packed tissue. It holds beautifully on a random point set, with a fitted constant of 1.64 against Lewis's 2. On a phyllotactic head it does not hold at all: the slope is 0.009, and area and side count are almost independent.

tissue · Lewis Cell area against side count on a 900-organ head, every organ displaced by 0.2 of a spacing. The joint distribution of cell area, as a multiple of the mean, and side count, over the 639 interior cells of a golden head with every organ displaced by 0.2 of a wall spacing, seed one. The dashed line is Lewis's law, a quarter of the mean area for each side; the solid line is the fit, at a slope of 0.113, and the side count explains 30 per cent of the variation in area. Classes: 4 sides, 16 cells, mean 0.75; 5 sides, 159 cells, mean 0.89; 6 sides, 295 cells, mean 1.00; 7 sides, 146 cells, mean 1.11; 8 sides, 22 cells, mean 1.22; 9 sides, 1 cells, mean 1.26.

Lewis's law needs the sides to vary

Lewis's law holds on a random set of points and fails on a golden-angle head. Walked from one to the other by displacing every organ independently, the head's Lewis slope reaches half a random set's at a fifth of a wall spacing and nine tenths by seven tenths, and in between it explains up to 41 per cent of the variation in cell area — more than the 31 per cent it explains in the random set. Moved instead by a smooth field correlated over eight spacings, the head's cell areas become nearly as varied as a random set's and its slope stays at nought, because its side counts stay the lattice's. The law is not about how varied the cells are. It is about how varied their sides are.

tissue · Lewis What a reading of 34/55 becomes when each family is closed at a mark of its own. Each axis is one family's closing error in degrees of the circle, out to 24° either way; every rectangle is the set of closing errors that give one rounded pair, keyed as read right, sharing a factor (so announcing itself), or sharing none (so passing silently). The diagonal is one mark shared by both families, where the first silent readings are 33/53 and its mirror at 9.82°; off it the nearest are 33/56 and its mirror at 5.29°, with the two marks erring on opposite sides. The ellipses are one and two spreads of the marks' joint distribution at 7.2° a mark and a correlation of 0.50: 21.0% of readings right, 58.6% announced and 20.4% silent.

Two marks chosen by one eye

A counter traces each family of spirals from a starting organ of its own, so a reported pair carries two closing errors, correlated because one eye chose both. Letting them differ costs 34/55 its ten-degree margin — 33/56 and 35/54 share no factor, and marks that err 5.3° in opposite directions reach them — while 21/34 keeps its margin whatever the marks do. And it decides the second annulus. At a spread of 7.2° the relation passes right readings 2.8 times as readily as silent ones when the marks are independent, 1.25 times at a correlation of 0.9, and stops telling them apart at 0.98; where it does work it keeps one reading in forty-six.

wrong · Count value

Named alongside it

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

Explained varianceHonest limitsLewis's lawOrder and disorderVoronoi cellsCoprimeDiscriminationDisorderDisplacementDivergence angleFibonacciIdentifiability

All concepts