Phyllotaxis
the spirals are counted, not admired
Concept

Weight triangle — where it appears

The triangle formed by the three weights a branch point is minimised against, one per tube, whose shape fixes both fork angles. Where it collapses towards a straight line the cost around the optimum flattens with it, which is why a twig's angle is at once the hardest to predict and the least worth measuring.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The fork the cost chooses at a daughter ratio of 1: 37.47 and 37.47 degrees. Three ends held fixed, three weights fixed by the radii, and the branch point put where the total cost is least. At the optimal radius the pumping term is exactly half the upkeep term, so a segment's weight is its own cross-section and the three weights here are 1.5874, 1.0000, 1.0000. Minimising directly over the position — a 41×41 grid re-centred and shrunk 220 times, told nothing about any formula — puts the daughters at 37.4673° and 37.4673° from the parent's own forward direction, against the closed form's 37.4673° and 37.4673°.

The angle the cost chooses

Murray's exponent falls out of minimising a cost over the radius of a tube. The same cost minimised over the position of the branch point instead fixes both fork angles, and 281 networks minimised on a grid told nothing about any formula agree with the closed form to under two ten-thousandths of a degree.

branching · Fork angle The three weights at an exponent of 2 and 3: one closes a triangle and one is a straight line. A branch point minimising a weighted sum of three lengths has an interior solution only when the three weights close a triangle, and the weight on a segment here is its own cross-section. At an exponent of 3 the three areas clear that condition by 0.4126, and the triangle they close is what the two fork angles are read off. At an exponent of 2 the parent's area is exactly the daughters' areas summed — that is what area conservation says — so the slack is -4.44e-16, the triangle collapses onto a line, and the fork closes to 0.0000 degrees. Leonardo's rule does not predict a different angle here; it predicts no angle.

A rule that predicts everything

Leonardo's rule says a fork conserves cross-section, and cross-section is exactly the weight the branch point is minimised against. So the three weights land on the boundary of the triangle inequality, the cosine comes out at one to the last bit, and the rule predicts no angle at all — and the free constant its own derivation leaves behind then walks the prediction across every angle a fork could have.

branching · Fork angle Two lopsided crowns sized by the larger of flow and stress, drawn at the angles their radii predict. Two crowns of 9 orders whose every fork sends a thick daughter one order down and a thin daughter 2 orders down, so the crown has 89 twigs; each branch is 0.794 of its parent's length times the shortening of its order, and each takes the larger of the radius equal bending stress asks for and the radius flow asks for, the two equal at order 4. Every fork is drawn with each daughter at the angle the transport cost predicts from that fork's three radii and tips. On the left the twigs are the radii the cost itself prefers, and no daughter anywhere opens wider than the angle Murray's law gives its own fork. On the right the twigs spend twice their upkeep on pumping: the thin daughters open furthest past Murray's angle, by 12.5°, at the fork one order inside the handover, and the thick daughters furthest, by 10.4°, at a fork 1 order nearer the trunk. Branches sized by stress and branches sized by flow are drawn in two colours.

Two daughters cross Murray's angle together

On a crown sized by the larger of flow and stress whose every fork is lopsided, the thin daughter's subtree a few orders shallower than its sister's, the handover stops being one generation and becomes a band of forks one wider than the offset. Every fork past the band still opens each daughter at its own Murray angle, to a ten-trillionth of a degree. The thin daughter never overshoots alone: at the first fork whose daughters are both sized by flow, only the parent's weight differs from Murray's, and one side of a weight triangle passes its Murray length at a single value, so both daughters cross together, at a twig share within two hundredths of the even crown's. Past it they peak apart, the thin one by more.

branching · Fork angle Two crowns whose forks are lopsided by different amounts, drawn at the angles their radii predict. Two crowns of 9 orders whose every fork sends its thin daughter's subtree nought to three orders shallower than its sister's, the offset drawn at each fork; 76 twigs, each branch 0.794 of its parent's length per order, sized by the larger of flow and stress with the two equal at order 4 for the median branch of that order. Every daughter is drawn at the angle the transport cost predicts from its fork's three radii. On the left the twigs are the radii the cost prefers and no daughter opens past its Murray angle; on the right they spend twice their upkeep on pumping. This crown first turns back at a twig share of 0.512, at a fork whose parent is 1.008 times as thick as flow alone asks.

Mixed forks turn back at the cost's own share

On a crown sized by the larger of flow and stress, a fork whose two daughters are sized by flow opens both past their Murray angles exactly when the twigs spend more than q⁴/(1 + q²) of their upkeep on pumping, where q is how much thicker than flow its parent is — a closed form that is the even crown's threshold and every one-offset crown's at once. A crown lopsided by one amount at every fork has one q at its handover and a margin above the cost's own one half: 0.64 at a length ratio of 0.794. A crown lopsided by different amounts has a parent at every q, some barely thicker than flow, and turns back at 0.500 to 0.514. A branch's weight is its cost per length, the cost is flat at its own radius, and a slightly thickened parent with slightly thin twigs is lighter than its daughters expect.

branching · Fork angle Read at an opening of a degree, a blend turns the mixed crowns back later than the larger of flow and stress does. Twelve mixed crowns of fourteen orders and the even crown, every branch sized by the blend of flow and stress radii at the exponent on the axis, and in the last column by the larger of the two. For each, the twig share at which some daughter first opens more than 1° past its Murray angle. The bars are the twelve mixed crowns' range and the rings the even crown. At k = 2 the mixed crowns turn back at 1.122 to 1.151, at k = 4 at 0.817 to 0.821, at k = 8 at 0.706 to 0.754, and under the larger of the two at 0.603 to 0.672; the even crown at 1.084, 1.061, 0.943 and 0.724. The faint line is the same crowns read strictly, whose mean falls to 0.5014 at k = 8.

A blend turns the twigs back first

Size every branch of a mixed crown by a blend of what flow and stress ask, rather than by the larger, and no branch is exactly flow-sized any more. Read strictly, that removes every margin the crowns had: the forks of two twigs open past Murray's angles first, on a closed form of their own, and by a blend exponent of eight every crown does so within two thousandths of the cost's own one half, the even crown's 0.643 included. But those openings are thousandths of a degree. Read at a degree a protractor could see, a blend restores a margin at every exponent tried, 1.12 at two, 0.82 at four, 0.71 to 0.75 at eight, against 0.60 to 0.67 for the larger, and the fork that opens first climbs from the twigs to the handover as the blend sharpens.

branching · Fork angle

Named alongside it

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

Fork angleMurray's lawClaim testingClosed formDaughter ratioMetabolic costBranching exponentHonest limitsBranch pointDa Vinci's ruleDegeneracyOptimisation

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