A whorl feels the far side of its stem

The window was not carrying it read the rate edge of a Lucas seed on a fine rate — the slowest decline of the rise at which a stem still keeps the seed’s pairs — and found it proportional to the seed above about sixty whorls, at a ratio of edge to seed that depended on one thing besides the seed. An ordinary stem, one organ at a time, settles at about 2.184. A bijugate stem settles at 2.216 and a trijugate stem at 2.215. The two whorled stems agree to a thousandth and the ordinary stem sits one and a half per cent below them.

That essay asked which property of a whorled stem the higher level follows. A bijugate stem differs from an ordinary one in its angle, which is half as large, and in its timing, since its organs arrive two at a time at one height. The proposed experiment was an ordinary stem grown with its organs forced to arrive in pairs: ordinary in its angle, jugate in its timing.

This essay does not run that experiment, because on the one stem where the difference lives — the folded one — timing turns out not to be a difference at all. What is a difference is something nobody had named, and it can be turned like a dial.

Folded, a whorled stem keeps ordinary time #

Every whorled stem here is read after folding. A k-jugate stem’s whorl of k organs, all at one height, is sent to one node: its azimuth multiplied by k, its height by k, its rise by k squared. On an ideal lattice that is exactly the ordinary lattice, which is why a bijugate stem’s ladder is the ordinary ladder at half the rise, and grown by the rule that grows a shoot the folded stem walks the ordinary ladder with a lag of the same hundredth of a transition.

In the folded picture a whorled stem places one node a whorl, at the ordinary folded rise, on the same schedule an ordinary stem keeps. Its organs arrive together, and folding makes “together” mean “one node”. So whatever separates the folded bijugate stem from an ordinary stem at the same folded rate, it is not when its nodes arrive.

What folding leaves behind #

The thing folding cannot make ordinary is the neighbourhood an organ is placed against. The placement rule puts each new organ where the repulsion from the organs already present is least, summing an inverse cube of the distance to each, and on a cylinder that distance is taken the short way round.

On an unfolded bijugate stem every whorl below the new organ is two organs, half a turn apart. Each is a real organ, and each is reached the short way round from wherever the new organ is tried. Multiply every azimuth by two and those two organs become one folded node, but the two distances do not become one distance. The near organ is the folded node’s nearest copy, and the other, half a turn away on the stem, is the same folded node a whole folded turn further on.

An ordinary stem’s repulsion counts one copy of each neighbour, the nearest. A folded bijugate stem’s counts two copies, a turn apart; a folded trijugate stem’s, three; a k-jugate stem’s, the k copies that lie within half of k turns either side. A whole number of folded turns is just a scale, and an inverse cube does not care about scale, so the place where the repulsion is least is the same on the folded and the unfolded stem.

Two images are a bijugate stem #

That argument is a prediction, and it is exact enough to test node for node. An ordinary stem is grown by the ordinary rule with one change — its repulsion counts each neighbour at two images a folded turn apart instead of one — and set beside a free bijugate stem grown from the same Lucas seed at the same folded rate, then folded.

The two-image stem coincides with the folded bijugate stem at all 467 nodes. The one-image stem, which is the ordinary stem, coincides at 62: the sixty organs of the seed, the first organ grown above it, and one more far up the stem by chance. From the second grown organ on it is a different stem. The same identity holds with four images against a free tetrajugate stem, at every one of 443 nodes on a shorter stem, because a whorl of four, like a whorl of two, sits exactly on its shares of the turn when it is grown free.

So the folded bijugate stem is not an ordinary stem that happens to be read twice round. It is an ordinary stem whose every organ feels its neighbourhood twice — once where it is, and once a turn away.

The question asked for an experiment that is not needed #

The proposed stem, ordinary in angle and paired in timing, was meant to hold the angle fixed and move the timing. Folding has already shown that the timing of a whorled stem is ordinary, and the identity above shows that the bijugate stem is completely described by an ordinary timing and a second image. Anything a paired-timing stem could add would be a property of a stem whose pairs are not exact whorls, which neither the bijugate nor the tetrajugate stem is.

The dial that does separate them is the one the identity hands over. The number of images an ordinary stem counts changes the far side of its neighbourhood and nothing else — not the seed, not the schedule, not the window, not the grid — and it can be set to values no whorled stem has, including fractions of a copy.

What the far copy is #

Drawn for the first organ grown above a sixty-whorl seed, the far copies are not small. The rise there is 0.077, which in folded units means the organs are about a third of a turn apart, and a copy a whole turn away is at most three times as far as the nearest one. An inverse cube divides by twenty-seven at three times the distance, and summed over all eighteen neighbours the far copies carry 12.1 per cent of the repulsion this organ feels.

A neighbour drawn here is 0.305 of a turn from the new organ the short way and 0.695 the long way. The ordinary stem counts the first and ignores the second. A bijugate stem counts both, because on its unfolded cylinder the second is a different organ at its own short distance.

One image to five #

Each edge is found as the earlier reading found it: a scan of twelve rates across a bracket set by the seed, a bisection on whole organs of rate, a half-organ scan from two whorls below the edge to two above to check that the stem changes from kept to lost exactly once, and a bisection on a real-valued rate inside that. All thirty stems change exactly once.

One image reproduces the ordinary stem’s published ratios to the digit — 2.076, 2.133, 2.175, 2.166, 2.186 and 2.184 at seeds of 15 to 80 whorls — and two images reproduce the bijugate stem’s: 2.065, 2.141, 2.175, 2.197, 2.215 and 2.216. The level over the two longest seeds goes from 2.1848 with one image to 2.2155 with two, 2.2237 with three, 2.2226 with four and 2.2231 with five.

So the second image is the whole of the step the jugacies differ by. A third moves the level a quarter as far again, and the fourth and fifth move it by less than a thousandth, back and forth. Copies two and three turns away are too far for an inverse cube to feel.

The short seeds go the other way #

The levels are a statement about long seeds, and below thirty whorls the images do something different. At fifteen whorls the ratio is 2.076 with one image, 2.065 with two, 2.075 with three and 2.073 with four: the far copy lowers it, the next restores it, and the spread among them is a hundredth, about what one organ of rate is worth at that seed. At twenty whorls the order is irregular again; at thirty the five image counts lie within 0.003 of one another.

Only from forty whorls up does the far side lift every edge, by 0.030 at forty and at sixty and 0.032 at eighty. A short seed is lost in the rise of about a tenth that every stem shows below thirty whorls, and in that range the far side is one small push among several larger ones.

Three organs a whorl, imposed and free #

A trijugate stem should be three images, and it is not quite. Grown with its whorls imposed exact — one member placed by the rule against the whorls below, the other two copied a third of a turn round — it keeps its sixty- and eighty-whorl seeds to ratios of 2.2237 and 2.2240, and the three-image stem to 2.2236 and 2.2239: the same to two parts in ten thousand, the remainder the window’s edge falling part-way through a whorl. The identity carries over to three exactly as the derivation says it should, once the whorls are made exact.

Grown free, as every earlier essay grew it, the trijugate stem is not exact, because a whorl of three misses its thirds when each member is placed against the ones already there. That stem keeps its eighty-whorl seed to 2.2155, eight thousandths below the three-image level, and at sixty whorls it flickers — kept, lost and kept again across a whorl of rate — so that its edge there is a band rather than a line, as the earlier reading found.

Five organs a whorl #

Five images put the level at 2.2231, beside three and four, and a stem of five organs a whorl with its whorls imposed exact keeps its sixty- and eighty-whorl seeds to 2.2225 and 2.2227, within five ten-thousandths of it. A free stem of five organs a whorl, whose whorls cannot sit exactly on their fifths any more than a whorl of three can, keeps its eighty-whorl seed to 2.2172 — between the bijugate level and the five-image level, nearer the first — and its fifteen- and twenty-whorl seeds to 2.073 and 2.111. At thirty, forty and sixty whorls it flickers, as the free trijugate stems do at most of their seeds.

So the earlier question, whether the level stays with the bijugate and trijugate stems at four and five organs a whorl, has an answer that reverses the question’s premise. A stem of four does not stay with the bijugate stem: its whorls are exact, so it is four images node for node, and four images sit at 2.2226, seven thousandths above two, beside three. A free stem of five lands between, for the same reason a free stem of three does: the far copies lift it and the missed shares let it down. There is no single whorled level for the ordinary stem to sit below — there is a level for each number of copies an organ feels, rising from one to three and flat after, and free whorls that miss their shares sit under their own.

Turning the far side up #

Because the far copy is a term in a sum, it can be counted at any fraction of its weight — something no whorled stem can do — and turned up from nothing in eighths, at six long seeds from fifty to a hundred whorls. The mean ratio over the six goes 2.1837 at no weight, 2.1821 at an eighth, 2.1837 at a quarter; then 2.1893, 2.1922, 2.1992, 2.2074 and 2.2145 in equal steps of weight; and 2.2152 at the full weight, where the stem is the bijugate one.

So the far side is not felt in proportion. A quarter of it changes nothing, the middle half carries almost the whole step, and the last eighth adds seven ten-thousandths. Seed by seed the ratio is rougher still: single seeds move in steps of up to a hundredth between neighbouring weights and fall back as often as not in the first quarter, which is the size of the grain the whole-organ edges were found on and a reminder that one seed’s edge is a single draw. Why the response waits for a quarter of the far side is not established here; the next sections say where along the stem the far side acts, which is where an answer would have to come from.

Where the far side is felt #

The far copies’ share is not a constant of the stem. It is set by how many organs fit round the folded circumference, which the rise decides. At the first organ grown above an eighty-whorl seed, at a folded rise of 0.077, the far side carries 12.2 per cent of the repulsion; at a rise of 0.040, 8.6 per cent; at 0.013, 4.0; and near the stem’s end, at a rise of 0.005, about 2 per cent. The two stems drawn, seeded with thirty and eighty whorls, lie on one curve: the share is a function of the rise and not of where on the stem the rise happens.

So a whorled stem feels its far side most at the coarse end, where few organs fit round, and hardly at all at the fine end. Whether that matters to the edge depends on where along the stem the seed is won or lost.

The seed is lost at the first transition #

Every ordinary stem grown just past its edge — a rate one per cent slower than the edge — keeps reading the seed’s pair, 1 and 3, until just after the seed ends, and then reads something else. At every seed from fifteen to eighty whorls the first reading off the seed’s pair comes within one counting window of the seed’s last whorl, at a folded rise between 0.063 and 0.075. At twenty and thirty whorls it reads 2 and 3 there, the Fibonacci branch; at forty, sixty and eighty, 2 and 4, a pair the Lucas ladder never visits; at fifteen, 1 and 4. Three of the six go on to end at 6 and 10 or 10 and 16 — counts with a common factor of two, on stems that place one organ at a time — and the others at 8 and 13.

Every seed is lost in the same place: at the first transition the stem has to make on its own, as the rise falls past the point where the pair 1 and 3 can no longer hold. That is the shaded band on the figure above, and it sits at the coarse end of the curve, where the far side carries an eighth of the repulsion. The far side is felt most exactly where the seed is decided.

The edge is a rise #

That turns the edge from a rate into a rise. A seed of s whorls on a stem whose rise falls by a factor of e every T whorls ends at a folded rise of $0.12\,e^{-s/T}$, and the edge is the rate at which that rise is still just above the first transition. Converted to whorls a rung, the ratio of edge to seed is then $2\ln\varphi / \ln(0.12/h)$, where $h$ is the rise at which the seed ends — the seed’s length drops out entirely.

Read that way, every stem with a long seed must still be holding the seed at a folded rise of about 0.077. The one-image stem must hold it down to 0.07723 at eighty whorls, the two-image stem only to 0.07772 and the three-image stem to 0.07784. Five ten-thousandths of a rise, two thirds of a per cent, is the whole of what the far side buys, and through the logarithm it becomes the one and a half per cent the ratios differ by.

The rise curves also say why short seeds are lost earlier. At fifteen whorls every stem has to hold its seed down to 0.0753 to 0.0755, a little further than a long seed; whatever a short seed lacks, it is made up by being carried a little further into the transition.

What this does to “an ordinary stem folded twice” #

A bijugate stem has been described here as an ordinary stem folded twice round, and for an ideal lattice that is exactly true: its positions folded are the ordinary lattice’s. It is also true of the transitions a grown bijugate stem passes, which come at half the ordinary rises to within a hundredth of a transition.

It is not true of the rule that grows it. A grown bijugate stem is an ordinary stem with a second image, and every grown organ after the first is placed differently from the ordinary stem’s at the same rate. That those differently placed organs pass the same transitions to a hundredth, and lose a Lucas seed at the same rate to one and a half per cent, says how little the far side moves the transitions — not that it is absent.

Why the trijugate stem agreed with the bijugate #

The earlier essay found the bijugate and trijugate levels equal to a thousandth, 2.216 and 2.215, and read the difference from the ordinary stem as one between one organ a whorl and more than one. The trijugate half of that equality is a coincidence of two effects of nearly the same size. The third image lifts an exact trijugate stem 0.008 above the bijugate level, to 2.224; the free whorls’ missed shares take it back down by 0.008. Imposed exact, a trijugate stem does not sit with the bijugate one at all.

That makes the ordinary stem’s lower level a statement about counting the far side, and the agreement between the two whorled stems a statement about two small errors cancelling. Neither was visible on the grain of two whorls of rate, and the second would not have been visible without the image dial: an imposed trijugate stem alone would only have shown that free and imposed whorls differ.

What a model’s cylinder decides #

The far side exists because the repulsion here is an inverse cube with no range of its own: every organ in the window is felt, however far round the stem it is, and on a folded whorled stem “however far round” includes whole turns. An inhibitor with a range of a few spacings would feel nothing a turn away at a fine rise, and at a coarse rise it would feel only the part of the far side that falls inside its range.

So the one and a half per cent is a property of the model’s repulsion as much as of the jugacy. A rule that cuts the neighbourhood at a distance would move it, in either direction depending on where the cut falls against the far copies, and nothing here measures that.

A model’s far side, not a plant’s #

It does not say anything about a plant. The far image is a property of a power-law repulsion summed round a cylinder, and a stem that counts a neighbour twice is an instrument built to take one property of the whorled stems apart from the others.

It does not derive the level of about 0.077, the rise at which the seed must still hold, or why the third image adds a quarter of the second’s step rather than an eighth. It does not explain the short seeds’ irregular order. The design is unchanged from the essays before it: a folded rise falling from 0.12 to 0.004, a counting window of twenty-six folded nodes striding four, an azimuth grid of 512 positions a folded turn, and a Lucas seed laid as exact lattice.

Three results that would undo the far side #

A two-image stem that differs from a folded free bijugate stem at any node, or a four-image stem from a folded free tetrajugate one. A level that does not rise with the far image’s weight. An ordinary stem grown past its edge that keeps the seed’s pair beyond the first transition after the seed. Each is checked whenever the stems are read.

One and a half per cent is one image #

The one and a half per cent between an ordinary stem and a whorled one on a long Lucas seed’s edge is the far side of the neighbourhood. Folded, a k-jugate stem keeps ordinary time; what it does not keep is an ordinary repulsion, because each lower whorl is k real organs and folded they are k copies of one node a turn apart. An ordinary stem made to count two copies is the bijugate stem at every node. The second copy is the whole of the step, it acts at the coarse end of the stem where the seed is decided, and it buys a few ten-thousandths of a rise.

Still open: a far side with a range #

Every repulsion here is an inverse cube with no range of its own, so a whorled stem feels the far copies of its neighbourhood at every rise. An inhibitor with a range of a few spacings would feel them only where few organs fit round the stem — at the coarse end, where the seed is decided — and not at all at the fine end. The next measurement grows the same stems under a repulsion cut off smoothly at ranges of two to six spacings and asks whether the one and a half per cent between an ordinary stem and a whorled one shrinks with the range, and whether it vanishes at the range below which the far copy lies outside the cut-off at the rise where the seed is lost.