Two at a time

There is an assumption in every counter this site has built and it has never been written down.

parastichy() measures the distance from point $i$ to point $i+m$ and asks which $m$ makes that short. cylCount() does the same on a stem. coneCount() does it on a cone. All three are handed an array of coordinates and none of them is handed the divergence angle, which is the discipline the site is built on — but all three are handed an index, and an index is not a neutral piece of bookkeeping. It is a claim that the elements arrived one at a time, in that order, along a single genetic spiral.

For most plants that is true. For a real minority it is not, and for those the machinery cannot be pointed at the specimen at all.

What a multijugate pattern is #

In a k-jugate pattern, $k$ primordia appear simultaneously, spaced $1/k$ of a turn apart, and the whole whorl is rotated from the one below it by the divergence angle. On a cylinder of circumference 1, whorl $w$ sits at height $wh$ and carries nodes at

$$x_{w,s} = \operatorname{frac}\!\left(w\delta + \frac{s}{k}\right), \qquad s = 0,\dots,k-1$$

$k = 1$ is the ordinary spiral lattice and the construction has to reproduce it node for node, which it does.

The condition is real and named. Teasel (Dipsacus) and Cephalaria are the standard bijugate examples; bijugate sunflowers exist and are counted 42 and 68 rather than 21 and 34; and the whole class was described by Bravais and Bravais in the 1830s, long before anything on this site was arithmetic.

The thing that makes it awkward is not the geometry. It is that there is no genetic spiral. In a bijugate plant two primordia are the same age, so “the next one” is not a well-defined object, and any counter that walks from element $i$ to element $i+m$ is asking a question the specimen does not answer.

What it is, once the coordinates are written down #

The construction looks like a new object and is not, and the argument that it is not takes two lines.

Take a $k$-jugate lattice at $(\delta, h)$ and apply the map $X = \operatorname{frac}(kx)$, $Y = ky$ — wrap the cylinder $k$ times around itself and scale the height to match. Then

$$X_{w,s} = \operatorname{frac}(k w \delta + s) = \operatorname{frac}\!\left(w \cdot k\delta\right), \qquad Y_{w,s} = w \cdot kh$$

The $s$ has vanished, because adding a whole number to a fractional part changes nothing. The $k$ nodes of each whorl land on top of one another, and what is left is the ordinary lattice at divergence $k\delta$ and rise $kh$.

The map multiplies every local distance by $k$ and it is $k$-to-one, so:

• every parastichy count of the $k$-jugate pattern is $k$ times the corresponding count of the ordinary one;
• every hop length is $1/k$ times it;
• every transition rise is $1/k$ times it;
• every fork sits at $1/k$ of the divergence and $1/k$ of the rise.

A k-jugate pattern is an ordinary lattice, seen k times over. Everything the site already knows transfers, divided by $k$. That is checked rather than argued: five lattices at four jugacies are built, counted from their positions, and required to return $k$ times what the ordinary lattice at $(k\delta, kh)$ returns. They do.

The trap: the counts cannot decide it #

Here is the part that makes multijugacy worth an essay rather than a footnote.

A $k$-jugate pattern’s counts share the factor $k$: a bijugate stem is counted 4 and 6, or 2 and 4, or 42 and 68. So the obvious test for multijugacy is to take the counted pair and look at its greatest common divisor.

That test does not work, and the counterexample is ordinary.

Take a single-jugate lattice — one node at a time, one genetic spiral, no whorls anywhere — at a divergence of 180.5°. The lattice vector for offset 2 is nearly vertical, because two steps of 180.5° is one whole turn plus one degree. The vector for offset 4 is twice that. At any small rise those two are the shortest, so the counted pair is 2 and 4, gcd 2, from a pattern with no jugacy in it at all.

Build a genuinely bijugate stem whose counted pair is also 2 and 4, at the same node density, and the two point sets are counted identically by machinery shown only their positions.

So the counts are not a diagnostic. They are necessary — a $k$-jugate pattern’s counts do share the factor $k$ — and they are nowhere near sufficient.

What does decide it #

What separates the two is a property of the point set that has nothing to do with counting: rotational symmetry.

A $k$-jugate pattern is unchanged by a rotation of $1/k$ of a turn about the axis. That is not a consequence of multijugacy; it is what multijugacy is. And it is measurable directly: rotate the whole set, and require every rotated node to land on an existing one.

Applied to the two patterns above, the bijugate stem has symmetry order 2 and the single-jugate one at 180.5° has order 1. Applied across the four jugacies built here, the measured order is the constructed $k$ every time.

There is one piece of care the test needs. The top and bottom whorls of a finite pattern have no partner to rotate onto, so the test is made on the interior; without that every pattern reports order 1, because the extreme rows always fail. That is the same class of edge effect the boundary cells that inverted Lewis’s law were: a population at the edge of a finite sample, answering a different question from the one asked, and large enough to decide the result.

Which real arrangements are in this family #

The construction covers more of botany than the word “bijugate” suggests, and it is worth naming the members, because two of them are the arrangements most people can already picture.

Decussate leaves — two opposite leaves at each node, each pair at right angles to the pair below — are the case $k = 2$, $\delta = 90°$. Mint, maple and most of the Lamiaceae are decussate, and it is one of the two commonest leaf arrangements there is.

Whorled arrangements with three or more leaves per node — Nerium, Galium, many Equisetum — are $k = 3, 4, 5$ at whatever divergence the successive whorls happen to take. When each whorl is rotated half its own spacing from the one below, which is the usual case, that is $\delta = 180°/k$ and the pattern is the alternating one a stem of bedstraw shows.

Bijugate spirals — teasel, Cephalaria, occasional sunflowers — are $k = 2$ at a divergence near 68.75°, which is the interesting case and the subject of the third essay in this group.

So the family is not exotic. The exotic member is the last one; the first two are ordinary and are usually discussed as though they had nothing to do with spiral phyllotaxis. The construction says they are the same object at a different divergence, and the counts say so too: a decussate stem is $k = 2$ at 90°, which under the wrapping map is an ordinary lattice at 180° — the distichous pattern, one leaf at a time on alternating sides. Decussate is doubled distichous, and that is a statement about coordinates rather than a metaphor.

A whorl is not a row #

There is a confusion available here that the site has already run into once from the other direction, and it is worth separating carefully.

An ordinary lattice at a divergence near a simple fraction has its nodes falling into visible rows. At 120° they fall on three radial lines; at 90° on four; at 180.5° on two, nearly. The site’s whorl-or-spiral figure draws exactly that, and the counted pair in each case shares a factor.

A $k$-jugate pattern has $k$ nodes at the same height, which is a different thing entirely. Its nodes are not on radial lines unless the divergence is also rational.

The two are told apart by the same test that decides everything here. A pattern with rows has no rotational symmetry — rotating a lattice at 120° by a third of a turn moves node 1 to where node 3 would be if node 3 were at the same height, and it is not, because it is three rises higher. A genuinely trijugate pattern rotates onto itself exactly.

Confusing the two is easy because the vocabulary encourages it: “whorled” is used in the literature for both, and this site’s own classifyPair returns the word “whorled” for any non-coprime pair, which is precisely the reading the census essay has to unpick.

Why the ordinary recovery refuses, and is right to #

The site’s cylindrical recovery — the one that closes to $10^{-13}$ in both parameters — begins by refusing any pair whose counts share a factor. It has done so since it was written, with the message “this is a whorled stem, not a spiral one”, and it turns out the refusal is load-bearing rather than cautious.

The reason is that the two families of a non-coprime pair are parallel. If the counted pair is 2 and 4, the 4-family’s lattice vector is exactly twice the 2-family’s — no extra wrap intervenes — so the second hop length is determined by the first and carries no new information. Two measurements that are one measurement cannot pin two unknowns. The recovery is not being timid; the data genuinely does not determine the lattice.

That is worth contrasting with the disc’s recovery, which returns an interval several degrees wide because a range of angles produces the same two offsets. There the data is weak. Here it is degenerate, which is a different failure and needs a different fix.

The fix is a second measurement of a different kind: the symmetry order. Given $k$, the pattern is an ordinary lattice seen $k$ times, so divide the counts by $k$, multiply the hop lengths by $k$, recover the ordinary lattice exactly, and divide both parameters by $k$ again. The next essay does that and the round trip closes as hard as the ordinary one.

The divergence is only defined modulo 360/k #

One consequence of the construction is worth stating on its own, because it changes what a reported number means.

Add $360/k$ degrees to a $k$-jugate pattern’s divergence. Then $w\delta + s/k$ becomes $w\delta + w/k + s/k$, and since $s$ runs over every residue, the set $\{(s+w) \bmod k\}$ is the same set as $\{s\}$. The point set is identical — not similar, not congruent: the same coordinates.

So a bijugate plant’s divergence is defined modulo 180°, a trijugate’s modulo 120°, and the statement “teasel has a divergence of 68.75°” is a statement modulo 180°. The larger representative, 248.75°, describes exactly the same plant, and nothing in the specimen distinguishes them.

This is not a subtlety about conventions. It is a statement about what is observable, and it is the reason the recovery here returns an angle with a stated period rather than an angle.

What the construction does not model #

The lattice built here is idealised in three specific ways, and each of them is the kind of idealisation that a real specimen breaks visibly rather than subtly.

The whorl is exactly simultaneous. In the model the $k$ members of a whorl are at identical heights. In a plant they are formed within a short window and then displaced by growth, so a real “whorl” is a tight helix rather than a ring. Where the displacement is comparable to the rise — which happens on a compressed shoot — the pattern is genuinely intermediate and the symmetry test will return order 1 for something a botanist would call bijugate.

The members are exactly evenly spaced. The model puts them $1/k$ of a turn apart. Real whorls are usually close to even and are not exactly so, and the deviation is again a continuous quantity that the symmetry test converts into a yes or no.

The jugacy does not change. Real plants change it: a shoot can begin decussate and become spiral, and the transition is one of the standard observations in the field. Nothing here describes that, and a model that did would need the time axis the rising essays introduce, plus a mechanism for changing $k$, which is a larger thing than a rate.

The honest summary is that the symmetry test is exact on exact patterns and needs a tolerance on real ones — it already carries one, set as a fraction of the local spacing — and that the tolerance is doing more work on a specimen than on a lattice.

Counting one by hand #

If the counters here cannot be pointed at a multijugate specimen, it is worth asking what a person does instead, because the answer is what the next essay automates.

A person counting a teasel head does not track which primordium came after which. They put a finger on a bract, follow the family of spirals it belongs to around the head, and count how many distinct spirals there are before returning to where they started. That is a count of chains, and it needs no order of arrival at all — only the positions and the ability to see which element is next along a given direction.

That is the instrument the site was missing, and it turns out to be strictly more general than the one it had: on a single-jugate lattice it must agree with the index counter, and on a multijugate one it works where the index counter cannot be run. Both halves are needed. An instrument that only worked on the hard case would be an instrument with nothing to check it against.

Where this leaves the counters #

Three things follow for the machinery, and the next two essays are about them.

An index-based counter cannot be pointed at a multijugate specimen. Not because it gives a wrong answer, but because the input it needs does not exist. Something that counts from positions alone is required, and it is the subject of the next essay.

A pair sharing a factor is a question, not an answer. It could be a genuinely $k$-jugate pattern or an ordinary one near a rational divergence, and the census that the wrong field built on this distinction turns out to have been reading it the second way throughout — which is what the census essay is about.

And the whole theory transfers divided by k, which means there is a family of limit divergences rather than a constant. The bijugate one is exactly half the golden angle, and it is where the bijugate plants are found.