A head photographed from the side

The positions read the twist’s shape found that a seed head’s organs carry the local divergence even though their birth order is lost: each counted family’s phase sum across an annulus peaks at the divergence with no organ named, and two annuli, an inner and an outer, tell a twisted head from one grown at a changed angle. The centre the spirals give then took away the head’s centre, which a photograph does not supply, and found that the centroid of the organs finds it to a hundredth of a spacing and costs the reading nothing.

Both readings assumed the head was photographed straight down its axis. A camera a few degrees off the axis sees the head foreshortened: every distance along one bearing is shortened by the cosine of the tilt and every distance across it is left alone, which is what a section seen from the wrong angle found for a shell. A seed head is a disc of spirals, and the squash moves every organ’s angle by an amount that goes round the head twice and changes the radius law from a circle’s to an ellipse’s. The measurement here asks how large a tilt reads as a twist, whether fitting the ellipse restores the reading, and whether the one twist the positions could never see becomes visible when the head is seen from the side.

The photograph #

The heads are the ones the two earlier readings used: Vogel heads of 900 and 2,400 organs at the golden angle, every organ displaced by a tenth of a spacing, ten heads of each size on ten seeds — the displacement at which a head displaced before it is counted found the counts still reliable and recovering the angle found the positions’ reading finer than the counts’. Each is photographed at a tilt along a bearing set by its seed, so that ten heads see ten directions round a half turn. The photograph is affine — the camera is far enough away that the image is a projection rather than a perspective — so the centroid of the organs is still the head’s centre, and every error below is the squash’s own.

Two readings are taken. Square on reads the photograph exactly as the centre-finding essay read a true photograph: the organs about their centroid, in the inner annulus from 0.35 to 0.6 of the radius and the outer from 0.7 to 0.95, the counts supplying the pair and the window, the positions the divergence inside it. Ellipse fitted first measures the head’s second moments about its centroid. A round head’s are the same in every direction; a squashed head’s are smaller along the bearing by the square of the cosine. So the square root of their ratio is the cosine of the tilt, the direction of the smaller one is the bearing, and the organs are stretched back along it before the same reading.

Twelve degrees read as nothing #

The first result is how little a modest tilt does. The twist threshold is the largest difference between the two annuli that any untwisted head shows when photographed square on — a thousandth of a degree on 900 organs, 0.34 thousandths on 2,400 — and a head read past it would be declared twisted. Read square on, not one of the ten 900-organ heads passes it at a tilt of 2, 5, 8, 12 or 14 degrees. The root-mean-square difference between the annuli stays at half a thousandth of a degree, exactly where it sits for a head photographed straight down. On 2,400 organs the same holds to twelve degrees.

The reason is the shape of the squash. A tilt $\tau$ along a bearing $\beta$ turns an organ at angle $\theta$ by about $\tfrac{1}{2}(1-\cos\tau)\sin 2(\theta - \beta)$ and scales its squared radius by a factor that also goes round the head twice. A family’s phase sum adds up $m\theta - b\,u$ over every organ in the annulus, and a term that goes round twice sums to nothing over a whole annulus: to first order the squash moves every family’s peak not at all. What it does do is spread the organs about the peak, which lowers its height without moving it, until the spread is large enough to drag the peak onto a side lobe. At twelve degrees $1 - \cos\tau$ is 0.022; at twenty, 0.060.

And the counts never notice. At every tilt read, to thirty degrees, every band of every head is counted as two consecutive Fibonacci numbers — 34 and 55 inside, 55 and 89 outside on 900 organs, one step higher on 2,400. The flag that fires when a band stops returning consecutive Fibonacci numbers does not fire for a squash, which is to say that a squash is not a changed divergence as far as the counts can tell.

Past the edge, a scatter #

What happens beyond the tolerance is not a steady drift. On 900 organs three heads of ten pass the threshold at sixteen degrees, nine at eighteen, six at twenty, eight at twenty-five and six at thirty. On 2,400 organs four pass at fourteen, one at sixteen, all ten at eighteen, seven at twenty, one at twenty-five and all ten at thirty. The count goes up and down with the tilt because each failure is one head’s phase sum caught by a side lobe — the peak has been spread flat enough that a neighbour a few hundredths of a degree away wins — and whether that happens to a given head at a given tilt depends on where the squash’s bearing falls against that head’s own displacements.

So a reading of a photographed head is not degraded by a tilt; it is either unaffected or wrong. The root-mean-square difference between the annuli moves from half a thousandth of a degree to a few thousandths once any head fails, and on 900 organs at eighteen degrees one head’s outer annulus is misread by nearly a tenth of a degree while its neighbours are exact. A worker with a single photograph has no way to tell from the reading which of the two cases the head is in.

A rougher head tolerates less #

The tolerance belongs to the head as well as to the camera. The heads above were displaced by a tenth of a spacing; a head whose organs sit more roughly has a flatter phase sum to begin with, and a squash that spreads it further finds the side lobes sooner.

Photographed at each tilt and read square on, heads displaced by five hundredths of a spacing behave like the heads displaced by a tenth: none passes its own threshold — 0.68 thousandths of a degree, set by untwisted heads photographed square on at that displacement — up to fourteen degrees, two pass at sixteen, all ten at eighteen, and then the same scatter. Heads displaced by a quarter of a spacing, with a threshold of 4.7 thousandths, hold to eight degrees and begin to fail at twelve: one head of ten at twelve, two at fourteen, five at sixteen, nine at eighteen. And when they fail they fail by more — the root-mean-square difference between the annuli jumps from a couple of thousandths of a degree to a tenth at sixteen degrees, because a rougher head’s side lobes stand nearly as tall as its peak.

So the working tolerance for reading a photograph square on is about eight degrees for a head as rough as a quarter of a spacing and about twelve for a head ten times better placed. Neither is a small angle to hold a camera to by eye: a photograph taken by hand over a flower head is routinely a few degrees off, and a head on a nodding stem can easily face the camera at twenty. The counts are no guide, since they returned consecutive Fibonacci numbers at every tilt and every displacement read. What decides whether a photograph can be read square on is a number the photograph does not show, and the next reading is the one that measures it.

The moments find the squash, above a floor #

The repair needs the squash to be found, and the second moments are the obvious instrument. They find it well where it matters and cannot find it where it does not. A 900-organ head photographed straight down has moments that imply a tilt of 2.8 to 4.0 degrees, and a 2,400-organ head 1.2 to 2.8: a finite spiral of organs is not round in its second moments, because the organs at the rim sit where the last turn of the spiral happens to leave them. Below that floor a real tilt is lost in the head’s own shape — at three degrees the moments read 1.6 to 5.0 on 900 organs, with the bearing wrong by up to seventy-eight degrees. Above it they close in: 7.0 to 8.9 at eight degrees, 11.4 to 12.6 at twelve, 19.6 to 20.4 at twenty, 29.8 to 30.2 at thirty, with the bearing to within a degree by twenty.

The floor would be a problem if it sat above the tolerance. It sits far below it. A squash the moments cannot see is a squash of a few degrees, and the square-on reading tolerates twelve.

The same arithmetic prices the moments’ error where they do see the tilt. A head photographed at twenty degrees and read by its moments as 19.6 is stretched back by a factor a fraction of a per cent too small, which leaves it squashed as a head photographed at about four degrees would be; one read as 20.4 is left stretched by the same amount. Across the ranges measured — 11.4 to 12.6 at twelve degrees, 19.6 to 20.4 at twenty, 29.8 to 30.2 at thirty — the squash left behind is equivalent to a tilt of 3.9 to 4.0 degrees, whatever the tilt being corrected. What the moments measure is the cosine of the tilt, from the ratio of two sums over the organs, and they measure it to the same precision at every tilt — the head’s own shape sets that precision, as it set the floor — so the squash they leave behind is the same size wherever they are used. Every one of those is inside the tolerance of the head being read, even the roughest, which is why the stretch that follows restores the reading exactly rather than approximately.

Fitted, the ellipse restores it exactly #

Stretched back along the short axis the moments find, every head reads as it reads square on. On 900 organs the root-mean-square difference between the annuli is 0.57 thousandths of a degree at every tilt from none to thirty, and no head passes the threshold; on 2,400 organs, 0.06 thousandths at every tilt, and none passes. The inner annulus’s reading is 0.50 thousandths of a degree off golden and the outer 0.12, at every tilt, to the second significant figure — which is what they were for a head photographed straight down.

Two things combine to make the repair exact rather than approximate. The stretch undoes an affine squash exactly when its axis and factor are right, and above a few degrees they are. Below a few degrees they are wrong, because the moments are reading the head’s own shape — but then the stretch is a spurious squash of a few degrees, and a squash of a few degrees is exactly what the reading tolerates. The two errors live in different ranges and never meet.

Twist shapes, through a tilted camera #

The last reading of the positions was that they separate a twist from a changed angle, at a twist size that depends on the twist’s shape, and that they do it no later than the count sees the twist first found the counts flag it. A tilt could shift that.

Photographed at twelve degrees, nothing moves: on 900 organs a twist growing as the square root of the radius, as the radius and as its fourth power is separated at a tenth of a radian, a tenth and five hundredths, exactly as square on, and on 2,400 organs at five hundredths for all three. At twenty and thirty degrees read square on, the separation coarsens by a step or two of the grid — to a quarter of a radian on 900 organs at thirty degrees, and on 2,400 at twenty for all three shapes — which is the side-lobe scatter again: a twisted head misread by a squash reads its annuli wrongly apart or wrongly together. With the ellipse fitted first, every cell of the table returns to its square-on value.

The one twist no camera can show #

The flag reads the angle, not the twist found one twist the counts and the positions can never see: a head turned in proportion to the square of its radius puts every organ where a head grown at the golden angle plus a/(N − 1) puts it, to a billionth of a spacing. The continuation suggested a squash might break that equivalence, since a tilted head is not a changed angle organ for organ.

It cannot, and the reason is not subtle once it is written down. The squash is applied to the positions, after the head is grown. Two heads whose organs sit at the same positions become two photographs whose organs sit at the same positions, whatever the camera does, and every reading of those photographs is the same reading. Measured, the largest difference between the twisted head’s readings and the grown head’s, over ten heads at every tilt, is zero. A squash cannot break an equivalence that holds organ by organ; nothing done to the picture can. The only thing that separates a square-law twist from a changed angle is something that is not in the positions at all — the order in which the organs were made, which the positions essay showed the positions do not keep.

What this does not establish #

That a real seed head is a flat disc. A capitulum is domed, and a domed head photographed off its axis is not an affine image of itself: its rim and its centre sit at different depths, so the squash changes across the head and the moments read an average of it. That the camera is far away. At a working distance of ten head diameters a perspective’s taper across the head is a few per cent, which is a squash that changes from one side to the other and is not modelled. And that the tilt is the only thing wrong with the photograph — a lens’s own distortion is radial, goes round the head once rather than twice, and would be read by each family’s phase sum differently.

What would overturn it #

An untwisted head photographed at under twelve degrees whose annuli, read square on, disagree past the threshold. A head stretched back along its moments’ short axis, at any tilt to thirty degrees, that reads differently from the same head photographed square on. A square-law twist that a photograph separates from the changed angle it equals. Each would mean the account of what a squash does to each family’s phase sum is wrong.

Still open: a head seen in perspective #

Every photograph here is affine. A camera close to a head sees it in perspective: the near side larger than the far side, so the squash is not one factor across one bearing but a factor that changes across the head, and the centroid of the organs is no longer the head’s centre. The next measurement photographs heads in perspective from a stated distance and angle and asks three things: how close the camera can come before the square-on reading fails where the affine one did not; whether the centre the spirals give, which found a cropped head’s centre when the centroid could not, finds a perspective head’s centre too; and whether a projective correction — four numbers rather than the ellipse’s two — can be fitted from the organs alone the way the second moments fitted the squash.