Systematic error — where it appears
Named by 21 essays across 4 fields — each of them below, with the objects they name alongside it.
The exponent an error moves
Every real measurement of a branch radius carries error and no synthetic tree does, so the question is what a symmetric error does to a fitted exponent. It does two things — a bias and a spread — and the bias runs downward at every error level and in every band, by an amount derivable from the daughter ratios alone.
The window that closes
The spread of a fitted branching exponent falls as the reciprocal root of the sample and its displacement does not fall at all, so there is a count past which every further junction buys confidence and no accuracy. Between three and five per cent of radius error the count arrives before the answer does, and no sample size both states a claim and contains the truth.
What one exponent reports
Fit a single shape exponent to an organ that has four of them and it returns a real quantity — the harmonic mean of what its individual steps report. Harmonic means sit below arithmetic ones, so the fit understates, systematically, in a known direction, and invisibly.
A swelling at the fork
A branch thickens where it forks, so a parent measured just below a junction and daughters measured just above it carry three different amounts of the same swelling. A swelling that fattens all three alike moves a fitted exponent by exactly nothing. A parent read one per cent fat moves it by as much as 3.6 per cent of random error on every radius, in a sign known in advance, and a tenth of a radius turns a tree built at Murray's three into one that reads Da Vinci's two with no noise at all. Added to the noise, it does not bring the two rules together any sooner: the two errors do not add.
A plateau the instrument should have had
The counting window decides nothing on the settling table, and it has two bounds that decide everything outside it. A window one organ too narrow returns the previous rung of the same ladder, a family past the offset ceiling is reported as a coarser rung at every width there is, and neither failure produces a refusal, noise or a wide error bar.
A count carries no error
Fitting r₀ᵏ = Σrᵢᵏ junction by junction puts a measured radius on both sides of every equation, and at twelve per cent of error a tree built at Murray's three and one built at Da Vinci's two stop being told apart, however the fit is corrected. Fit the same measured radii against the number of tips each branch carries instead — a count, which nobody measures with error — and the two trees read 2.996 and 1.997 at twelve per cent and 3.015 and 2.001 at thirty, never overlapping. The twelve per cent belonged to the junction fit, not to the tree. The count fails in its own way, and the way is stated.
How many organs a pair needs
A count taken over too few organs does not fail. It returns the rung below, which is a perfectly good pair, and nothing anywhere says so. The window that avoids it is not a constant but the counter's own arithmetic, and 384 settled runs sit exactly where that arithmetic puts them.
A count that has lost tips
Radii read against the tips each branch carries keep Murray's three apart from Da Vinci's two where junction fits cannot, because a count has no measurement error in it. A count of the tips a tree has is not a count of the tips it grew. Losing them lowers both trees' readings by one factor that belongs to the losses and not to the rule, so the count stops being right long before it stops telling the trees apart: on fifty junctions at twelve per cent of error, to seventy per cent of the tips lost one at a time, and only to about a quarter lost in whole limbs. Counting scars repairs single losses exactly. Nothing countable repairs a shed limb.
A count that is not exact
Reading a deposition law off two whorls' growth-line counts divides one by the other, so a miscount that is the same in both divides out: four lines in five missed at random moves the answer by five thousandths and costs only scatter. What biases it is a miscount that varies along the shell, and there is one that always does. The arc between successive lines rises or falls with the radius according to whether the law is shallower or steeper than a length clock, so a section's resolution limit eats the inner whorls of a shallow shell and the outer whorls of a steep one, and eats evenly at exactly p = 1. Where the limit binds, a whorl's surviving count is its arc over the limit, and whorl arcs stand in the ratio W — so a shell too worn to read reports a length clock whatever law it had.
A section seen from the wrong angle
A photograph of a shell section taken off the normal is the coiling plane compressed along one direction by the cosine of the angle, and nothing in the picture says so. The fit that recovers a growth factor is moved by it — half a turn seen twenty degrees off gives a band of answers 23.7 per cent wide as the span's starting point moves round the shell, centred almost exactly on the right answer, so it is a spread and not a bias. The caliper measure is exactly immune at every tilt and every aim, because a projection scales all three points on a line through the centre by the same factor. And the fit's residual names the tilt to three decimal places, which makes this the rare error a section reports about itself.
Whether a section can see its own limit
A shell section that cannot resolve growth lines closer than some distance reads every deposition law as nearer a length clock, and says nothing about it. Given an animal whose days vary, the section can often catch itself: a limit changes how irregular successive spacings are in the whorls where it binds, and one animal is not steady in one whorl and irregular in the next. On an angular clock a test comparing whorls flags every limited section, before the reading has even moved. On a volume clock it can miss a limit that has pulled the reading from 3 to 1.87 — when the animal's own days vary by a tenth, which is exactly as irregular as the limit leaves the whorl it binds.
A limit is a length, and a short day is a share
A shell section that cannot resolve its finest growth lines names a steep law as a shallower one, and so does a shell whose animal really did change its law. Matched to name the same law from the same whorls, the two agree in their counts, and the test that compares how irregular each whorl is goes blind on both across the same band of limits. What separates them is what a limit is. Every spacing it leaves is at least one length, wherever on the whorl it sits, while an animal's short days are short in proportion to the spacing around them. Asked whether the shortest spacings along the outer whorl follow the local spacing, forty sections of a drifting or stepped law are flagged at most twice, and a limited volume clock in the blind band thirty-three to forty times.
A photograph squashed along the axis is another shell
A cut shell is photographed, not measured with calipers, and a camera a few degrees off the face's normal squashes the whole picture across one bearing. Read as a shell with elliptical openings, a squash straight across the axis or along it is absorbed without a trace: the picture is exactly that of another shell, its openings ellipses and its translation off by the squash — six per cent at a twenty-degree tilt — and no fit can tell. At any bearing between, the squash turns every opening the same way on both sides of the axis, which a real opening, mirrored across the axis, never does: a ten-degree tilt at fifteen to seventy-five degrees from the axis is caught on every section. The common photograph, the shell upright in the frame and the camera off to one side, is the one that cannot be caught.
A limit that loosens along the whorl hides from the floor test
A shell section that cannot resolve its finest growth lines was told from an animal that changed its law by the floor test: a limit is one length, so along the outer whorl its shortest spacings fall behind the local spacing, and an animal's short days do not. A real section's resolution changes along a whorl. Let the limit loosen outward by half again a turn and the floor test goes blind on every animal and limit — not because the limit has stopped binding, but because a limit that binds everywhere sets the very spacing the test compares it with. The limit still removes two thirds of the outer whorl and still drags a volume clock's reading to 1.5–2.4. A trend against position recovers most of it, and the spacings themselves recover all of it: a limit leaves a run's spacings piled against its shortest, which no animal's days do.
A cut seen in perspective is not another shell
A photograph of a cut shell squashed straight along the axis is exactly the picture of another shell, with nothing to tell them apart. A camera close enough to see the cut in perspective is different. The near whorls come out larger than the far ones, and a whorl's size then changes across the picture in a way no shell's own expansion does — so the reading that fitted a distant photograph misses a near one by more than the noise, even straight along the axis. Before it is seen the error is small, W read too large by about the sine of the tilt over the distance. A camera fitted beside the shell, three numbers more, takes it back entirely: W, D and T within a third of a per cent at every bearing and every distance down to three rim radii.
A free opening lets the camera trade again
A camera fitted beside a shell recovered every photographed cut whose openings were round. Give the opening its own shape — an aspect and a turn, as a snail's has — and the twelve numbers recover everything only when the camera's tilt runs at a bearing between the axis and the line across it. Straight across or along the axis, a pinhole tilted τ at L rim radii is exactly a pinhole tilted τ′ at L′ followed by a squash, whenever tan τ′/L′ = tan τ/L, and that squash is another shell. Every camera on the line fits a noiseless picture to rounding. The fit keeps W and the perspective's strength, and nothing else about the camera: T and the aspect come back scaled by a ratio of cosines, a turned opening's turn and its D move with them, and on noisy pictures T ran from six per cent low to a quarter high with every picture fitted inside the noise.
Two photographs pin the shell only from two sides
A photograph of a cut shell whose opening has its own shape, taken straight across or along the axis, fits every camera on a line, and each camera on the line reads a different shell. A second photograph of the same cut shares the shell and brings a camera of its own. Taken at the same bearing — farther away, or steeper — it does not help: the two cameras slide together along a family set by one closed form, cos τ₁′/cos τ₁ = cos τ₂′/cos τ₂, each picture fitted to rounding at every point of it, and the shell slides with them. Turned so that its tilt runs across a different direction, it pins the shell, and on noisy pictures a turn of ten degrees is already enough.
A tilt undone beats the calipers
A shell section photographed off its plane reports its own tilt in the fit's residual, but not the axis the tilt was taken across, and a correction needs both. Read together from the whole outline — the tilt and axis whose undoing leaves the straightest line — they come back on half a turn of shell to a fifth of a degree and under a degree. Undone, the fit no longer merely matches the calipers that a tilt cannot touch: on a traced outline it beats them, by three times on half a turn and by twelve to forty from one turn up, because the calipers read three points and the fit reads all of them. Correcting a section that was square on costs a little, and a shell whose growth slows is mistaken for a tilted one without harm.
A loss that picks its tips can read high
A tree that has lost tips at random reads its branching exponent low, by one factor shared by every rule. Let the loss choose which tips go and the factor is still shared — the ratio of three to two survives any choice — but it is no longer below one. On fifty junctions with half the tips gone, the choice of which half moves Murray's tree from 2.23 to 3.52, and past about half the tips a single tree's reading can no longer say which rule built it. The ways a crown plausibly chooses stay within a few hundredths of a fair loss, all of them low, and counting scars repairs every one.
Two free frames cost one right angle
Two photographs of one cut shell pin it only when their cameras stand at different bearings, and the pair at right angles read best of all. Each of those fits was handed both pictures in one frame, and no real pair comes that way. Give the second picture a frame of its own, rotated and scaled and shifted, and leave the reading to find it: nineteen numbers, not fifteen. It costs almost nothing. The rotation comes back to a twentieth of a degree, the scale to under a tenth of a per cent, and the shell's numbers as well as before, except at one bearing. At exactly ninety degrees a free scale swallows one camera's squash whole, the fit slides, and the pair that had been best reads the translation twenty-three times worse than a pair at forty-five.
The handovers corrected
Six recorded handovers, relocated to the grid against where a one-per-cent sweep put them: all six sit on the fine side of a crossing and all six inside a single sweep step. Nothing about the rung explains the size of the discrepancy, which is what a sampling artefact is supposed to look like.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsMeasurement errorClaim testingIdentifiabilityResidualWhorlRound tripSilent failureBranching exponentDa Vinci's ruleMurray's lawResolution