Cartographic Projection
every map is wrong, and by exactly how much
Theme

The thread: Measured, not named — page 14

A projection is usually called conformal because that is its name. Here it is called conformal only after the angular deformation has been computed at several hundred points and found to be zero. Essays 313 to 336 of 372.
Beside a bending river, the fan's chords cross water; a route drawn from the river's own outline never does. Everywhere within a 10 km walk of the centre, shaded pale, with a river 3 km north meandering ±2.0 km on a 4 km wavelength and bridges every 2.5 km. Left: the polygon 16 bearings draw, with the ground it claims and should not in solid ink — chords spanning the bends. Averaged over where the first bearing falls, it misses 35.0% of the true ground and claims 0.15% that is not there. Right: every point reachable by a route with at most one crossing — straight out, or straight to a bridge and straight on — worked out from the river's outline and the bridges alone, with no fan and no shortest path. It misses 6.4% and claims nothing. Paths and directions

A reach set drawn from the river's outline is a bound again

Beside a bending river, a fan of bearings drawn with straight chords claims a strip of water, and the drawing stops being a lower bound. The fan's own numbers cannot repair it: what a bearing records about the river is where the bridges are, not where the bends are, and the best a rule built from them does is cut the over-claim by seven eighths while giving up twenty times as much ground. The river's outline repairs it exactly. Cutting each bearing at the first bank costs three and a half times the over-claim it removes; keeping the chords wherever a route over one bridge reaches costs one and a half. Drawn from the outline and the bridges with no fan at all, that route set misses 6.4 per cent of the reachable ground where the fan misses 35.

 Along a meridian a good sphere's error adds up triangle by triangle; a bad base does not. A strip of thirty-kilometre equilateral triangles running north from 45°, each side computed from the last by Legendre's theorem on a substituted sphere. The scale error of the side just computed, against how many triangles lie between it and the measured base. The conformal sphere adds 7.90 ppm a triangle and reaches 158.1 ppm after 20; the geocentric, 158.9 ppm; the authalic, −52.4 ppm; the rectifying, 0.29 ppm. Shaded: three standard deviations of the scale error one-second angles leave, 3.96 ppm a triangle, growing as the square root of the count. Dashed: a base measured 8 ppm long, which stays 8 ppm at every triangle. What is taught wrongly

A strip of triangles carries the sphere's error to the next base

One triangle cannot tell a substituted sphere from a bad base: both leave the computed side a few parts per million long. A triangulation is not one triangle but a strip, each computed side the next one's base, and along it the two come apart completely. A bad base is one factor, the same at every triangle. A sphere's error is made afresh in each triangle and the strip adds it up, and what it adds up to at the far end is exactly the change in that sphere's own meridian scale between the two ends — 158 parts per million for the conformal sphere over twenty triangles north from 45°, predicted to half a part. Along a parallel it is nothing; at a check base three triangles on it is already three times the noise of one-second angles.

 Twelve cells of the grid's pole row are one map. The pole row of the aspect grid for Robinson over a 10° square region centred at 38° N: 24 pole longitudes across, 12 turns of the third angle down, with the projection's pole on the south geographic pole. There, moving the pole in longitude and turning the third angle are the same rotation, so every cell whose longitude less its turn is the same is the same map. The twelve filled cells are one map, the lowest on the row at 1.71 times the region's best score, and they score alike to 2e-11. No two of them are neighbours. The hollow cells are the maps one 15° step round the axis, the lowest at 2.33 times the best, and on the grid as it was the copies could join only through them. What each projection optimises

The pass was a map meeting itself

Four measurements fitted a power law to the depth of the pass that splits a near-optimal set of aspects, and found it tight — R-squared of 0.98 for Robinson, 0.996 along most of a family of maps. On the grid those aspects were searched over, a pole row holds every normal-aspect map twelve times at cells that are not neighbours, and in 28 of Robinson's 36 regions the published pass was two copies of one map meeting through the map one step round the axis. Joined as the one map they are, Robinson's curve falls to R-squared 0.008, and the sinusoidal's famous refusal turns out to be the one measurement the copies never reached.

 The lobes the ocean would choose, against the lobes the land chose. The water in two-degree cells, with the land's own layout dashed and the ocean's solid; a short tick at the equator marks each lobe's central meridian. The land's, at two lobes north and four south: edge at 20°W, northern cut at 165°E, southern cuts at 70°E, 135°E, 180° — 14.40° of mean angular deformation on the land outside Antarctica and 19.75° on the water. The ocean's: edge at 40°E, northern cut at 110°W, southern cuts at 135°E, 140°W, 60°W, meridians 175°E and 40°W north and 85°E, 180°, 105°W, 10°W south — 13.31° on the water and 26.18° on the land. The families

The ocean centres its lobes on the land's cuts, and cannot keep its own cuts short

Run on the water instead of the land, the same exact search gives an interrupted homolosine whose lobes sit on the Pacific, the Atlantic and the Indian Ocean — five of its six central meridians within twenty degrees of a cut the land's own layout chose — and it deforms the water by 13.31° against the 19.75° the land's layout gives it. The inversion stops at the cuts. Every meridian on Earth crosses at least 7,172 kilometres of water, so the ocean's map has no layout whose cuts stay under seven thousand, where the land's went below nine hundred. And no layout serves both: none comes within 2.57° of both optima at once, though a single wide Pacific lobe buys the water 2.8° for a tenth of a degree of land.

 The mountain's gravity follows the ground and the root's does not. A profile 260 km long across the stated mountain patch. Top: the height of the ground. Bottom: the gravity at the ground from the terrain's own rock, a sheet at 2,670 kg/m³ (solid), and from a root at 30 km carrying the same mass with the opposite sign (dashed). Over the whole patch the rock's gravity correlates with the height at 0.99, so of its 54.0 mGal rms only 7.8 is not a function of the height under it. The root's, smoothed over thirty kilometres, correlates at −0.85, and of its 10.0 mGal 5.3 is free of the local height. A levelling loop misses by exactly that free part, which crosses the ground's contours. The impossibility

A levelling loop weighs the root, and a triangle weighs the mountain

A mountain range is carried partly by a root of light crust beneath it, and a survey that knows the ground can ask how much. Put a root thirty kilometres down under a stated range and it moves a levelling loop's misclosure by 85 per cent as much as the mountains' own rock does, and a triangle's excess by 16 per cent: the root's gravity is a smoothed copy of the ground, and a smoothed copy crosses the contours. Read against their noise, pinning the root's share to a tenth takes about 2,700 fifty-kilometre loops, a million triangles or 550 astronomic stations — and if the rock's density is unknown too, a triangle cannot tell a compensated mountain from a lighter one.

 The sea surface, against the surface heights are measured from. The stated mean dynamic topography — how far the sea stands above the geoid — drawn on Mollweide, with the eight tide gauges marked. It ranges over 2.09 metres, it is smooth, and it is what every national datum's zero is sitting on. Nothing here is a model's output: it is a stated closed form, chosen to have the observed sign structure and the observed size. What the numbers refer to

Every country's zero is a different surface

Ten rungs measure a height against a geoid and treat the geoid as one object. No national datum is on it: each is pinned to the mean sea level at one tide gauge, the sea surface stands up to two metres from the geoid, and the eight European zeros measured here spread over 462 millimetres — a step no levelling can remove and nobody's error.

 The straight line between two correct maps passes through an incorrect one. Both ends of this sequence are exact cartograms of four cities: the triangular, x first construction at s = 0 and the triangular, y first at s = 1, each meeting the density to arithmetic noise. The panels between them are the straight-line blend of the two, which is what an animation between two maps computes. The cells drawn solid have turned inside out — their signed area is negative, so the map has folded over itself there and two places on the sphere are drawn at one place on the page. Measuring distortion

A map that meets its target can fold

The flow a diffusion cartogram integrates is a diffeomorphism at every instant and cannot fold. Every discretisation of it can, and the point at which one does is a root of a quadratic — written down rather than searched for.

 Adjusted to close on the second base, a sphere's drift of hundreds of ppm becomes a bow of under one. A strip of 40 thirty-kilometre equilateral triangles north from 45°, computed on the conformal sphere, between a base at each end. Unadjusted (dashed), the side error climbs to 315.0 ppm at the second base. Adjusted to close there, each triangle's share of the misclosure equal, what is left (solid) is the sphere's stretch less a straight line between the bases: 0.63 ppm at its largest, at triangle 23. Shaded: three standard deviations of what one-second angles leave in the adjusted strip, 37.6 ppm at the middle and nothing at either base. What is taught wrongly

An adjustment hides the sphere only in a strip of equal triangles

A strip computed on a substituted sphere drifts by the sphere's own stretch, 315 parts per million over forty triangles north from 45°, and an adjustment that closes it on a second base takes almost all of that away: in a strip of equal triangles what is left is the stretch less a straight line, 0.63 parts per million, a third-order bow sixty times below the noise of the old angles. A real chain is not made of equal triangles. The adjustment spreads the misclosure by how uncertain each triangle's angles are, the sphere put it in by how far north each triangle reaches, and the difference is a first-order residual that never clears the noise at any one side and that the old network's own geometry predicts exactly — so a re-observation matched against it reads the sphere at seven standard deviations.

 Two solved maps and the line between them keep what the whole family of exponents keeps. Area inside a band, per cent, against the band as a fraction of each region's least worst departure, from 0.3 to 1. Heavy: the best map on the line between the p = 1 map and the least worst map, found by a search along the line alone. Thin: the best of twelve exponents from 1 to 32, each solved separately. Dashed: the rule taking p = 1 below 0.9 and the least worst case above it; shaded, what that rule gives up. On the square the line trails the best exponent by at most 0.00 points and beats it by up to 1.3; the rule gives up 0.9. On the ellipse the line trails by at most 1.1 and the rule gives up 5.9. What each projection optimises

Two maps and the line between them do the work of every exponent

A conformal map's log scale is linear in its coefficients, so every blend of the map that minimises the mean departure and the least worst map is itself a conformal map, available the moment the two are solved. Searched along that one line for each band, the blend keeps within 0.17 points of the best of twelve exponents on every compact region and beats all twelve by up to 1.35 on the square; it trails by 1.5 at most on the thinnest ellipse and 0.74 on four regions with a bay, where the rule of switching between the two ends gave up as much as 7.1. The family's middle members lie up to two thirds of their own departure off the line. The line does not reproduce them; it does their work.

 Nineteen forecasts of one crossing, and the floor each says must be met. One crossing, with the forecast's error in where the streak starts at 150 km and in its drift speed at 50 km/h. The line is the worst floor over a four-hour crossing against where the streak starts, at the assumed drift of 250 km/h. Dots: nineteen ensemble members, each drawn with its own start and drift speed and placed at the worst floor its own crossing window meets. Open circle: the truth, 660.3 km. Square: the ensemble's centre, 640.3 km. Dashed: the centre's floor plus the fixed margin that covers 95 per cent of crossings at these errors, 653.3 km. Solid: the highest member, 663.2 km — the plan the ensemble gives with no margin added. Paths and directions

An ensemble carries its own margin, if its spread is honest

A fixed margin of about eight kilometres for every hundred of forecast error covers the diversion floor in nineteen crossings of twenty, but it needs the error's size and it is spent on every crossing alike. Nineteen forecasts of the same crossing need neither. If the truth is one more draw from the spread they come from, the highest of them covers the floor met in nineteen crossings in twenty by rank alone — whatever the error, the streak or the drift — and it holds a fifth to two fifths less in hand than the fixed margin, because it spends nothing on the crossings the streak cannot reach. What it cannot do is notice that its own spread is too small: at half the true spread it covers 88 in a hundred.

 A triangle never sees the root against its own noise; a levelling loop sees it until it is deep. For a root at each depth carrying all of the terrain's mass, the rms of each reading's response to it over 160 placements, divided by that reading's noise — the background the root sits in plus the observing error. A 25 km loop: 34.2 per cent at 10 km, 15.6 at 30, 3.9 at 60. A 50 km loop: 37.6 per cent at 10 km, 20.1 at 30, 5.5 at 60. A 100 km loop: 34.9 per cent at 10 km, 20.2 at 30, 6.1 at 60. A 50 km triangle: 3.1 per cent at 10 km, 1.0 at 30, 0.2 at 60. The triangle's response falls with depth more steeply than the loop's, which is what made their ratio look like a gauge of depth, but at every depth it is a few per cent of its noise or less. The impossibility

A loop reads a root's depth from its own pattern, and a triangle reads none of it

A levelling loop and a geodetic triangle respond to a buried root in proportions that change with its depth, so their ratio looked like a gauge of how deep a range is compensated. It is not, because a triangle's response to a root at 30 km is a hundredth of its own noise. The depth is there in the loops alone: a root changes the pattern of misclosures from loop to loop as it deepens, and 1,600 loops of 50 km place a 10 km root within a factor of 1.5 three times in four and a 30 km root about half the time. A 60 km root is not placed even by 6,400, and when the depth is fitted rather than stated, such a root is read as a light shallow one carrying 25 per cent of the mass where it carries 60.

 The water moves the join higher than the land does, to seventy-five degrees. Goode's composite joined at each latitude, two lobes north and four south. On the water to 88°, with the layout searched afresh at each join (solid): 14.10° at 25°, 13.87° at 30°, 13.60° at 35°, 13.37° at 40°, 13.31° at 40.7°, 13.14° at 45°, 12.89° at 50°, 12.66° at 55°, 12.52° at 60°, 12.40° at 65°, 12.29° at 70°, 12.27° at 75°, 12.27° at 80°, 12.29° at 85°, 12.30° at 89°; with the ocean's layout found at Goode's join held (dashed), within 0.05° of it everywhere. On the land outside Antarctica, the land's own layout searched at each join: best at 60° with 13.83° against 14.40° at Goode's; the water's best is 75° with 12.27° against 13.31°. The families

The water would join Goode's map at seventy-five degrees

Goode joined the sinusoidal to the Mollweide at 40° 44′, and the land, given its own lobes, moves that join up to sixty degrees. The water moves it further, to seventy-five, and gains twice as much by it — a full degree of mean angular deformation against the land's half — though less of the northern water than of the northern land lies above Goode's join. The gain is in the polar water, north and south, where Goode's Mollweide cap shears even the lobes' own meridians; the water just above his join, the northern edge of the Southern Ocean among it, pays part of it back. A separate join for each hemisphere is worth a few hundredths of a degree, and with the join free the one-layout-for-both question comes out the same way it did with the join fixed, only worse: no layout comes within 3.3 degrees of both surfaces' best at once.

 Seven published numbers, twenty-eight actual ones. The correlation matrix of a seven-parameter fit to 64 common points over a region 9° across. Only the diagonal is ever published — seven standard deviations — and the twenty-one off-diagonal entries are not small: the strongest is ty against rx at 0.940. A translation and the rotation that mimics it over a small patch are very nearly the same parameter, so the fit cannot tell them apart and its errors in the two are locked together. What the numbers refer to

The parameters are not independent

Rung seven gives the seven parameters their own uncertainty and stops at seven numbers. There are twenty-eight, and the twenty-one nobody publishes are not small: a translation and the rotation that mimics it correlate at 0.94, the normal matrix has a condition number of 4 × 10¹⁶, and propagating from the diagonal alone overstates the transformation's uncertainty by up to a factor of thirty-six.

 A ruler on a cartogram is not measuring anything. Twelve pairs of places, each measured on the ground and on the page of the triangular, x first cartogram of four cities, with the page scaled so that the median pair reads exactly right — the most generous calibration available. The line is that calibration. The worst pair, Tokyo to Quito, reads 21598 km for a ground distance of 14439 km, an error of 49.6%. The open marks are the same pairs on the equal-area map the cartogram was built from. Measuring distortion

Everything else on the page pays for the areas

A cartogram gets one quantity exactly right and every other reading a page supports is collateral. A ruler on it is out by 35 per cent after the most generous calibration available, and ten of sixty triples of places change which one is in the middle.

 The same labels, placed once and placed per tile. 140 stated label boxes on a 1024-pixel page cut into 256-pixel tiles. The pale boxes are placed the same way by both rules. The dark ones are placed by the tiled renderer and suppressed by the global one — labels that should have lost a collision with something in the next tile and did not, because the tile that drew them could not see it. nine of them, against 104 labels the global rule keeps. What a machine does with it

A label belongs to no tile

Nine rungs price the tile as a piece of geometry. A label is not geometry — it is a page object placed by collision against other page objects, and collision is a global relation while a tile is rendered alone. Cut a page into tiles and 10.3 per cent of the labels are placed differently; the buffer that closes the gap is half a tile at moderate density and a whole one when it is crowded.

 With the drift's length known, the count comes back. A compiled coast of 360 points traced with a drifting error — each point's error mostly the last one's, over a correlation length of 18 points, three ten-thousandths of the map in size — and the number of seams BIC chooses once the error is whitened at an assumed length, eight tracings each. No copy (0 seams made): 0/8 right at none, 1/8 right at 2, 8/8 right at 6, 8/8 right at 12, 8/8 right at 18, 8/8 right at 30, 0/8 right at 60; one copy (2 seams made): 0/8 right at none, 6/8 right at 2, 8/8 right at 6, 8/8 right at 12, 8/8 right at 18, 7/8 right at 30, 0/8 right at 60; two copies (4 seams made): 1/8 right at none, 7/8 right at 2, 8/8 right at 6, 8/8 right at 12, 8/8 right at 18, 5/8 right at 30, 0/8 right at 60. Unwhitened the criterion buys pieces for the drift in every compilation; whitened at 6, 12, 18 points it counts every compilation right; assumed 3.3 times too long, it takes the most it is offered. What is taught wrongly

A second tracing tells a drifting hand from a copied stretch

A hand tracing a coast drifts rather than jitters, and a drifting error is a smooth departure no projection explains — so the criterion that counted copied stretches correctly under independent noise counts four or five seams where two were made, and three or four on a coast with no copy at all. Undo the drift before fitting and every count comes back, for no copy, one and two, in every tracing. That needs the drift's correlation length from outside the coast, known to within a factor of three on the short side and much less on the long, and a second tracing of the same coast supplies it: the difference between two tracings holds nothing but the two errors, and read that way the count is right in 22 tracings of 24, and never too high.

 Three solved maps span a plane of maps, and the band's best map lies well outside their triangle. Every conformal map of a 25° spherical square whose ln k is the p = 1 map's plus s times its difference from the least worst map plus u times its difference from the mean-square map, for s and u from −4 to 4, contoured by the share of the square it keeps inside a band of ±0.74 per cent — 0.5 of the least worst departure. The three maps are the corners of the shaded triangle at (0, 0), (1, 0) and (0, 1); the line between the first two is its lower edge, and the exponents from 1 to 32, dashed, run from one corner to the other through the mean-square map and out past it. The best map of the twelve exponents keeps 66.4 per cent, the best on the line 66.5, the best inside the triangle 66.5. The best in the plane sits at s = 0.72, u = −2.81, outside every one of them, and keeps 75.8. What each projection optimises

Three maps make a plane, and the band's best map lies outside their triangle

Every conformal map whose ln k is a fixed blend of three solved maps — the p = 1 map, the least worst map and the mean-square map — is itself a conformal map, so three solved maps are a plane of them. Inside their triangle the third map recovers most of what the line between the first two gave up on thin regions: 0.63 points short of the best exponent on the thinnest ellipse, where the line was 1.48 short. But the band's best map in the plane is not between the three. On 172 of 182 regions and bands it lies outside the triangle, and it keeps up to twenty points more than any exponent; started from it, a search over all twenty-five coefficients adds about a point on four regions of five and seven on a hexagon. Its price is a worst departure up to nearly five times the least worst.

 Eight hundred surveys, each a poor estimate of the scale, find the patch where the published covariance is four times too optimistic. The middle one of nine agencies, by how much of the patch its map flags: a region covered by 800 local surveys at random, each tied to five control stations of a regional network whose published covariance is honest everywhere except a disc covering 15% of the region (dashed), where the control's true errors are 4 times its published variance; one survey in ten has a monument moved five centimetres. Filled dots: the 357 surveys whose own residuals put the scale above 2. Shaded: the 18 of 441 grid cells where the count of such surveys within 0.2 of the region's width exceeds what an honest region exceeds there one time in twenty, over the whole map; 18 of them lie in the disc. Grids, and what a survey does

An agency can map where its published covariance is optimistic

The local surveys tied to a regional network each estimate, badly, how far the network's published covariance understates the truth, and the agency that receives them can pool what they say. When the understatement is confined to one part of the region, pooling it into one number dilutes it: a patch covering 15 per cent of the region with errors four times their published size reads as 1.30 for the whole. Counted place by place, the same surveys find the patch — two agencies in three find it with 400 surveys and nearly every one with 800, flagging almost nothing outside it. Fed back as one number the scale removes little of the false alarms inside the patch and costs nearly a third of the test's power outside; fed back as a map it cuts the false alarms inside from 20 per cent to 3 and costs little outside.

 Beside a bending river, every reachable point is reached by a route that meets the bank at most three times. Everywhere within a 10 km walk of the centre, beside the stated river meandering ±2 km on a 4 km wavelength with bridges every 2.5 km, shaded by the fewest times a shortest route to it has to meet the bank — a contact being one run along the bank, however many corners it passes. 70.3% of the ground is seen from the centre across dry land. One contact reaches 28.0% more, 85% of it beyond the river and the rest in bays of the near bank hidden from the centre by a spur; two reach 1.18% and three the last 0.47%, 70% of those two beyond the river, in the lee of the far bank's bends beside the outer bridges. Nothing needs a fourth. Paths and directions

A layer is a meeting with the bank, not a corner turned

A reach set drawn in layers — rays from the centre, then from every place a route can leave the bank — ends in the exact answer, and how deep it must go depends on what a layer is. Counted by corners turned, the depth belongs to the polygon the bank is drawn as: five turns on a bank drawn every 400 metres, eighteen on one drawn every 50. Counted by meetings with the bank, it is three at every resolution. One meeting anywhere leaves 1.6 per cent of a ten-kilometre walk unreached, where rays from the bridges alone leave 6.4; two leave 0.47; three leave nothing. A longer walk needs more — up to six meetings at twenty kilometres across thirty-six rivers — and the bend's steepness, which prices the first layer, ranks the depth at 0.24.

 One smooth term beats the water's best join; the land needs three. Equal-area spacing curves with no join — the exponential of a cosine series in latitude — fitted to each surface on that surface's own lobes at its own best join, two lobes north and four south, each map at its own best proportions; the gain is over Goode's composite rejoined at the surface's best latitude (zero line). On the water to 88°, against the composite joined at 75° (12.27°): 1 term 12.03°, 2 terms 11.89°, 3 terms 11.73°, 4 terms 11.70°. On the land outside Antarctica, against the composite joined at 60° (13.83°): 1 term 14.31°, 2 terms 14.31°, 3 terms 13.71°. The families

A smooth curve beats the water's best join from its first term

On the land, a smooth equal-area curve with no join needed three terms before it beat Goode's composite rejoined at the land's best latitude, and on the land's own lobes it needs three again. On the water a single term beats the water's best composite, a second and third add more, and the best curve is 0.54 degrees of mean angular deformation better than the composite joined at seventy-five and 1.6 better than Goode's. It is nothing like the composite in shape: its parallels never close to less than seven eighths of the equator's spacing, so the map the water wants is the sinusoidal reshaped by a few per cent at every latitude, not the sinusoidal with a cap. The lobes do not move under it. And where a separate join for each hemisphere was worth three hundredths of a degree, a separate curve for each is worth a quarter of a degree.

 A legend saying "illuminated from 315°" is true at one longitude. A hillshade's azimuth is measured from the top of the sheet, because the shading is computed on the projected raster. The top of the sheet is grid north, so the compass bearing the light comes from is the declared azimuth plus the meridian convergence, and that varies across the sheet. On a conic it swings by 55.3 degrees over eighty degrees of longitude. On a cylindrical projection in its normal aspect it does not swing at all, which is the flat line — the only case the legend is right everywhere. Measuring distortion

The light comes from a page direction

A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.

 A section through a body with a neck, and the rays that leave it twice. An equatorial section of a stated contact binary — two lobes of radius 1 centred at ±1.5, joined by a neck of radius 0.35, with the origin in the neck. The lines are rays from the origin and the marks are where each one crosses the surface. two of the 25 drawn cross more than once, so along those directions there is no such thing as "the" radius, and a longitude and a latitude do not name a place. What the numbers refer to

A ray from the centre hits the surface twice

Eleven rungs map bodies that are lumpy, triaxial and turning at a drifting rate, and every one assumes the surface is star-shaped about the centre — which is what makes a longitude and a latitude a coordinate at all. A contact binary is not: on a stated body with a neck a third of a lobe wide, 10.9 per cent of the sky has no single radius, and the shape model everybody publishes fills the neck in and adds 1.67 per cent of the volume.

 Three selection rules, and what each one keeps. Keeping one feature in ten from a stated population whose size distribution has a Pareto exponent of a half — the exponent Töpfer's law is a theorem about. Keeping the largest carries 99.99 per cent of the total size and inflates the median feature by a factor of 95. A random sample keeps the median to 1.068 and carries 5.0 per cent of the total. The two rules are right about different things and there is no rule that is right about both, because the total lives in the tail and the median does not. What a machine does with it

Which features survive is not a sample

How many features a scale can carry is answered and treats the population as a number. Which ones survive is a different question: keeping one feature in ten carries 99.99 per cent of the total length and inflates the median feature by a factor of 95, and the shape of the size distribution survives both exactly.

 Given an easing to fit, the criterion counts an eased copy as one copy until the two easings run well into each other. One copied stretch, eased into place over a stated share of the coast either side of each seam, traced to a ten-thousandth of the map. Hollow: the rigid count, offered up to six seams — two for a rigid copy, then 4, 5, 6, 6, 6, 6, 6, 6 from 5.0% of easing. Filled: the count when each seam may be eased over a width the fit finds and pays for, offered up to four — two at every easing to 17.5%, then 4, 4. The dotted line is half the copy's length, 15% of the coast: past it the two easings overlap, and the copy is nowhere drawn as itself. What is taught wrongly

An easing is a seam with a width, and a drifting hand hides the width

A copy eased into a compiled coast is not a projection, so a count that explains the coast in rigid pieces buys four, five or six seams for one copy. Give each seam a width — the copy blended into the host over that width, the same linear fit at every width, the width charged as one more number — and the count is two seams at every easing up to 17.5 per cent of the coast, and the width it reports is the one drawn, wherever the copy departs from the host by many times the digitising error. Where it departs by seven, the width is a guess. On a tracing made by a drifting hand the easing disappears: whitened for the drift, the rigid count already calls two seams in every tracing, because an easing as long as the drift is a stretch of drift, and its width can no longer be read until it is twice the drift's length.

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