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

Gaussian curvature, and what it forbids

surface-curvature computes it two ways on six surfaces and requires the answers to agree, which is Theorema Egregium exercised on every build. developable is the zero-curvature case, curvature-field the case that changes sign, curvature-integral and triangle-excess the integrated forms, holonomy-split the direction carried round a loop, same-curvature two surfaces that share a number and not a shape, and plane-survey the practical question of how small a region has to be before the curvature stops mattering. The distance- modes make the same argument with a ruler instead of a derivative: a set of distances measured on the sphere is the distance set of no flat picture in any dimension — distance-determinant and distance-spectrum are the test, distance-diameter the square law, distance-counting the 2n − 3 a flat picture can hold, and distance-curved and distance-curvatureaxis the one page that holds them all, which is the sphere at its own radius. distance-omit and its neighbours ask which places the error belongs to, distance-extent what size a place must be before a flat picture is exact, and distance-bearingpair and its neighbours the directions a flat picture cannot hold.

surface-curvature computes it two ways on six surfaces and requires the answers to agree, which is Theorema Egregium exercised on every build. developable is the zero-curvature case, curvature-field the case that changes sign, curvature-integral and triangle-excess the integrated forms, holonomy-split the direction carried round a loop, same-curvature two surfaces that share a number and not a shape, and plane-survey the practical question of how small a region has to be before the curvature stops mattering. The distance- modes make the same argument with a ruler instead of a derivative: a set of distances measured on the sphere is the distance set of no flat picture in any dimension — distance-determinant and distance-spectrum are the test, distance-diameter the square law, distance-counting the 2n − 3 a flat picture can hold, and distance-curved and distance-curvatureaxis the one page that holds them all, which is the sphere at its own radius. distance-omit and its neighbours ask which places the error belongs to, distance-extent what size a place must be before a flat picture is exact, and distance-bearingpair and its neighbours the directions a flat picture cannot hold.

Every one of these is the same construction answering a different question, which is why they share their drawing rules, their colours and their checks. Changing it changes all 78.

36 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

What it draws

One picture per question the family answers. Each has a page of its own.

Six surfaces and their Gaussian curvature. Curvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not.

surface-curvature

Six surfaces and their Gaussian curvature A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere.

developable

A cylinder unrolls exactly; a sphere does not Gaussian curvature across a torus. Curvature along a cross-section of a torus. curvature of both signs — positive outside, negative inside, zero on two circles. Where the curve crosses zero the surface is momentarily flat in the intrinsic sense, and a strip along that circle could be unrolled without stretching.

curvature-field

Gaussian curvature across a torus The angles of a triangle, and the curvature inside it. A triangle on the sphere with sides that are great-circle arcs. Its three angles sum to 212.26°, overshooting the flat 180° by 32.26°. Integrating the curvature over the interior gives 0.56306 against an excess of 0.56306 — the same number by two routes that share no arithmetic. That is Gauss–Bonnet, and it is how a flatlander measures the curvature of a world it cannot step outside.

triangle-excess

The angles of a triangle, and the curvature inside it Total curvature is a counting number in disguise. The Gaussian curvature of each surface, integrated over the whole of it. Gauss–Bonnet fixes the answer at 2πχ, where χ is the Euler characteristic — vertices minus edges plus faces, a quantity with no geometry in it at all. The sphere gives 12.5664 against 4π = 12.5664. The torus gives -2.4e-11 against zero, and it does so by cancellation: its outer half is positively curved and its inner half negatively, in exactly equal measure.

curvature-integral

Total curvature is a counting number in disguise What a loop round a parallel turns a direction by, and what it encloses. Two quantities against the latitude of the loop. The turning — the angle a transported direction comes back rotated by, relative to the local north — is 2π sin φ, measured. The curvature the loop encloses is the area of the cap above it, 2π(1 − sin φ). They sum to 2π at every latitude, drawn as the flat line across the top, and neither is the other: at 40° the turning is 4.039 radians and the enclosed curvature 2.244. The difference is the winding of "north" itself around the pole, which is exactly one revolution however small the loop.

holonomy-split

What a loop round a parallel turns a direction by, and what it encloses Three surfaces with the same curvature, one of which is a sphere. Every one of these is a surface of revolution whose Gaussian curvature is 1 at every point, built by solving r″ + r = 0 for the meridian rather than by writing a shape down. The spindle closes to a point with an angle deficit, the sphere closes smoothly, and the bulge does not close at all — it ends in two circular edges. A surveyor confined to a patch of any of them, measuring angles and distances, cannot tell which one it is.

same-curvature

Three surfaces with the same curvature, one of which is a sphere How large a patch can be treated as flat, at 10 parts per million. The smallest scale distortion any map of a circular patch can have, against the radius of the patch, on logarithmic axes. The line is straight with a slope of 2.00: the error grows as the SQUARE of the size, so a patch ten times wider is a hundred times worse. A tolerance of 10 ppm is reached at a radius of 40.3 km — 81 km across — and that is the number behind the boundary between plane surveying and geodesy.

plane-survey

How large a patch can be treated as flat, at 10 parts per million The four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure built on them — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes.

distance-quadmap

The four places, and every distance between them The best flat picture of them, and what it still gets wrong. The four places placed on a plane so that the WORST of the six distances is as nearly right as any flat arrangement can make it. There is no projection here and no map: the dots carry no coordinates, only separations. Each edge is labelled with how far its drawn length is from the truth once the picture has been given its one free scale. The best any arrangement achieves is 0.629% on the worst edge — London to Tokyo — and no rearrangement lowers it, because the six numbers are not the distances of four points in a plane at all.

distance-flat

The best flat picture of them, and what it still gets wrong The spectrum that decides whether a picture exists. The four eigenvalues of the double-centred matrix of squared distances, for the same four places measured two ways. Drawn straight through the rock, the chordal distances give three positive eigenvalues and no negative one, which is Schoenberg's statement that they are the distances of four points in three dimensions — and so they are, because the places are already there. Measured along the surface, one eigenvalue is negative: -2.165e+6 against a largest of 1.879e+8. A negative eigenvalue is not a large error. It is the statement that no Euclidean space of any dimension holds these distances.

distance-spectrum

The spectrum that decides whether a picture exists Every three of them fit, and the four together do not. Each subset of the four places, with the smallest eigenvalue of its own double-centred distance matrix — the quantity that is zero exactly when a flat picture exists and negative when none does. Every one of the four triangles sits at zero to rounding, which is the triangle inequality doing what it always does. The one four-place subset does not, and the bar it draws is the whole of the impossibility this ladder is about. Nothing about the places was chosen to make this happen; it happens to any four points on a sphere that are not on one great circle.

distance-triples

Every three of them fit, and the four together do not The determinant that has to be zero, and is not. The Cayley–Menger determinant of the four places, divided by the sixth power of their diameter so that only the shape is left, against how much of the sphere they span. Zero is what a flat picture requires; the chordal distances give it to rounding at every size, and the geodesic ones never do. The fitted slope is 2.149 on the log axes, so the departure falls as roughly the square of the span — the same exponent every other measurement in this ladder finds, arriving here through a determinant rather than through an optimisation.

distance-determinant

The determinant that has to be zero, and is not The least error a flat picture can have, against how much sphere it spans. The same configuration of places, shrunk about its own centroid so that every bearing is kept and only the span changes, with the least worst-case relative error of the best flat picture at each size. Both axes are logarithmic. The fitted slope over the rows below ninety degrees is 2.0246: the error falls as the SQUARE of the diameter. That is Gauss's theorem arriving as a number for a finite set — curvature is a second derivative, so its first effect on a distance is quadratic in the separation — and it is why a county fits on a sheet and a hemisphere does not.

distance-diameter

The least error a flat picture can have, against how much sphere it spans The same arrangement at three sizes. The best flat picture of London, New York, Tokyo, Sydney, with the configuration shrunk about its own centroid so that every bearing from the centre is unchanged and only how much of the sphere it covers varies. At 152.8° across the worst edge is out by 0.629%; at 39.7° it is 0.0299%; at 9.9° it is 0.0018%. The pictures are indistinguishable to the eye and the numbers under them fall by a factor of sixteen for each factor of four, which is what a square law looks like when it is drawn rather than plotted.

distance-shrink

The same arrangement at three sizes Three ways of scoring the same picture, and one exponent. The same ladder of spans scored three ways: the worst edge of the best possible arrangement, the worst edge of the arrangement classical multidimensional scaling produces, and the root-mean-square error over all the edges. The three curves are parallel and are not the same curve — classical scaling sits a factor of 1.31 above the optimum at every span, and the root-mean-square sits below both because averaging hides the edge the minimax is about. The fitted exponents are 2.0246, 2.0223, 2.0111: the constant in front is a property of the estimator and the exponent is a property of the sphere.

distance-estimators

Three ways of scoring the same picture, and one exponent The exponent belongs to the sphere and the constant to the arrangement. The square law fitted separately on five configurations of places — from five towns inside one country to sixteen cities on every continent — each shrunk through its own ladder of spans. The exponents agree: 2.025, 2.016, 1.999, 2.013, 1.978. The constants in front of them do not, and span a factor of 17. That separation is the point of running five sets rather than one: how badly a particular arrangement of places resists a flat sheet is a fact about the arrangement, and that the resistance falls as the square of the span is a fact about the surface they sit on.

distance-sets

The exponent belongs to the sphere and the constant to the arrangement Every projection against a picture that is not a map. The worst relative distance error over London, New York, Tokyo, Sydney, for each projection in the library and for the best flat arrangement of the same distances. Each azimuthal member is centred on the set's own centroid, which is the fair comparison. The free picture reaches 0.63% and the best projection, Azimuthal equidistant, reaches 8.28% — a ratio of 13.17. The free picture cannot lose, because every projection's own layout was handed to the search as a starting point; what the figure measures is how much the freedom is worth, and it is worth different amounts at different sizes.

distance-library

Every projection against a picture that is not a map The same places, arranged by a map and arranged by nothing. On the left, London, New York, Tokyo, Sydney where Azimuthal equidistant puts them, centred on the set's own centroid. On the right, the arrangement that makes the worst of the six distances as nearly right as any flat arrangement can. The edges that carry more than half the worst error are drawn heavier. The projection reaches 8.28% and the free picture 0.63%. The free picture is better and is not a map: it has no graticule to hang a coastline on, nothing to say about any place not in the set, and adding a seventeenth city moves every dot in it.

distance-against

The same places, arranged by a map and arranged by nothing What abandoning the projection is worth, and how fast it stops being worth it. For each set of places, the worst-edge error of the best projection in the library divided by the worst-edge error of the best flat arrangement of the same distances. The free picture cannot lose, because every projection's own layout is handed to the search as a starting point. On four world cities the freedom is worth a factor of 13.2; on sixteen it is worth 1.08, which is eight per cent. What collapses it is arithmetic rather than cartography: a picture of n places has 2n − 3 free numbers against n(n−1)/2 distances, so the freedom available per constraint falls as 4/n, and by sixteen places the optimiser has almost nothing left to spend that the projection has not already spent.

distance-advantage

What abandoning the projection is worth, and how fast it stops being worth it The share of its distances a flat picture can hold. A picture of n places on a plane has 2n coordinates and is unchanged by two translations and a rotation, so 2n − 3 numbers in it are genuinely free; each distance held exactly is one equation. So at most 2n − 3 of the n(n−1)/2 distances can be right, whatever the places are and however hard the map maker tries. The share falls as 4/n: five of six for four places, thirteen of twenty-eight for eight, and 197 of 4,950 — 4.0% — for a hundred. The dashed curve is 4/n, which the count approaches from below and never crosses.

distance-counting

The share of its distances a flat picture can hold The distances a construction holds, and the rest it decides. The construction a surveyor would use, on London, New York, Tokyo, Sydney. The first place is put down anywhere, the second at its true distance along an axis, and every further place is trilaterated from those two — which holds 5 distances exactly, drawn heavy, and that is 2n − 3 for n = 4. Nothing is fitted. The remaining 1 distances, drawn thin, are then decided rather than chosen, and on places taken from a sphere they are decided wrongly: the worst is out by 6.87% and the closest to right by 6.867%.

distance-trilateration

The distances a construction holds, and the rest it decides What a map centred on a place spends, and what it leaves unspent. An azimuthal equidistant map centred on London is exactly right about every distance FROM London — measured here at 6.7e-16 on three of them, which is rounding — and about nothing else: the worst of the other 3 is out by 32.3%. Those n − 1 exact distances are the star, and the plane allows 2n − 3. The gap between the two curves is n − 2 distances a flat picture could have held and no projection centred on a place does, because a projection is a rule about the whole sphere and cannot spend its freedom on the places that happen to be in the set.

distance-star

What a map centred on a place spends, and what it leaves unspent The distances a centred map holds, drawn on the page that holds them. London, New York, Tokyo, Sydney on an azimuthal equidistant projection centred on London. The three heavy spokes are exactly right: the projection's radial coordinate IS the angular distance from its centre, so those separations are true to the last bit of the arithmetic — measured at 6.7e-16. Every thin chord is wrong, the worst by 32%, and the reason is that a straight line between two points of this page is not a route on the ground. 3 exact of 6: the map has spent n − 1 of the 2n − 3 a flat picture is allowed.

distance-starmap

The distances a centred map holds, drawn on the page that holds them What each extra dimension buys, which is nothing. The worst relative error of the best k-dimensional picture of London, New York, Tokyo, Sydney, for every k the set allows. Going from a line to a plane buys almost everything — from 180.2% to 1.27%. Going from a plane to a space buys nothing at all: the two numbers agree to every digit, because the third eigenvalue the extra dimension would spend is 2.85e-8 — zero — and the eigenvalue that would help is negative and cannot be spent in a Euclidean space of any dimension. The flat sheet is not the obstruction. Euclid is.

distance-dimensions

What each extra dimension buys, which is nothing How much of the spectrum has the wrong sign. The signature of the double-centred squared-distance matrix for five sets of places, one row each, one cell per eigenvalue, ordered largest first. Filled cells on the left are positive, pale cells are zero, and the cells on the right are negative — the eigenvalues that no Euclidean space of any dimension can hold. Four places give one; sixteen cities give 8 of 16. The chordal distances of every one of these sets give three positive, no negative, at every size, which is the refusal that makes the geodesic column mean something rather than merely being a picture of a spectrum.

distance-signature

How much of the spectrum has the wrong sign The page's curvature is not a free parameter. The same trilateration, carried out on a sphere of stated radius instead of on a plane, with the error of the distances the construction decides rather than holds. At the radius the distances were measured on it is 3.3e-16 — exact, by construction, which is the refusal this figure exists to make. Two per cent either side of it the worst decided distance is already out by around 4%. Below 90 per cent of the Earth's radius the construction cannot be completed at all: two circles that must cross do not, and the page is simply too small to hold the places. The flat sheet is the right-hand limit, at 7%.

distance-radius

The page's curvature is not a free parameter Five pages of different curvature, and the one that works. The worst decided distance on a page of each stated radius, for London, New York, Tokyo, Sydney. A page too small is refused rather than fitted — the construction reaches a step where two circles that must cross do not, which is a geometric fact rather than a failure of a solver. A page too large is drawn and wrong. Exactly one radius is right, and it is the radius the distances were measured on, which is the whole of what "the escape is a curved page" amounts to: the page has to be the sphere.

distance-curved

Five pages of different curvature, and the one that works One axis for every page a picture could be drawn on. The three escapes of this rung are one parameter. A hyperbolic page of curvature radius k has Gaussian curvature −1/k², a flat sheet has zero, and a sphere of radius R has +1/R²; the axis runs through all of them in units of the Earth's own curvature, so the Earth sits at +1. The same construction runs at every point — hold 2n − 3 distances exactly, read the rest — and the curve has exactly one zero, at +1, where the error is 3.3e-16. The left branch never dips below the flat sheet's 7%; it approaches it from above as the curvature goes to zero, which is also a check that two constructions written separately with cosines and with hyperbolic cosines agree where they must. Below about +1.2 on the right the sphere is too small to hold the places and the construction refuses rather than fits.

distance-curvatureaxis

One axis for every page a picture could be drawn on Sixteen cities, each left out in turn. The least worst-case distance error any flat picture of these sixteen cities can have is 53.3%; the best projection in the library, Azimuthal equidistant, manages 57.6%. Each bar is that least error with one city left out. Leaving out Sydney brings it to 25.5%; leaving out Cape Town, the next most costly, to 46.7%; and leaving out any of five — New York, Tokyo, Anchorage, Singapore, Los Angeles — changes it not at all. The note on a row counts the pairs holding the picture at its worst that the city belongs to.

distance-omit

Sixteen cities, each left out in turn The pairs that hold the best flat picture at its worst. The flat arrangement of sixteen cities whose worst distance error is as small as any can be, 53.3%. Every pair is drawn faintly; the six drawn heavily are the pairs at that worst error, three drawn too long and three too short, and no move of any dot can improve one without making another worse. Sydney, Santiago and Reykjavík each belong to three of them. The picture is not a map and carries no coordinates, only separations.

distance-binding

The pairs that hold the best flat picture at its worst Four cities left out, one at a time. The least worst-case error of a flat picture as cities are left out one at a time, each time the one whose absence helps most. Without Sydney it is 25.5%; without Singapore it is 20.4%; without Honolulu it is 16.9%; without Los Angeles it is 13.3%. The dashed line is the best projection in the library on all sixteen, 57.6%; the best flat picture beats it by 8%, and leaving out the first city beats the best picture by a factor of 2.09. Greedy rather than exhaustive: the best four to omit together need 1,820 searches and these took fifty-eight.

distance-greedy

Four cities left out, one at a time Which city matters depends on which have already gone. Left: the least error with each city left out of all sixteen. Right: the same, once Sydney has already gone. Most cities barely move. Singapore moves most — from 53.3%, where leaving it out changed nothing, to 20.4%, the best of all fifteen — because the pairs holding the picture at its worst changed when the first city went, and Singapore is in the new ones. The order in which places matter is not a ranking.

distance-reorder

Which city matters depends on which have already gone Four cities cannot be drawn to scale, and any three can. The least worst-case error of a flat picture of London, New York, Tokyo and Sydney is 0.629%. Leave any one of them out and the three that remain have an exact flat picture — the error is zero to the arithmetic in all four cases — because three distances that satisfy the triangle inequality are always the sides of a triangle. That is the one set where leaving out a place is guaranteed to buy everything, and it is the check that the instrument returns zero when zero is the answer.

distance-omitfour

Four cities cannot be drawn to scale, and any three can What choosing the places buys, against what choosing the picture buys. How many times smaller the worst distance error becomes, against the best projection in the library on all sixteen cities. Abandoning projections for the best free arrangement buys a factor of 1.08. Leaving out Sydney as well buys 2.26, and leaving out four cities buys 4.34. The largest single decision in the picture is not how it is drawn but what is in it.

distance-omitgain

What choosing the places buys, against what choosing the picture buys A radius of 37.7 km makes a flat picture of four cities exact. Each row is one pair of London, New York, Tokyo, Sydney. The bar is the range of distances between two places of radius 37.7 km — the true centre-to-centre distance, plus or minus 75.4 km — and the solid dot is where the best picture for places of that size draws the pair. Every dot is inside its bar, and the pairs at the bars' ends are the ones that fix the radius. The hollow dots are the picture that is best about ratios instead: its worst pair is 103.3 km off, so it would need places of 51.6 km. The best picture of points is not the best picture of places.

distance-extent

A radius of 37.7 km makes a flat picture of four cities exact The radius a place needs grows as the cube of the set. How large every place has to be for some flat picture to be right about all of a set's distances, against the set's diameter, as the four-city and eight-city arrangements are shrunk about their own centres. Both axes are logarithmic. The four cities' fitted slope is 3.003 and the eight's 3.008: halve the set and the radius falls eightfold. Four cities 552 km across need places of 0.77 m. The two lines are parallel and apart — the exponent belongs to the sphere, the constant to the arrangement.

distance-extentladder

The radius a place needs grows as the cube of the set The radius each set needs at its own size. How large every place must be for a flat picture of each set to be right about all its distances, on a logarithmic bar. Five towns, 623 km across: 6.2 m. Four cities, 16,994 km across: 38 km. Five cities, 16,994 km across: 575 km. Eight cities, 17,685 km across: 1,198 km. Five towns and four cities differ by a factor of twenty-seven in diameter and six thousand in the radius they need — about the cube of the one, with a factor of three belonging to their different arrangements. Four cities and five cities span the same diameter and need radii fifteen times apart, which is the arrangement rather than the size. The world sets' values are the best a search found, so they are upper bounds; the smallest set's agrees from every starting picture.

distance-extentsets

The radius each set needs at its own size The picture best about ratios needs larger places. For each set, the radius the picture that is best about relative error would need, divided by the radius the picture built for places of equal size needs: 1.77 times for five towns, 1.37 times for four cities, 1.26 times for five cities, 1.63 times for eight cities. The two pictures answer different questions — one charges an error in proportion to the distance, the other charges every kilometre the same — and the arrangement that wins one loses the other by between a quarter and three quarters.

distance-extentratio

The picture best about ratios needs larger places Places along one great circle need no size at all, until one moves off it. Five places on the equator, between 0° and 88° east, have the distances of five points on a line, so a flat picture of them is exact with places of no size — the radius measured is 5e-13 km, which is the arithmetic. Lift the middle one off the equator and the set stops being a line on the sphere: a quarter of a degree needs 232.7 mm, one degree 3.8 m, eight degrees 245.5 m — growing as the square of the lift, sixteen times for four times the angle.

distance-extentline

Places along one great circle need no size at all, until one moves off it One line cannot point both ways between New York and Tokyo. The great-circle bearing from New York to Tokyo is 333.0°, and from Tokyo back to New York 25.1°. A flat picture draws one straight line between them, whose two directions differ by exactly 180°; the two bearings differ by 180° and another 127.9°. The dashed direction is the best a straight line can do for both ends at once, and it is 63.9° from the truth at each — which is a floor on the bearing error of every flat picture that contains these two cities.

distance-bearingpair

One line cannot point both ways between New York and Tokyo Up to a span, the best picture sits exactly on the floor. The four cities shrunk about their centre, with the worst bearing error of the best flat picture, the floor that no picture can go under, and Mercator's layout, against the span of the set. At every span up to 56° the best picture reaches the floor to the arithmetic — 77.22° at 56°, 39.52° at 40°, 13.56° at 20°, 5.68° at 10°. By 71° it has lifted clear of it and stays there, at 116° to 118° against floors of 64° to 70°.

distance-bearingladder

Up to a span, the best picture sits exactly on the floor The floor, the best picture and Mercator, set by set. For each set, the floor on the worst bearing error, the best flat picture's worst bearing error, and Mercator's. Five towns in Britain: 2.13°, 2.13° and 2.13° — all three the same. Four world cities: 63.9°, 116.1° and 137.5°. Sixteen: 87.1°, 158.4° and 169.1°. Five places along the equator keep their bearings reciprocal and have a floor of 0°, which a picture reaches.

distance-bearingsets

The floor, the best picture and Mercator, set by set The rhumb line runs midway between the two bearings, to first order. From London along a great circle at an initial bearing of 60°, to distances of 100 to 4,000 km. The upper line is how far the bearing out and the bearing back depart from being reverses of each other — 11.05° at 1,000 km — which grows in proportion to the distance (fitted slope 1.05). The lower line is how far the rhumb line's constant bearing sits from the mean of the two: 0.126° at 1,000 km, growing as the square of the distance (slope 1.98). So a straight line on Mercator splits a pair's two bearings evenly over any regional distance, which is what puts it on the floor.

distance-bearingmidway

The rhumb line runs midway between the two bearings, to first order Five bearings of four cities, held exactly. A flat picture of London, New York, Tokyo, Sydney holding five one-way bearings exactly, each drawn as an arrow from the city it is measured at: London to Tokyo, Sydney to London, New York to Tokyo, New York to Sydney, Tokyo to Sydney. Five is the most any flat picture of four places can hold, and of the 192 ways of choosing five pairs and a direction on each, this is one of eight that can be drawn with every arrow pointing forwards; all eight leave London–New York, dashed, unheld. The price is paid by the other seven bearings: the worst, London to New York, is 165.0° from the truth.

distance-bearingheld

Five bearings of four cities, held exactly Every slope right, and one arrow pointing backwards. The same four cities holding a different five bearings: Tokyo to London, London to Sydney, Tokyo to New York, New York to Sydney, Tokyo to Sydney. The equations that say a line is drawn at a stated bearing are satisfied by a line pointing either way along it, and the picture they give has all five slopes exactly right. Four arrows point forwards. The fifth, Tokyo to Sydney, is drawn along the right line in the wrong direction: Sydney is drawn on the opposite side of Tokyo from the one its bearing points to, 180° from the truth. The short dashed arrow at Tokyo is where the bearing points.

distance-bearingbackwards

Every slope right, and one arrow pointing backwards Of 192 ways to hold five bearings of four cities, eight can be drawn. Every way of holding five of the four cities' six pairs, with one direction chosen on each pair, grouped by the pair left out. Each choice has exactly one picture, and the bar says how many of its arrows that picture draws backwards. Eight of 192 draw every arrow forwards; 48 draw one backwards and 136 draw two. Every drawable choice leaves London–New York unheld, and leaving out any other pair makes every choice undrawable.

distance-bearingchoices

Of 192 ways to hold five bearings of four cities, eight can be drawn Drawable on a region, almost never on the world. The share of choices of 2n − 3 one-way bearings whose one picture draws every arrow forwards, as each set is shrunk about its centre. Four cities: every one of the 192 choices at each size, 4.2 per cent at their full span and 100 per cent at 10°. Eight cities: 33 per cent at 31°. Sixteen cities: none at 32° and 13 per cent at 8°. Five towns in Britain, spanning 5.6°, the dot: every choice drawable.

distance-bearingshrink

Drawable on a region, almost never on the world Every bearing from two places, and how many point backwards. For each set, every ordered pair of centres: hold every bearing out of the first and every bearing out of the second, 2n − 3 in all. The bar is the fewest arrows any pair of centres leaves pointing backwards, as a share of those held. Five towns in Britain: none, and all 20 pairs of centres drawable. Four cities: 1 of 5. Eight: 3 of 13. Sixteen: 8 of 29, from Santiago and Buenos Aires, and not one of the 240 pairs of centres drawable.

distance-bearingcentres

Every bearing from two places, and how many point backwards What holding 2n − 3 bearings exactly costs the rest. For each set, three worst bearing errors. The floor, half the largest mismatch between a pair's two bearings; the best flat picture, which holds no bearing exactly and minimises the worst; and the best picture found that holds 2n − 3 bearings exactly with every arrow forwards. Britain: 2.13°, 2.13° and 3.53°. Four cities: 63.9°, 116.1° and 165.0°. Eight: 73.7°, 128.1° and 152.2°. Sixteen: 87.1°, 158.4° and 172.6°.

distance-bearingprice

What holding 2n − 3 bearings exactly costs the rest Twenty-nine bearings of sixteen cities, every arrow forwards. A flat picture of sixteen cities on every continent that holds twenty-nine one-way bearings exactly, the most any flat picture of sixteen places can hold, with every arrow pointing forwards. It was built rather than chosen: starting from Nairobi and Tokyo, each further city is placed where two bearings to or from cities already placed meet in front of both. No choice of twenty-nine made at random in two hundred tries can be drawn. The frame holds 15 of the cities: Santiago is placed by two bearings that cross at a shallow angle and lands 19 times the rest of the picture's own radius away, so its arrows are drawn dashed to the edge. The other 211 bearings pay for the twenty-nine: the worst, New York to Los Angeles, dashed and heavy, is 172.6° from the truth.

distance-bearingbuilt

Twenty-nine bearings of sixteen cities, every arrow forwards Two pairs of ten are in every picture that can be drawn. London, New York, Tokyo, Sydney, Cape Town, drawn holding seven of their ten pairs' bearings exactly — the most any flat picture of five places can hold. Of the 15,360 ways of choosing seven pairs and a direction on each, 152 give a picture whose held arrows all point forwards, and London–Tokyo and Sydney–Cape Town, drawn heavy, appear in every one of them. The three pairs this particular choice leaves out are dashed. The price is paid by the rest: the worst, New York to Sydney, is 142.1° from the truth.

distance-mandatory

Two pairs of ten are in every picture that can be drawn Every pair of ten is in some picture that can be drawn. For London, New York, Tokyo, Sydney, Cape Town, the share of the 152 drawable choices of 7 pairs that hold each pair, with the pair's bearing asymmetry beside it and the rows ordered by that asymmetry. None is held by none: the share runs from 37% to 100%, and the two at the top are in every drawable choice. At five places there is no forbidden pair, so the four-place case is a statement about how little room five constraints leave rather than about the geography. A filled marker means the pair's arrow has a forced direction in the drawable choices that hold it.

distance-heldshare

Every pair of ten is in some picture that can be drawn Three of the five arrows may point either way, and two may not. A flat picture of London, New York, Tokyo, Sydney holding five bearings exactly. All eight drawable choices hold the same five pairs and differ only in direction, and they agree on two of them: London to Tokyo and New York to Tokyo, drawn heavy, point that way in all eight. The other three — London–Sydney, New York–Sydney, Tokyo–Sydney — are drawn light and appear four times each way, which is what makes eight rather than two. The two forced pairs are the two of largest bearing asymmetry, 124.6° and 127.9°, against 78.5° for the largest of the free ones. London–New York, dashed, is held by none.

distance-forcedfour

Three of the five arrows may point either way, and two may not Asymmetry says which way an arrow must point, and nothing about which pairs may be held. Every pair of four and of five cities that some drawable picture holds, with its bearing asymmetry against how one-sided its arrow is over the drawable choices — 1 when every such picture points it the same way and 0 when both directions appear equally. Nothing below 78.5° of asymmetry shows any one-sidedness at all, in either set; every one-sided pair is at 78.5° or above. The pair exactly at 78.5° is one-sided among five cities and free among four, which is where the threshold's own width is. The share held, which is the quantity the question was about, shows no such order.

distance-imbalance

Asymmetry says which way an arrow must point, and nothing about which pairs may be held One pair of six is in no picture that can be drawn, and it is not the one asymmetry names. For London, New York, Tokyo, Sydney, the share of the eight drawable choices of 5 pairs that hold each pair, with the pair's bearing asymmetry beside it and the rows ordered by that asymmetry. London–New York is held by none of them, and its asymmetry of 57.1° is fourth of six: the quantity that sets the floor on the whole problem does not say which pair has to be let go. A filled marker means the pair's arrow has a forced direction in the drawable choices that hold it.

distance-heldfourbars

One pair of six is in no picture that can be drawn, and it is not the one asymmetry names The room runs out faster than the count of constraints suggests. The share of choices of 2n − 3 pairs and a direction on each that give a picture with every held arrow pointing forwards, against the number of places, over 20,000 seeded choices at each size. It falls from 9.2% at four places to 2.75% at five, 0.68% at seven and 0.070% at eight. The count of bearings held rises only as 2n − 3 while the count available rises as n(n − 1)/2, so a picture is asked to satisfy a shrinking fraction of the pairs and finds it harder rather than easier — because the difficulty is in the directions, and every held pair adds one more arrow that has to come out forwards.

distance-drawshare

The room runs out faster than the count of constraints suggests What goes wrong is arrows, and their number grows with the picture. The average number of held arrows that come out pointing backwards, against the number of places, over the same 20,000 choices. It rises from 1.51 at four places to 4.92 at eight, about 0.85 of an arrow for each place added — while the count of arrows rises by two. A picture is drawable only when that number is nought, and a quantity whose mean grows steadily is nought less and less often, which is the whole of the fall in the previous measurement. Nothing here is about a pair being impossible; it is about a tally having to come out empty.

distance-wrongway

What goes wrong is arrows, and their number grows with the picture Six of eighteen, where either kind alone can hold only five. London, New York, Tokyo, Sydney, drawn holding five distances exactly — the thick edges — and one bearing exactly, the arrow from London to New York. Six exact constraints, where five distances is the most any flat picture of four places can hold and five bearings is the most it can hold of those. All six are satisfied to 1.9 × 10⁻¹¹. The sixth is not free room found by luck: a picture is blind to where it sits, how it is turned and how large it is, distances see the last of those three and bearings see the second, and a picture holding both kinds is blind only to where it sits. Tokyo–Sydney, dashed, is the distance left out; the picture is wrong about it by 6.9%.

distance-mixedmap

Six of eighteen, where either kind alone can hold only five No set of 6 constraints of one kind is independent, and thousands of mixed sets are. Every way of choosing 6 of the eighteen constraints four cities offer — six distances and twelve one-way bearings — sorted by how many of each, with the share of each split whose constraints are independent. 0 of the 924 sets of 6 bearings are independent, and the one set of 6 distances is not independent either. Every split with at least one of each kind has some, and the split with five distances and one bearing has all 72. In total 7,596 of 18,564 choices can be held exactly.

distance-mixsplit

No set of 6 constraints of one kind is independent, and thousands of mixed sets are The extra constraint is one, and stays one however many places there are. The largest number of constraints a flat picture can hold exactly, against the number of places, for distances alone, bearings alone and the two mixed. Distances give 5, 7, 13 at four, five and eight places and bearings give the same, which is 2n − 3 in both cases. Mixed sets give 6, 8, 14 — exactly one more, at every size. The gain does not grow with the picture because it is not a share of anything: it is the single freedom that neither pure kind can see, and there is only ever one of it.

distance-mixmax

The extra constraint is one, and stays one however many places there are The share that can be held falls, and then it is nothing. The share of all choices of k of the eighteen constraints four cities offer whose constraints are independent, against k. It is 77% at four, 62% at five and 41% at six. At seven it is nought of 31,824 choices — not rare, not unlikely, but impossible, because a picture of four places has six numbers a measurement can reach and a seventh demand has nothing left to be satisfied with.

distance-mixcount

The share that can be held falls, and then it is nothing The sixth constraint chooses a turn, and not the turn the other bearings would choose. Five distances fix the shape of a picture of four cities and leave it free to turn. The worst of its twelve bearing errors, against how far it is turned. The best turn is 342°, where the worst bearing is 132.1° out and the mean is 65.5°. The turn that holds London to New York exactly is 83°, where the worst is 165.9° and the mean 96.3°. So the sixth constraint costs nothing in distances — the shape does not move — and it is not free in bearings: it spends the turn on one of them, and the eleven others pay 33.8° more in the worst case than they need have.

distance-mixrotate

The sixth constraint chooses a turn, and not the turn the other bearings would choose And the mirror: one distance chooses a size, and not the size the others would choose. Five bearings fix the shape of a picture of four cities and leave it free to grow. The worst of its six relative distance errors, against how large it is drawn. The best size makes the worst distance 78.8% out and the mean 49.5%. The size that holds London–New York exactly is 0.41 times that, where the worst is 88.1% and the mean 48.7%. The two cases are the same statement twice: a pure problem leaves one freedom unmeasured, one constraint of the other kind spends it exactly, and what it buys is exactness on that constraint rather than accuracy anywhere else.

distance-mixscale

And the mirror: one distance chooses a size, and not the size the others would choose What the picture gets wrong is everything it was not asked about. The twelve constraints the six-constraint picture was not asked to hold, with what it makes of each — a relative error for a distance and an angle for a bearing, both drawn against the same bar so the two kinds can be seen together. The worst distance is Tokyo–Sydney at 6.9% and the worst bearing Tokyo → Sydney at 169°. Six exact constraints are not six-eighteenths of a correct picture: they are six statements that happen to be true about a drawing that is otherwise as wrong as any flat picture of four world cities has to be.

distance-mixfree

What the picture gets wrong is everything it was not asked about A picture of four places, and the one constraint that is about the sheet. The best flat picture of four world cities, with the six distances it could be asked to hold drawn between the dots and the twelve bearings implied by them. Every one of those is a statement about how two places stand to one another, so every one is blind to where the whole picture sits on the paper. The cross is a constraint of a third kind: a stated position for one place — this dot goes here — which is two numbers and is blind to nothing. Adding both of them takes the largest independent set from 6 to 8.

distance-pinmap

A picture of four places, and the one constraint that is about the sheet Only the sets holding both pins reach the end. How many independent constraint sets there are of each size, split by how many of the two pins they hold. Sets with no pin stop at 6, which is the mixed budget; sets with one pin stop at 7; sets with both reach 8 and there are 7596 of them. Nothing at all is independent at 9, which is the statement that a picture of four places has exactly eight numbers in it and they have all been spent.

distance-pinladder

Only the sets holding both pins reach the end Every count is a choice of pins times a choice of place constraints. For each size and each number of pins, how many independent sets there are, beside what a direct sum predicts: the number of ways to choose that many pins from two, times the number of independent sets of PLACE constraints of the remaining size. All 12 agree exactly. That is a much stronger statement than a budget of 2n — it says the pins and the place constraints do not interact at all, so a picture's eight numbers are two separate accounts rather than one, and nothing can be moved between them.

distance-pinsum

Every count is a choice of pins times a choice of place constraints Two more constraints, and not one more of them is about the places. What a largest picture holds, split into constraints about the places and constraints about the sheet. Without pins it holds 6 place constraints and nothing else. With both pins it holds 8 constraints in all — and still exactly 6 about the places. The budget grew by two and the part of it that says anything about where the cities are did not move. A pin is not more capacity of the same kind; it is a different kind, and the arithmetic that adds them into one total is arithmetic about the page rather than about the geography.

distance-pinaccount

Two more constraints, and not one more of them is about the places The same pairs, held the same often. For each pair of the four cities, the share of largest pictures that hold a constraint relating those two — a distance between them, or a bearing either way — computed without the pins and with them. Every share is identical: 80.0% in both columns, for all 6 pairs. There are 7,596 largest pinned pictures and 7,596 largest unpinned ones — the same number, because each of the first is one of the second with two pins added.

distance-pinpairs

The same pairs, held the same often At five places the same three curves, shifted by the number of pins. The share of random constraint sets that are independent, at five places, against the size of the set and split by how many pins it holds — 20,000 draws each. The exhaustive count at five places is sixty-four million subsets, so this is a sample; what it shows is the same structure. Each curve is the no-pin curve moved right by its own number of pins: the share at size 10 with both pins is 0.3225 and the share at size 8 with none is 0.3235. Nothing reaches 11.

distance-pinfive

At five places the same three curves, shifted by the number of pins A pin moves the whole curve along by one and changes nothing else. Each pinned share against the unpinned share of the set one size smaller for each pin held, at five places. If the pins and the place constraints do not interact, every point lies on the diagonal, because the factor the pins contribute cancels in a share. The worst departure is 0.0048, at a set of 8 holding one pin — which is sampling noise on 20,000 draws and not a structure. The exhaustive count at four places makes the same statement exactly.

distance-pinshift

A pin moves the whole curve along by one and changes nothing else Four freedoms, and the kind of statement that reaches each. A flat picture has four freedoms: it can be moved two ways, turned, and scaled. A constraint holds only what its own blindness leaves, so the budget of each kind is 2n less the number of freedoms it cannot see. A distance is blind to position and turning; a bearing to position and size; the two together only to position. A stated position for one place is blind to none of the four, and two of them take the budget to 2n — after which a picture of n places has no numbers left to spend.

distance-pinfreedoms

Four freedoms, and the kind of statement that reaches each Pins at one place move where the picture sits; pins at four reshape it. The best flat picture of four cities, fitted to every distance with a tolerance of 1 per cent, one bearing a pair to 0.01 radians, and pins to the sheet good to 50 km. Each ellipse is the error in a city's position relative to the picture's centre — its shape, which no shift changes — drawn 30 times its size. Solid: the part the distances and bearings put there. Dashed: the part the pins put there. Pinned at London alone, the pins put 0.00 km into the shape and the places 63.0 km. Pinned at all four, the pins put 30.5 km and the places 29.7 km, and the shape as a whole is 42.6 km against 63.0.

distance-pinfitmap

Pins at one place move where the picture sits; pins at four reshape it The pins' share of the shape is nothing at one place, and grows with every place pinned. How much of the error in the picture's shape comes from the pins, as a share of its variance, against the pins' tolerance, for pins at one to four of the four cities. At one place it is exactly nothing at every tolerance. At two places it peaks at 4.6 per cent near 40 km; at three, 20.7 per cent near 40 km. At four, pins good to a kilometre supply 100 per cent of the shape's error, pins good to fifty 51 per cent and pins good to three hundred 3 per cent. As the pins become useless every curve falls to nothing, because the fit stops listening to them. As they become exact the curves for two and three places fall to nothing too, because an exact pin has no error to give, and the curve for four rises to all of it, because four exact pins leave the distances and bearings nothing to decide.

distance-pinfitshare

The pins' share of the shape is nothing at one place, and grows with every place pinned Weighted as they deserve, more pins never make the shape worse. The root-mean-square error in each city's position relative to the picture's centre, against the pins' tolerance. With one place pinned it is 63.0 km at every tolerance, all of it from the distances and bearings. With two places pinned it falls to 57.3 km when the pins are good to a kilometre; with three, to 40.8; with four, to 1.2, because four pins good to a kilometre are a better drawing of the cities than the measurements between them. At fifty kilometres the four figures are 63.0, 61.1, 53.5, 42.6 km, and every curve returns to 63.0 as the pins become useless.

distance-pinfitshape

Weighted as they deserve, more pins never make the shape worse Held exact, a pin helps the shape only while it is better than the shape already is. The error in the picture's shape when the pins are held exact but really have the error on the horizontal axis, for pins at one to four places. At one place the shape is 63.0 km whatever the pins' error. At two, three and four places, exact pins start from 57.3, 40.8, 0.0 km, and each kilometre of real pin error adds 0.71, 1.03, 1.22 km to the shape in quadrature. They make the shape worse than a single pin once their real error passes 37 km, 46 km, 51 km — between 0.58 and 0.82 of the single pin's own shape error.

distance-pinfithard

Held exact, a pin helps the shape only while it is better than the shape already is Wherever the pins are, the picture's place on the sheet takes no error from the places. The error in where the pinned cities sit on the sheet, on average, split by where it came from, with every pin good to 50 km. Pinned at one place, 70.7 km, all from the pins; at two, 50.0; at three, 40.8; at four, 35.4 — the square root of two times the pins' tolerance, divided by the square root of how many places are pinned. The part from the distances and bearings is zero in every case, to rounding. Fifteen hundred seeded fits at full nonlinearity with two pinned places put the registration 49.8 km off and the shape 61.8 km off, against 50.0 and 61.1 computed.

distance-pinfitregister

Wherever the pins are, the picture's place on the sheet takes no error from the places

Where it is called

Changing this changes every one of these figures.

Six surfaces and their Gaussian curvature. Curvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not. The impossibility

No map is faithful

Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.

 A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere. The impossibility

What can be unrolled

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.

 The angles of a triangle, and the curvature inside it. A triangle on the sphere with sides that are great-circle arcs. Its three angles sum to 212.26°, overshooting the flat 180° by 32.26°. Integrating the curvature over the interior gives 0.56306 against an excess of 0.56306 — the same number by two routes that share no arithmetic. That is Gauss–Bonnet, and it is how a flatlander measures the curvature of a world it cannot step outside. The impossibility

Measuring curvature from inside

A flatlander with a ruler can find out that its world is round, by drawing a triangle and adding the angles. Gauss–Bonnet turns that into an exact statement, and the total curvature of a closed surface turns out to be a number obtained by counting.

 How large a patch can be treated as flat, at 10 parts per million. The smallest scale distortion any map of a circular patch can have, against the radius of the patch, on logarithmic axes. The line is straight with a slope of 2.00: the error grows as the SQUARE of the size, so a patch ten times wider is a hundred times worse. A tolerance of 10 ppm is reached at a radius of 40.3 km — 81 km across — and that is the number behind the boundary between plane surveying and geodesy. The impossibility

How small is flat enough

A builder works in plane coordinates and a national mapping agency does not, and the line between them is not a convention. The unavoidable error of treating a patch of the Earth as flat grows as the square of its size, and the size at any stated tolerance is a number.

 The radius of equal curvature, from equator to pole. The radius of the sphere that has the same Gaussian curvature as the ellipsoid does, at each latitude. On WGS84 it runs from 6357 km at the equator to 6400 km at the pole — the polar region is the flattened part of a squashed ball and is therefore the LEAST curved — so K itself varies by 1.35%. A surface whose curvature varies cannot be laid on one whose curvature does not, so no map from the sphere to the ellipsoid is faithful either, and the least a conformal one can vary in scale is 6739 parts per million. The impossibility

The curvature of the Earth is not one number

An ellipsoid's Gaussian curvature varies by 1.35% from equator to pole, so the sphere is not developable onto the ellipsoid any more than the plane is onto the sphere. The spherical approximation is a projection with an irreducible cost, and the cost is 6,739 parts per million.

 A direction carried once round the sphere. A vector transported round a closed loop on the sphere, kept as parallel to itself as the surface allows at every step — the component that leaves the tangent plane is removed and nothing else is done to it. The heavy arrows are its direction at the start and at the finish, drawn from the same point; the light ones are its direction along the way. It comes back turned through 282.0 degrees, which is 0.783 of a revolution, and nothing in the transport turned it. Drawn in an orthographic projection of the embedding, which is itself a map and has its own distortion. The impossibility

A direction carried round a loop

Carry a bearing round a circuit, keeping it as parallel to itself as the surface allows, and it comes back turned. Round a parallel at 45° the turn is 4.443 radians and the cap enclosed is 1.840, and they sum to exactly one revolution — so the turning is not the curvature, and on a cone it is all of one and none of the other.

 A triangle of 25.0° excess, drawn on Mercator. The three geodesic sides as curves and the three straight sides a ruler draws, with each vertex labelled by how far the ruler's angle is from the true one. The straight-sided triangle's angles sum to exactly 180° because it lies in a plane, and the real one's sum to 205.02°, so the three errors have to account for the whole 25.02° of spherical excess between them — and they do so on this conformal projection exactly as they do on any other. What is taught wrongly

Conformal does not mean the angles are right

A conformal projection preserves angles between curves at a point. Draw a triangle on one with a ruler and its angles are wrong by degrees — and the total error is fixed before the projection is chosen, because a plane triangle sums to 180° and the real one does not.

 The image of a circle on Mercator, against its own indicatrix. A circle of three radii on the ground at 30°, 40°, projected exactly — the solid curve — against the ellipse the indicatrix predicts for it, dashed. The filled dot is the image of the circle's centre and the hollow one is the centre of area of what was actually drawn, which is not the same point. The departure runs from 3.14 per cent at 4° to 15.80 per cent at 16°, so it grows in proportion to the radius rather than to its square: halving the circle halves the relative error and does not quarter it. Measuring distortion

The indicatrix is a limit

Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.

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