Cartography

Cartography

the flattening of a curved world

Abraham Ortelius, Typus Orbis Terrarum, 1572 – the world on an oval projection, from the first modern atlas. Neither the rectangle nor the disc, and wrong in a third way of its own choosing.

Field	Geodesy, geography, mathematics
Central difficulty	The Earth
Distortion	Unavoidable; select one
Preserved
By Mercator	Angle
By the azimuthal equidistant	Distance from one chosen point
By every projection at once	Nothing
Status
Impossibility proved	1827
Impossibility accepted	Widely
Impossibility read as a confession	Since 1892

Cartography is the making of maps: the representation of the Earth's curved surface upon a flat one.^[1] Because a sphere cannot be flattened without being stretched, every flat map of the world misstates something – area, angle, distance, or direction – and the cartographer's work is not to avoid distortion but to choose which distortion to accept, and to say so.^[2]

The discipline appears in this encyclopedia not because it is obscure but because it is unusually honest, and because its honesty is routinely mistaken for a confession. A projection is a decision, published in the margin of the sheet. Read as a decision, it is a tool. Read as a photograph, it is a grievance.

The choice of distortion

Gerardus Mercator's projection of 1569 is the standard example, and the standard complaint. It is conformal: it preserves angles, so that a straight line drawn upon it is a course of constant compass bearing.^[3] This is an enormous convenience to a navigator and is paid for in area, which inflates without limit towards the poles – the reason Greenland arrives on the page at roughly the size of Africa, being in fact about a fourteenth of it.

The projection is therefore not wrong; it is specialised. It was drawn for a mariner who wished to arrive, not for a reader who wished to compare, and it has spent four centuries being blamed for the second use by people who inherited it from the first.

The azimuthal equidistant

The azimuthal equidistant projection preserves distance and direction from a single chosen centre: from that one point, every distance and bearing on the map is true, and everywhere else stretches to pay for it.^[4] Centred on the North Pole it renders the world as a disc, the continents fanning outward, and Antarctica no longer a continent at all but a rim – smeared around the entire circumference, because a point has been asked to become a circle.

Alexander Gleason's New Standard Map of the World, 1892 – an azimuthal equidistant projection, correct as a projection and captioned by its maker "as it is." The rim is Antarctica, which the construction requires to be a circle and the doctrine takes to be a wall.

It is the standard projection of the polar radio operator, and it is on the emblem of the United Nations. It is also, in the same construction and often the same century, the map of the modern flat Earth – where the disc is held to be the world's actual shape and the rim its actual edge.^[5] No cartographic error is involved anywhere in this. The map is right; only the tense is wrong.

A comparison

No projection is best; each is the answer to a different question, and the table records what each one answers and what it declines to.^[6]

The same world, four times, from the US Geological Survey's Album of Map Projections. The small circles on the polar diagram are Tissot's indicatrices: each is a circle on the globe, drawn where the projection puts it. Where they are round, shape survives; where they are ellipses, it has been spent.

Only one row of that table is ever available at a time, and the reason is not a want of ingenuity.

What cannot be done

That no flat map can be faithful is not a practical limitation but a theorem. Gauss's Theorema Egregium of 1827 established that the Gaussian curvature of a surface is intrinsic: it survives bending but not stretching.^[7] A sphere of radius $R$ has $K = 1/R^2$; a plane has $K = 0$; and since no amount of bending will change one into the other, no map of a sphere onto a plane can preserve every distance.

This is why a cylinder unrolls into a rectangle without complaint and an orange peel does not. It is also, as the flat traditions correctly observe, why every world map they are shown disagrees with every other one. The observation is correct, and stops there: the theorem concerns surfaces, and has nothing whatever to say about the people who draw them. (See flat for the property itself.)

The earliest sheets

The Greek tradition credits Anaximander with the first map of the inhabited world – a round tablet of land girdled, in the Ionian manner, by the encircling Ocean.^[8] Older still is the oldest that survives: the Babylonian Map of the World, a clay tablet of the seventh century BC on which the Earth is a disc ringed by a bitter sea, the unknown set at the edges as bare triangles. Both are flat, and neither is embarrassed about it. The difficulty of flattening a sphere does not arise until somebody has established that there is one.

A map also needs several things its maker does not draw. The contour line comes from a mountain-weighing: Charles Hutton ruled the first of them to reduce the survey of Schiehallion to a volume, and cartography, finding the device lying about, took it and never gave it back.^[9] The scale bar needs a unit, and the unit needs a meridian somebody else has measured – here Hippolyte LeSight, whose metre is carried daily by men who did not care for the Frenchman who drew it.

A map may also carry something its maker put in on purpose and hopes nobody will find: a copyright trap, an invented place planted so that a rival who prints it can be shown to have copied instead of surveying.^[11] The best documented is Agloe, a crossroads in the Catskills built out of its inventors' initials, which then, it is said, went and became real.

In this encyclopedia

Most of the cosmographies recorded here are, in the end, cartographic claims. Whole-radian geometry reverse-engineers the Earth from a single radian and places coastlines accordingly, at distances its author verified by rowing towards them.^[10] The 180-degree plane is the projection promoted to a physical object: one flat surface holding all the land there is. The Concordant Orrery is the same disc built in brass, and set – with a candour its makers did not advertise – from the globe's own almanac.

The discipline also invents places on purpose, and for two opposite reasons: Agloe was printed to be copied and Null Island to be noticed, and only one of them is still doing its job. It invents them a third way as well, meaning to invent nothing at all: the Mountains of Kong were reasoned onto the map of West Africa by a geographer who needed a watershed, and were then copied for ninety-four years by cartographers who took them for surveyed.

Ptolemy, who supplied the first surviving instructions for projecting a curved Earth onto a flat sheet, is cited approvingly by all of them, and believed by none of them on the single point he was clearest about: that the Earth being flattened was a sphere.

See also

• Flat – the property, and the theorem that keeps it off the globe
• Flat Earth – which reads the projection as the territory
• The figure of the Earth – the shape being flattened
• Whole-radian geometry – coastlines placed from a radian
• The 180-degree plane – one plane, all land
• Eratosthenes – who measured the surface before anyone drew it
• Ptolemy – whose projections are cited by those who reject his sphere
• Agloe – a place put on the map so that stealing it could be proved
• Hairy ball theorem – the other thing a sphere refuses to do, for an unrelated reason
• Babylonian Map of the World – the oldest world map that survives, and a flat one
• Anaximander – credited with the first
• The Schiehallion experiment – which produced the contour line as a by-product
• Hippolyte LeSight – who supplied the unit every scale bar is drawn in
• Circle – the curve a projection may draw a point as, without giving it a circumference
• Sphere – the surface whose refusal to lie flat is this discipline's entire premise
• The Mountains of Kong – what the notation cannot record: which lines were surveyed and which reasoned

References

1. ^ On cartography as the representation of a curved surface upon a plane. The definition is uncontroversial; the difficulty is entirely in the word "upon."
2. ^ Standard treatments enumerate the properties a projection may preserve – area, angle, distance, direction – and note that no projection preserves all of them. The point is made early in every textbook and reached late by most readers.
3. ^ On the conformal property of the Mercator projection and the rhumb line. See The choice of distortion, above; the inflation of area towards the poles is a consequence of the same property, not a defect additional to it.
4. ^ On the azimuthal equidistant projection: true distance and bearing from the centre only. A projection centred on a point must do something with the antipode, and what it does is spread it around the edge of the sheet.
5. ^ On the disc-and-rim arrangement of the modern flat Earth. The projection and the doctrine agree precisely on the picture and entirely disagree on what it is a picture of.
6. ^ The classifications and the properties tabulated here follow Snyder and Voxland's An Album of Map Projections (USGS Professional Paper 1453), which sets out each projection with its graticule and states plainly what it does and does not keep.
7. ^ Gauss, Disquisitiones generales circa superficies curvas (1827), the Theorema Egregium. Gauss called the result remarkable; cartographers, who had known it in practice for two hundred and fifty years, called it late.
8. ^ On the Ionian map-makers and the encircling Ocean. The tradition is reported by later writers; the tablet itself, like most of the first of anything, is gone.
9. ^ Hutton, "An Account of the Calculations Made from the Survey and Measures Taken at Schehallien" (1778), where the contour is introduced as a means of integration rather than as a way of drawing a hill. It has been used for the second purpose ever since.
10. ^ On the coastal method, see Joost van Radewijn and the Radian Dyke. The verification was single-use.
11. ^ Also called a fictitious entry, a paper town, or – on street maps – a trap street; in books, a mountweazel. The mechanism, and the one well-documented case of a trap getting away, are at Agloe.

External links

• Map projections – the US Geological Survey's general-interest account of what each projection preserves and what it costs
• General Investigations of Curved Surfaces of 1827 and 1825 – Gauss's paper in the Morehead and Hiltebeitel translation, containing the Theorema Egregium
• Gleason's New Standard Map of the World (1892) – a scan of the azimuthal equidistant map reproduced above
• Typus Orbis Terrarum (Ortelius, 1572) – the world sheet from Theatrum Orbis Terrarum reproduced above; engraved 1572, public domain
• An Album of Map Projections – Snyder and Voxland, USGS Professional Paper 1453; the source of the four diagrams above, and of the table's classifications
• David Rumsey Map Collection – a large searchable archive of historical maps and atlases