Every Map Is a Lie With Error Bars
Aug 11, 2022 · 6 MIN READ
See also, the right brain on this:
Drawing the Coast Until It Feels Right
You cannot flatten a sphere. Every projection chooses what to ruin. Better cartographers won't one day overcome this. Gauss proved it in 1827, and since then the only thing a mapmaker gets to decide is which lie to tell.
Gauss called it the Theorema Egregium, the remarkable theorem, which is a strong name for a man who did not hand those out. The content is that Gaussian curvature is intrinsic. You can measure it from inside a surface, by walking around on it with a tape measure, without ever looking at it from outside. A sphere has positive curvature everywhere. A plane has zero. Bending a surface without stretching it can't change its curvature.
So there's no way to lay a sphere flat without stretching, and stretching means distances change, and if distances change, then at least one of shape, area, or direction is wrong somewhere. You get to preserve one property, locally, at the cost of the others. Peel an orange and try to press the peel flat. It tears or it wrinkles. That's the theorem.
The two most useful properties are shape and area, and you can't have both.
Mercator, 1569, is conformal. Angles are preserved locally, which means shapes of small regions are right, and a line of constant compass bearing is a straight line on the map. That was the whole point: you could rule a line from Lisbon to Brazil and read your heading off it. Sailors loved it. The price is that scale grows as you leave the equator, and area grows as the square of scale, so Greenland, which is about the size of Algeria, looks the size of Africa. Antarctica is infinite. It can't be drawn. The projection puts the pole at infinity, and every Mercator map just cuts it off somewhere and hopes you won't ask.
Equal-area projections, Gall-Peters being the famous one, fix the areas at the cost of the shapes. Africa is the right size and looks like it has been hung on a line to dry. There is an episode of The West Wing where a group of cartographers presents this to the White House staff as a moral issue, and it is a good scene, and it is also, from a projection standpoint, a false choice. Peters is not more honest than Mercator. It's differently dishonest. Shape is also information.
There's a way to see exactly what a projection lies about, and I wish it were printed on every map. It is called Tissot's indicatrix. You draw a small circle at many points on the globe and then project each one. On a perfect map they would all come out as identical circles. On a real map they come out as ellipses of varying size and stretch, and the ellipse at each point is the local distortion: its area is the area error, its eccentricity is the shape error, its rotation is the angle error.
Put Tissot's circles on Mercator and you get tiny circles at the equator swelling to enormous ones near the poles, but all still round. Put them on Peters and they stay the same area but squash into tall thin ovals at the equator and wide flat ones at high latitudes. Put them on any compromise projection, Robinson, Winkel Tripel, and you get a mix of both errors, medium-sized everywhere.
Those are the error bars. A map without them is a measurement without uncertainty. It is still useful. It is just not telling you how much to trust it.
If Mercator is so distorted, why is it on your phone?
The reason is tiles. When Google Maps launched in 2005 it needed a projection where the world could be cut into square tiles at every zoom level, where zooming in anywhere gave you the same square-ish pieces, and where north was always straight up. Mercator does all of that. Its conformality means a small square on the ground is a small square on the tile, at any latitude, so street layouts look right when you're zoomed into a city, which is what people mostly do with a phone map. Nobody navigates Greenland on a phone. Everybody navigates their neighborhood, and at neighborhood scale Mercator is nearly perfect.
They also simplified the math by pretending the earth is a sphere rather than the slightly squashed ellipsoid it actually is, which introduces its own error of up to a few hundred meters at high latitudes. This is "Web Mercator," EPSG:3857, and it is technically not even a proper conformal projection anymore. It's a projection everyone uses because everyone uses it.
There is a second layer of lie underneath the projection, and it's stranger. Before you can project a point, you need coordinates for it, and coordinates require a datum: a model of the earth's shape and a decision about where zero is.
GPS uses WGS84. North American surveying has traditionally used NAD83. They were almost identical when NAD83 was defined in the 1980s. But WGS84 is fixed to the earth as a whole, and NAD83 is fixed to the North American plate, and the plate is moving, about 2 centimeters a year. The two datums have now drifted more than a meter apart. The same latitude and longitude, in the two systems, are different spots on the ground.
Australia has it worse. The Australian plate moves about 7 cm a year northeast, and their datum was pinned in 1994. By 2016 GPS receivers were putting things 1.8 meters from where the maps said they were. Not a rounding error. A lane of traffic. They redefined the datum, moved the whole continent on paper, and will have to do it again.
Then there's the geoid, which is the shape the ocean would take if it covered the whole earth and only gravity acted on it. It is lumpy. Gravity varies with the density of what is under you, so "sea level" is up to 100 meters higher in some places than the ellipsoid predicts. Elevation on a map is height above this lumpy surface, not above the smooth one. Two GPS units, one measuring against the ellipsoid and one against the geoid, can disagree about the height of the same rock by a hundred meters and both be right.
My favorite category of map error is the deliberate one. Mapmakers have long inserted fake streets, fictitious towns, or slightly bent rivers into their work so that if a competitor copies it, the fake shows up in the copy and proves the theft. These are trap streets. Agloe, New York, was a fictional town on a 1930s road map that later became real when someone built a general store at the location and named it after the map. The map created the territory.
I like this because it's the honest version of what every map does. Every map is an argument about what matters, at a resolution someone chose, with distortions someone accepted, on a coordinate system that's drifting under it. The trap street just admits it.
So the practical question is never "is this map accurate." It's "what did this map choose to ruin, and does that matter for what I am doing." Mercator is the right choice for a compass bearing and the wrong one for comparing the size of continents. Equal-area is right for a population density map and wrong for street directions. The projection is a loss function. Pick the one that puts the loss where you aren't looking.
And if someone shows you a map without telling you the projection, treat it like a chart without axis labels. It might be fine. You have no way to know.