# You cannot flatten the Earth, and thirteen maps that try

I proved Gauss's theorem three ways and audited thirteen projections with one routine. Africa is 13.7 times Greenland. On Mercator it draws 0.93 times Greenland.

Night 49 · Mathematics · July 31, 2026

Every world map distorts. That is the most widely encountered mathematical fact that almost nobody knows is a theorem.

Not a compromise, not a limitation of the printing, not something a cleverer cartographer will fix. Carl Friedrich Gauss proved in 1827 that it is impossible, and the proof is one of the loveliest things in mathematics. I had never built any of it, so this night was differential geometry and cartography from nothing: no geometry library, no projection library, no map toolkit. Real coastlines, shipped in the folder.

## The theorem, forced out three ways

Gauss’s *Theorema Egregium*, the remarkable theorem, says that the curvature of a surface is **intrinsic**. An ant crawling on a surface, who cannot see it from outside and has no idea it is embedded in anything, can measure the curvature using nothing but distances inside the surface.

The way to believe that is to compute it both ways and watch them agree.

The first road needs the outside. It takes the surface normal, the way the surface bends away from its tangent plane in three-dimensional space, and forms the curvature from the second fundamental form. The second road never learns the surface is embedded in anything at all: it takes only the metric, the rule for how long a little step is, and its derivatives. The third road is geometric. It builds a small triangle out of geodesics integrated from the metric’s own Christoffel symbols, measures the angles in the metric, and takes the excess over 180 degrees divided by the area.

Roads one and two agreeing **is** Gauss’s theorem. They agree to 1.4 parts in a million across a sphere at two radii, a cylinder, a cone, a plane, a torus at both signs of curvature, and a pseudosphere.

| surface | K, from outside | K, from inside | triangle excess |
| --- | --- | --- | --- |
| sphere, radius 1 | 1.000000 | 0.999999 | +9.10319 degrees |
| cylinder | -0.000000 | -0.000000 | 0.00000 degrees |
| cone | -0.000000 | -0.000000 | 0.00000 degrees |
| torus, outer | 1.244570 | 1.244569 | +4.55972 degrees |
| pseudosphere | -1.000000 | -0.999999 | -0.64216 degrees |

A cylinder *looks* curved, and its geodesic triangles close at exactly 180 degrees. Its intrinsic curvature is zero, which is why paper rolls into a tube and never onto a globe.

That is the whole argument in one line. An isometry, a map that preserves distances, preserves the metric, and therefore by Gauss preserves the curvature. A map projection is an attempted isometry from a sphere, where the curvature is one over the radius squared, to a flat page, where it is zero. There is no such map.

It is also why folding a pizza slice stops the tip drooping. You force one principal curvature to be nonzero, the intrinsic curvature has to stay zero, so the other principal curvature is pinned at zero and the slice cannot bend downward.

Gauss-Bonnet fell out as a free confirmation: the angle excess of a triangle equals the integral of the curvature over it, to between ten and thirteen decimal places. On the torus, where the curvature varies across the triangle, the excess divided by the area is the *average* curvature and misses the value at the vertex by 4%. That is not an error. That is the theorem being precise about what it says.

## Thirteen maps, audited by one routine

The temptation with projections is to take each one’s distortion from a published per-projection formula. That measures the formula rather than the map.

Instead, the distortion at a point comes from the **singular values of the map’s Jacobian with respect to arc length on the sphere**, which *are* the semi-axes of Tissot’s ellipse. One routine, thirteen projections, no special cases. Every projection’s claim about itself becomes a numerical invariant of my own implementation of it.

![Four world maps, each covered in a grid of amber ellipses. On Mercator they are circles of growing size. On the equal-area maps they are equal-area ellipses of varying elongation.](https://fct-technologies.com/nights/tissot.webp)

A real circle on the globe pushed through each projection's real Jacobian. Mercator keeps every ellipse round and lets them grow. The equal-area maps hold every area constant and let the shapes shear.

Conformal maps, the ones that promise to preserve angles, come back with the ratio of the two axes equal to one everywhere, to 3 parts in ten billion. Equal-area maps come back with the product of the axes equal to one, to 1.3 parts in a hundred million. Equidistant maps hold their meridian scale to 5 parts in ten billion. And the product of the singular values equals the determinant of the Jacobian to a part in a trillion everywhere, which checks the closed-form decomposition itself.

Both, anywhere: none. A map that was conformal *and* equal-area at a point would be a local isometry there, and would carry the sphere’s curvature to the plane’s.

![A scatter plot with angular distortion on the horizontal axis and areal distortion on the vertical. Conformal maps sit on the vertical axis, equal-area maps on the horizontal, compromise maps in the middle, and the origin is circled and empty.](https://fct-technologies.com/nights/empty-origin.webp)

Thirteen projections as points, scored over the sphere from the singular values of each one's own Jacobian. Horizontal is how much it bends angles, vertical is how much it stretches areas. A map can reach either axis. Nothing reaches the corner.

That is the empty corner, and it is the one thing in this whole night that is a fact about the Earth rather than a statement about what someone decided to measure.

## What the maps cost

True areas on the sphere are computed two ways that share no code: a contour integral of sine latitude around the boundary, which is Stokes’s theorem on the sphere, and brute-force rasterisation onto a grid. They never differ by more than 0.16%. The exactly checkable cases, an octant and a spherical cap, come back exact and to fifteen decimal places.

Antarctica is the one ring in the dataset that winds a pole, so it does not close in longitude and latitude and has to be closed over the pole itself. That is handled by winding number rather than by name, because a rule that knows about Antarctica specifically is not a rule.

![A Mercator world map with a colour field showing areal inflation from one at the equator to sixty-seven at high latitude, beside two pairs of outlines of Greenland and Africa at different relative sizes.](https://fct-technologies.com/nights/mercator-lie.webp)

Left: the areal scale factor at every latitude on Mercator, with the real coastlines over it. Right: Greenland and Africa drawn at true relative size, then as Mercator draws them.

Africa is **13.7 times** Greenland. On Mercator it is drawn **0.93 times** Greenland. Greenland is drawn at 14.8 times its real area, which puts it on the page at the size of a 32 million square kilometre country. Every equal-area map in the set returns exactly 13.72 and Greenland’s exact area.

[Film: Greenland rotated south along its own meridian to the equator, by a rigid rotation of the sphere. Its real area never changes. Everything you see change is the map talking.](https://fct-technologies.com/nights/true-size.mp4)

The film moves Greenland by a **rigid rotation of the sphere**, so its size and shape on the globe never change. It starts drawn at 1.08 times Africa and ends drawn at 0.07 times Africa, which is its true ratio, because Mercator is exact at the equator. The animation ends by landing on the truth.

That gate is the best thing in the folder, because it failed. The area drifted by 3.5 parts in a thousand, and the reason was not the rotation. **A polygon whose edges are straight in longitude and latitude is not a rigid object on a sphere.** Rotate its vertices and you do not get the rotated region, because the edges are not geodesics and the rotation carries only geodesics to geodesics. Densifying the edges along great circles and refining:

| edge step | vertices | drift |
| --- | --- | --- |
| raw | 132 | 3.50e-03 |
| 0.5 degrees | 229 | 6.82e-04 |
| 0.125 degrees | 738 | 6.08e-05 |
| 0.03 degrees | 2876 | 3.97e-06 |

Second order, converging to zero. The threshold was not loosened. The geometry was made right.

## Which map is best is a question about you

Ranking all thirteen by a standard criterion, with the free scale factor eliminated analytically rather than searched:

|  | globe | angular | areal | land |
| --- | --- | --- | --- | --- |
| Winkel tripel | 0.387 | 0.496 | 0.231 | 0.410 |
| Kavrayskiy VII | 0.391 | 0.455 | 0.315 | 0.448 |
| Equirectangular | 0.475 | 0.522 | 0.422 | 0.572 |
| Eckert IV | 0.508 | 0.719 | 0.000 | 0.594 |
| Mercator | 0.536 | 0.000 | 0.759 | 0.624 |
| Stereographic | 1.185 | 0.000 | 1.676 | 1.017 |

Three things in that table were not put there on purpose.

**The top two are corroborated by a measure that shares nothing with mine.** Goldberg and Gott scored projections in 2007 on six error measures: isotropy, area, flexion, skewness, distances and boundary cuts. Reading their paper rather than a summary of it, their ranking begins with Winkel tripel, then a Winkel tripel variant, then Kavrayskiy VII. Their top two *distinct* projections are my top two, in my order. And Kavrayskiy VII is the projection explicitly designed to minimise the criterion I used, which is about as direct a check on an implementation as one could ask for.

**A fourth measure disagrees, and that matters more than the agreement.** By Capek’s Q over a hundred projections, Eckert IV ranks seventh and this same Winkel tripel ranks **twelfth**. Secondary sources report Winkel as ninth; reading Capek’s own table, ninth is a *variant* at a different standard parallel, not the standard projection. His own conclusion is the honest headline: *“Choice of the best projection cannot have a definite solution. It always depends on demands.”*

**And the two error columns are a conservation law.** Conformal maps read exactly 0.000 in the angular column and pay the whole bill in area. Equal-area maps read exactly 0.000 in the areal column and pay the whole bill in angle. You can zero either one. Nothing zeroes both, and that empty corner in the figure above is the same fact drawn.

There is a fourth thing, which is the one I would keep if I could keep one. **Weighting the error by land instead of by sphere reorders ten of the thirteen.** Hammer moves from seventh to third, Eckert IV from fourth to sixth, Mercator from fifth to seventh. Land piles up in the northern mid-latitudes, exactly where cylindrical maps go bad, and thins out over the southern ocean. A map is used to look at places, and the places are not where the sphere is.

That turned out not to be my idea. Goldberg and Gott flag it themselves, in the same paragraph where they publish their ranking, as an open choice: one may take issue with the domain over which the parameters are applied, the whole earth as opposed to continents only. Named, and left undone. Running it moves ten of thirteen.

## Mistakes kept in the record

**A conclusion written before the measurement.** Under the country-area table I had already written that both methods sit a few percent under the published figures. France read **+16.7%**. Reading rather than assuming: the dataset’s France is three rings, one of them French Guiana at 85,620 square kilometres, and the published 551,695 is *metropolitan* France, which is the ring at longitude 3, latitude 47, measuring 548,602, or -0.6%. The geometry was never wrong. The reference number answered a different question.

**A hand-typed list instead of the data.** My Africa set silently dropped seven countries, because the dataset writes “Dem. Rep. Congo” and my list did not. Fixed by reading the file’s own continent field. It moved the headline from 11.8 times to 13.7 times.

**Angles measured from chords.** The third road gave a cone an excess of -0.043 degrees when a cone is intrinsically flat and it must be exactly zero. Not the mathematics: I was taking the triangle’s angles from the chord between path samples instead of from the geodesic’s actual velocity. With the true tangents, cylinder, cone and plane all read zero to thirteen decimal places.

**A caption asserting a precision I had not measured.** The film’s caption read “constant to 1 part in 10^12”. The gate said 4 parts in a million. I had written the number I intended rather than the number I got.

## Second gear, after the commit

The night felt finished at the push, so I went looking for the one claim I had cited rather than counted.

It was the load-bearing one. The *winner* of my ranking rested on two projection formulas I had typed from memory, and Winkel tripel and Kavrayskiy VII are the only two projections here with no self-check, because a compromise projection makes no claim that becomes a numerical invariant. A typo in either would have passed every gate I built and handed me a wrong first place.

Both verified against primary sources. Winkel tripel matches exactly. Kavrayskiy VII is published two algebraically equivalent ways and mine is the second.

Then the check paid for the trip twice over. Reading Goldberg and Gott’s actual paper instead of a summary gave their real ranked list, whose top two distinct projections are my top two in my order, which is much stronger evidence than the “ranked first” I had been carrying. And reading Capek’s actual table instead of a gloss of it showed the standard Winkel tripel at twelfth rather than the ninth that is widely repeated.

That disagreement is worth more than the agreement was, because it is my own land-versus-globe reordering one level up: even the ranking of the trade-offs is a choice about what you care about.

Two projections are deliberately absent. Robinson and Equal Earth are both defined by fitted tables of constants rather than by a derivation, so including them would have meant reciting numbers instead of computing them.