Every world map is lying. You get to pick the lie.

A flat map can keep every country's size honest, or keep every country's shape honest. It cannot do both — not by being cleverer, not ever. One dial slides between the two, and you can watch the error you just fixed reappear somewhere else.

What you're seeing

The globe has been chopped into patches that all cover exactly the same true area — the bands are spaced so that each one contains the same slice of the sphere's surface, which is why they look uneven in latitude but identical in size on an honest map. So every difference you see between two patches on screen is not geography. It is the map lying to you. Two of them are outlined: one sitting on the equator, one up near the pole. Watch the ratio between them as you drag.

The little circles are the other half of the story. Imagine drawing a tiny circle on the actual globe, then seeing what the map does to it. If the map keeps angles honest, a circle stays a circle — it may grow, but it never squashes. If the map keeps areas honest, every circle keeps the same amount of ink but gets flattened into an ever-thinner ellipse as you go north, and a coastline meeting a river at a right angle no longer looks like a right angle. Cartographers call these Tissot's indicatrices.

You may have noticed there are no coastlines. That is deliberate. Drawing a Greenland here would mean drawing a Greenland I hadn't verified, on a page whose whole argument is that maps mislead — and it would invite the wrong reaction ("Greenland looks odd") instead of the right one ("that patch is the same size as the one at the equator, and it is four times bigger on screen"). The patches are honest by construction. Nothing here needs you to take a coastline's word for it.

Drag the top dial and the two errors trade places, perfectly and unavoidably. At the far left every patch is the right size and the circles are crushed. At the far right — Mercator, the projection behind almost every map app you have ever used — every circle is a perfect circle at every latitude, which is exactly why it is so good for navigation: a compass bearing is a straight line on it. The price is on the other chart. Greenland and Africa are the famous casualty. Africa is genuinely about 14 times Greenland's area. On Mercator, Greenland's latitudes are inflated about 12.2× and Africa's only about 1.15×, so the pair is drawn at roughly 1.3 to 1 — near enough that generations of schoolchildren have grown up believing they are the same size.

The dial between the ends does not rescue you; it just splits the damage. The honest reason is older than cartography: a sphere is curved and a sheet of paper is not, and curvature is a property a surface keeps no matter how you bend it without stretching. That is Gauss's Theorema Egregium — his "remarkable theorem" — and it is why an orange peel will not lie flat without tearing, and why every map on your wall has already chosen which truth to give up.

The rule, exactly. A cylindrical projection of the unit sphere is x = longitude, y = f(latitude) with two local scale factors: h = f′(φ) along a meridian, and k = sec φ along a parallel — the second forced by the cylinder itself. Keeping area means h·k = 1, so f′(φ) = cos φ and y = sin φ (Lambert). Keeping angles means h = k, so f′(φ) = sec φ and y = ln(sec φ + tan φ) (Mercator). Both at once demands cos φ = sec φ, i.e. cos²φ = 1 — true only at the equator. The dial sweeps f′(φ) = cos^1−2tφ, passing through the plain equirectangular map at t = ½. Verified in node (improve/verify/78-map-projection.js, 48 checks): the family reproduces all three named projections to 1 part in 10⁹; the area factor at t=0 is 1 to within 10⁻¹² at every latitude and the angular error at t=1 is 0 to within 10⁻⁹ at every latitude, while each end fails the other test (Mercator's area factor is exactly sec²φ — 4× at 60°N); a 20,001-point sweep of the whole family finds the best possible compromise at 60°N still draws patches 1.53× too big and bends angles by 53°, so no member escapes both. Two negative controls: a zero-curvature cylinder unrolls with area factor exactly 1 and zero angular error everywhere — proving the culprit is the sphere's curvature, not cylindrical projections as such; and the Greenland–Africa numbers are computed from the projection integrals rather than quoted, with the equal-area member reproducing the true band ratio to 1 part in 10⁶ while Mercator distorts it by more than 8×.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 78.