Why Greenland looks as big as Africa
Tonight's idea
Try to flatten an orange peel: it tears or stretches. You cannot lay a piece of a sphere flat without distorting it somewhere, so every flat map of the Earth lies about something. The only question is what.
In 1569 Gerardus Mercator chose to keep angles true: draw a straight line between two ports on his map, measure its compass bearing, sail that bearing, and you arrive. The price is area. To keep angles right, Mercator had to stretch east–west distances as you move away from the equator, and then stretch north–south by the same amount so shapes stay right.
How much stretch? A circle of latitude is shorter than the equator. Compared with the equator, it is this fraction as long (this fraction is the cosine of the latitude):
| Latitude | 0° | 30° | 45° | 60° | 70° | 80° |
|---|---|---|---|---|---|---|
| Fraction of equator | 1 | 0.866 | 0.707 | 0.5 | 0.342 | 0.174 |
The map draws every circle full width, so east–west is stretched by 1 ÷ fraction, north–south by the same, and area by 1 ÷ fraction². At 60° north the fraction is 0.5, so areas are shown four times too big. At 80°, about 33 times. At the pole, infinite, which is why Mercator maps simply stop before they get there.
The shortest route between two points on a sphere is a great circle, and on a Mercator map it looks like a curve bending toward the pole. That is why a flight from New York to Tokyo goes up over Alaska. The pilot is not lost; the map is.
Watch one: how big is the lie at 70° north?
At 70° the fraction is 0.342. Each direction is stretched by \(1 / 0.342 \approx 2.92\). Area is stretched by \(2.92 \times 2.92 \approx 8.5\). So a country at 70° north is drawn about eight and a half times too large compared with one at the equator.
Your turn
- Using the table, compute the Mercator area exaggeration (1 ÷ fraction²) at 30°, 45°, 60° and 70°. Round to one decimal place.
- Africa is 30.4 million km² and Greenland is 2.2 million km². (a) What is Africa's area divided by Greenland's? (b) Greenland's centre is near 70° north, so the map draws it about 8.5 times too big, while Africa, on the equator, is about true size. Divide your answer to (a) by 8.5. (c) In one sentence, what does that number say about how the two look on the map?
- Antarctica is a roundish continent about the size of the USA and Mexico together. On a Mercator map it appears as a long white strip across the entire bottom edge. Explain in two sentences why.
- London is at 51° north and Los Angeles at 34° north. On a globe, is the shortest path between them along a line of latitude, or does it bend north of both cities? Explain using what you know about great circles.
Stretch: how wide is one degree?
The Earth's circumference is about 40,000 km, so at the equator one degree of longitude is \(40{,}000 / 360 \approx 111\) km. At 60° north the circle of latitude is only 0.5 times as long (from the table). How wide is one degree of longitude there? At 89° north, where the fraction is 0.0175? At the pole? Then explain why equal one-degree steps in latitude and longitude make cells that get narrower toward the poles on the globe, while on a Mercator map the cells keep the same width and get taller.
Answers
- 30°: \(1/0.866^2 \approx 1.3\). 45°: \(1/0.707^2 \approx 2.0\). 60°: \(1/0.5^2 = 4.0\). 70°: \(1/0.342^2 \approx 8.5\). (Each one is 1 divided by the fraction squared.)
- (a) \(30.4 / 2.2 \approx 13.8\): Africa is about 14 Greenlands. (b) \(13.8 / 8.5 \approx 1.6\). (c) On the map Africa looks only about one and a half times Greenland, and much of Greenland is even further north than 70°, which closes the gap further. That is why they look similar.
- Antarctica surrounds the South Pole, where the Mercator stretch becomes infinite. The map cuts off before the pole, so the continent is sliced across and its remaining edge is stretched to the full width of the map.
- It bends north. The shortest path on a sphere is a great circle, and a line of latitude (other than the equator) is not a great circle. The great circle from London to Los Angeles passes over Greenland and northern Canada, further north than either city.
Stretch. At 60°: \(111 \times 0.5 \approx 56\) km. At 89°: \(111 \times 0.0175 \approx 1.9\) km. At the pole: 0 km, all the lines of longitude meet at a point. On our spherical Earth, successive one-degree lines of latitude are about 111 km apart along a meridian, but lines of longitude close in, so each cell gets narrower toward the pole. Mercator keeps longitude lines equally spaced and increases the spacing between latitude lines toward the poles. The one-degree cells on the map therefore get taller, not narrower.