A Sphere Will Not Lie Flat

The Peeled Skin

Globe makers print the map in tapering strips and paste them onto a ball. The strips are called gores.

On the model the skin is cut along the meridians into 12 gores, each 30° wide, and laid on the board. Each gore meets its neighbours on the equator and narrows towards the poles, leaving gaps between them.

To close the gaps the paper would have to be stretched sideways, more and more towards the poles. Thinner gores are each closer to flat, but the gaps never go away.

Gauss’s Remarkable Theorem

On the sphere, walk from the North Pole down the 0° meridian to the equator, turn left by 90°, walk along the equator to 90° E, turn left by 90° again and walk back to the pole, where the angle is 90° too. The triangle’s angles add up to 270°; on a plane they always make 180°.

The extra 90° is exactly the triangle’s area divided by the square of the radius (Girard’s theorem). Circles agree: on the sphere a circle of radius r has circumference 2πR·sin(r/R), shorter than the flat 2πr.

Gauss proved in 1827 that a surface’s curvature is fixed by distances measured within the surface, without looking at how it bends in space. This is the Theorema Egregium, the “remarkable theorem”. The sphere’s curvature is 1/R² everywhere; a plane’s is 0.

So a map that kept every distance would have to keep the curvature too, which cannot be done. Euler had already shown in 1777 that a sphere cannot be drawn on a plane exactly.

Surfaces That Do Unroll

A cylinder or a cone can be cut along a line and unrolled flat with no length on it changing. They look curved, but their curvature is 0: through every point there is one direction that is straight.

So many projections take two steps: put the sphere’s points onto a cylinder, a cone or a tangent plane, then unroll it. All the distortion happens in the first step; unrolling stretches nothing.

On the model the skin is laid in turn on a cylinder, a cone and a plane touching the globe.

Mercator Keeps the Angles

In 1569 Gerardus Mercator published a world map for navigators. The meridians are evenly spaced vertical lines, the parallels horizontal, and the gaps between the parallels grow towards the poles.

At latitude φ the east–west direction is enlarged by 1/cos φ, that is sec φ. Mercator stretched north–south by the same factor, so near every point all directions are enlarged alike and small shapes and angles are kept. Such a projection is called conformal.

The price is area. With a scale of sec φ, area grows by sec² φ: 4 times at 60°, 33 times at 80°. The poles lie at infinity, so the map stops at 84°.

The red enamel discs are all the same size on the sphere, small circles of radius 6.5°. On Mercator they stay circles and grow towards the north.

Why Sailors Wanted It

A ship that holds one compass course sails a rhumb line. It crosses every meridian at the same angle, and on the globe it is a spiral winding towards the pole.

Mercator’s map is conformal and its meridians are parallel, so a rhumb line on it is straight. A ruler between start and finish, and the angle it makes with the vertical, is the compass course to steer.

From New York to London the rhumb line holds a course of about 78° east of north and is about 5,790 km long. The shortest way is the great circle, about 5,570 km, which starts out at about 51° and keeps changing course. On Mercator it bends north.

How fast the gaps between the parallels must grow to keep angles was worked out by Edward Wright in 1599, by adding up sec φ step by step; it is ln tan(45° + φ/2).

Greenland and Africa

Greenland covers about 2.17 million km², Africa about 30.37 million km²: Africa is 14 times larger. On Mercator they look about the same size.

The model moves Greenland’s outline over the sphere to the equator. On the sphere the outline does not change, only its place; at the equator, Mercator shows it at its true size.

This is Mercator’s choice, not his mistake: a conformal map cannot keep area as well. A sailing chart needs courses, not areas.

Keeping Area

Archimedes found that the area of the sphere between two parallels equals the area between the same heights on the cylinder round it. Push each point of the sphere straight out to the cylinder and every area is kept. Unrolled, that is Lambert’s cylindrical equal-area projection (1772).

Area kept, shape must give: near the equator shapes are stretched upright, at high latitudes squashed flat. The discs turn into ellipses, each with exactly the area of the circle it came from.

Mollweide’s projection (1805) fits the whole sphere in a 2 : 1 ellipse, and Equal Earth (2018) gives the edge more of a globe’s outline. Both keep area too; they only put the distortion in different places.

A Lamp at the Centre

Light a lamp at the centre of a glass globe and the land throws its shadow on a glass screen touching the globe. That is the gnomonic projection, one of the oldest.

It has one property no other map has: every great circle becomes a straight line. A great circle’s plane passes through the centre, through the lamp; it meets the screen in a straight line.

Great circles are the shortest ways on a sphere. A ruler between two cities on a gnomonic map gives the shortest route, which navigators then copied piece by piece onto Mercator to find their courses.

The price: the scale grows fast away from the point of contact, and 90° away lies at infinity, so a gnomonic map cannot hold even half the Earth.

Distances from the Centre

The azimuthal equidistant projection keeps two things: directions from the centre, and distances to it. A straight line from the centre to any point on the map has the true length on the sphere.

Centred on the North Pole, the parallels are evenly spaced circles. The map on the United Nations emblem is this projection, centred on the North Pole and drawn to 60° S.

The point opposite the centre is stretched round the whole outer circle. The rings on the map are 2,000 km apart.

Cut Into Twenty Faces

Buckminster Fuller put the map on an icosahedron: the sphere is projected onto twenty flat triangles, which are cut along their edges and laid flat. His first version, in 1943, used a cuboctahedron; the icosahedral one followed in 1954.

Each face holds only a twentieth of the Earth, so the distortion within any face is small. The cuts all run through the sea; the land holds together as one piece, with no “up”.

The model’s net is chosen from the coastlines: it keeps the edges that cross the most land as hinges and cuts the rest. No edge unfolding of the icosahedron overlaps itself.

An Honest Ruler

In 1859 Nicolas Tissot gave a way to measure distortion: draw a tiny circle on the sphere and see what the map makes of it. Every projection turns it into an ellipse (or keeps it a circle).

The ellipse’s half-axes a and b tell everything: a = b means angles are kept (conformal); a·b = 1 means areas are kept (equal-area); a length of 1 in some direction means distances are kept that way.

No projection makes every ellipse the same circle; that is Gauss’s theorem again. The readouts are the largest area factor and the largest angle distortion of the map between 60° N and 60° S.