As is well known, you can’t squash the surface of a sphere (or of a hemisphere) flat without either tearing it or (if it’s stretchy enough) stretching it. The mathematics of this comes from Gauss’s Theorema Egregium (his “remarkable theorem”) which says, in effect, that you can only map surfaces to each other without distortion if they have the same curvature. The Gaussian curvature of a plane and a cylinder are both zero, so you can map a cylinder onto a plane (imagine unrolling the surface of a cylinder, like a paper towel from its roll). But the surface of a sphere of radius has positive curvature so it’s not possible to map a sphere to a plane without distortions. See the nice discussion at nrich.maths.org for more information and details.
One of the outcomes of this theorem is that any attempt to map the earth’s surface onto a plane – to create a map projection – will require some compromise. You have to give up any of shape, size, angles. This hasn’t stopped cartographers trying for several thousand years to find the best compromise, and there are now many many different projections.
Note, for simplicity I will speak of the earth as a sphere, even though it’s not: it’s flattened slightly north-south, and bulges slightly at the equator, making it an oblate spheroid. But it’s pretty close to a sphere: its flattening, the value of with and being the minor and major axes of an ellipse from a cross-section through the poles, is only about .
Standard projections, seen in atlases, on classroom walls, and on the web, include the ancient equirectangular projection, where the earth’s surface is flattened onto a cylinder, thus vastly distorting the regions away from the equator, and the Mercator projection, where as well as flattening onto a cylinder, it is also stretched upwards – so preserving angles. This makes the Mercator projection excellent for navigation, which is what it was designed for. Mercator himself realized that there were unavoidable distortions of shape and size, and that his map was not useful for depicting landmasses. By a curious quirk of fate, this is what it is now mostly used for!
Efforts to decrease the distortions near the poles include the Robinson and Winkel tripel projections (which look superficially similar) – the latter is now the projection of choice by the National Geographic Society.
All of those projections, and many others, project the earth’s surface onto a single unbroken map. Other projections split the map to reduce distortions. One of these is the Goode Homolosine projection, which has been described as “abominable”, and a “travesty”: not only are land masses distorted, but lines of longitude bend all over the place.
Here’s a picture taken from geoawesomeness.com showing some of these standard projections:
Although you can’t map the entire sphere onto a plane, you can map small bits of it with manageable distortions. So one approach to mapping the earth was to project the sphere onto a polyhedron, and then flatten the polyhedron. Buckminster Fuller had a go at this with his Dymaxion world map, using an icosahedron. The result is certainly excellent for reducing distortion, but has an ugly, jagged look to it:
Also, many of the landmasses are (unavoidably) in curious places and at odd angles: Australia and South America are at opposite ends of the map, as the oceans have been chopped into bits to preserve the landmasses. I don’t believe this map has never got much love.
Although the icosahedron would seem to be the best choice of polyhedron because of its large number of faces (and no doubt this was Buckminster Fuller’s reasoning), the best results seems to have been obtained using an octahedron. Here is Waterman’s “butterfly projection” first developed in 1996, which is in many ways a magnificent example of good cartography:
The green circles here are Tissot indicatricies: they show the local distortions by means of small circles. As you can see, the distortions are very small indeed. And of course you can break the world up into an octahedron in such a way as to minimize distortions over particular regions. You can see more projections at the map’s own page.
Most Waterman maps show Antarctica as a separate entity, and another approach was provided many years earlier in 1909 by Bernard Cahill; Cahill’s map has been redeveloped, starting in 1975, by Gene Keyes, and Keye’s own website is a treasure house of cartographic information, as well as stern critiques of many standard projections. (This is where you’ll find Goode’s homolosine projection soundly trashed.) Keye’s version of Cahill’s map: the Cahill-Keyes projection, is as of now the best projection available:
Keyes has discussed Waterman’s map against his own, and provides various reasons why he believes the Cahill-Keyes projection is the better of the two. Remarkably, the landmasses are placed in positions and angles not vastly different from standard (Mercator, Robinson) projections we are all used to, so it doesn’t appear too strange. As with all maps, there are compromises: I’d love it if Australia and New Zealand were next to each other rather than at opposite edges. But that’s just they way the octahedron has been placed.
I believe that properly constructed polyhedral projections are the way to go, and of the several in existence, the Cahill-Keyes is the best. Even if you don’t care about the mathematics and the cartography, it simply looks terrific.
Gene Keyes very kindly responded to this post, and recommended – in view of my comments about the placements of Australia and New Zealand – that I check out another projection against a “starry night” background, and available at http://www.genekeyes.com/DW-STARRY/C-K-DW-starry.html. However, the map’s designer – Duncan Webb – was dissatisfied with this map and asked that it not be published. So I won’t include the picture here, but invite you to view it on Gene Keye’s site. But notice Australia and New Zealand in chummy proximity!