Most common world maps are highly size distorted, with any land masses located toward the poles tending to look much bigger than they really are, which, in some ways, makes this weird-looking projection a lot more accurate. How it's made is a fascinating process – now, what is it actually useful for?
Creating 2-D maps of the entire Earth is a hard job. It's a sphere, and trying to nicely unwrap it into a neat rectangle leads to varying degrees of distortion. The map you're familiar with, that pretty much everyone has a copy of somewhere, is called the Mercator Projection, which was first presented by cartographer Gerardus Mercator in 1569.
It works well for nautical navigation, but it also massively distorts the size and shape of land masses, particularly near the poles. Take Canada as an example – it looks absolutely enormous, but if you remove the cylindrical distortion, you find it's not actually that much bigger than Australia (if you don't consider an extra 2.3 million km2 (888,000 mi2) much):
Another example: Greenland looks massive, but here it is in between Africa and South America with the size distortion removed. You can play around with this stuff yourself at TheTrueSize.
There are other projections of the globe that seek to represent the size of land masses more fairly; take the Gall-Peters projection, which tries to even things up by vertically compressing the areas near the poles:
Or the Mollweide projection, which throws out the idea of fitting everything onto a nice neat rectangle, and bends continents around in order to try to represent land mass as accurately as possible:
Or the Boggs eumorphic projection, which takes something like the Mollweide idea and slices it up to try to undo some of that bending and show both the correct shape and land mass size, sacrificing any usable navigation abilities in the ocean areas:
There are many more, and now another has joined the list. Winner of the Japan Institute of Design Promotion's Good Design Award for 2016, meet the Authagraph World Map:
Clearly, nobody is using this thing to plot a course. Lines of longitude and latitude bend, twist and curl, and figuring out which way's north is a challenge in some spots. But in terms of representing the size and shape of land masses, it's actually very accurate.
Created by Hajime Narukawa of the Keio University Graduate School of Media and Governance, the Authagraph map fits perfectly into a nice rectangular shape with a neat 3:4 aspect ratio thanks to its ingenious method of construction.
First, the sphere is divided into some 96 regions and then these regions are mapped onto an inflated tetrahedron shape. The inflated tetrahedron is flattened into a regular tetrahedron, then the tetrahedron is cut out and flattened into a big triangle and lined up with a whole bunch of other ones. From there, you can choose where to cut your map out from, depending on what you need to highlight and whether you're comfortable cutting land masses in half.
It's not officially yet an equal-area map in the eyes of the cartography community, which is rightly picky about such things, but with an increased number of subdivisions it could attain that rating.
Now, to figure out what it's useful for!