Topology tells you a sphere can't be perfectly flatted - and that's a problem with 2d world maps. Now Richard Gott, Dave Goldberg and Robert Vanderbei have used a new mathematical technique that turns the sphere into a two sided flat map with adjustments made to the features on both sides. Here's a descrption.

a tip of the hat to Greg

And the paper is on the arXiv

**Flat Maps that improve on the Winkel Tripel**

J. Richard Gott III1, David M. Goldberg2, and Robert J. Vanderbei3 1. Department of Astrophysical Sciences, Princeton University, Princeton, NJ, USA

2. Department of Physics, Drexel University, Philadelphia, PA, USA

3. Department of Operations Research, Princeton University, Princeton, NJ, USA

**ABSTRACT**

Goldberg & Gott (2008) developed six error measures to rate flat map projections on their verisimilitude to the sphere: Isotropy, Area, Flexion, Skewness, Distances, and Boundary Cuts. The first two depend on the metric of the projection, the next two on its first derivatives. By these criteria, the Winkel Tripel (used by National Geographic for world maps) was the best scoring of all the known projections with a sum of squares of the six errors of 4.563, normalized relative to the Equirectangular in each error term. We present here a useful Gott-Wagner variant with a slightly better error score of only 4.497. We also present a radically new class of flat double-sided maps (like phonograph records) which have correct topology and vastly improved error scores: 0.881 for the azimuthal equidistant version. We believe it is the most accurate flat map of Earth yet. We also show maps of other solar system objects and sky maps.