Unexpectedly Intriguing!
05 March 2021

Here's a challenge. Find a three dimensional sphere whose surface iscovered by intricate patterns. Now, figure out how to project those patterns onto a two-dimensional sheet of paper, so that the resulting image remains as true as possible to the original patterns as they appear on the sphere.

Solving that exact geometrical problem has challenged mapmakers for centuries, because no matter how you project a 3D surface onto a 2D one, you'll end up distorting the true picture. In the following six minute video, Vox explains why all world maps get it wrong:

Mathematicians have an answer for minimizing that problem, which involves treating the surface of the globe you want to project onto a 2D surface like an orange peel. In the following Numberphile video, Hannah Fry demonstrates that concept, called the Euler Spiral, which takes 13 minutes to explain, and which requires scissors and a lot of patience to execute.

While the Euler Spiral represents what might be the optimal way to convert a three dimensional spherical surface into a two dimensional representation with minimal distortion, there is a new method for presenting a world map that trades its complexity for relative simplicity and a surprisingly small increase in distortion.

For centuries, mapmakers have agonized over how to accurately display our round planet on anything other than a globe.

Now, a fundamental re-imagining of how maps can work has resulted in the most accurate flat map ever made, from a trio of map experts: J. Richard Gott, an emeritus professor of astrophysics at Princeton and creator of a logarithmic map of the universe once described as "arguably the most mind-bending map to date"; Robert Vanderbei, a professor of operations research and financial engineering who created the "Purple America" map of election results; and David Goldberg, a professor of physics at Drexel University.

Their new map is two-sided and round, like a phonograph record or vinyl LP. Like many radical developments, it seems obvious in hindsight. Why not have a two-sided map that shows both sides of the globe? It breaks away from the limits of two dimensions without losing any of the logistical convenience—storage and manufacture—of a flat map.

Better still, here's a very short video that presents their mapmaking innovation:

The way they did it was to first score existing maps for six different types of distortions, with a score of 0.0 representing a perfect representation. They then applied their expertise in polyhedra geometry, the shapes of multi-sided solids, to work out how to project the world's 3D surface onto them to beat the best score of their cartographic competition. But then, their work took on a very different shape:

In a recent paper, Gott began considering "envelope polyhedra," with regular shapes glued together back-to-back, which led to the breakthrough idea for the double-sided map.

It can be displayed with the Eastern and Western Hemispheres on the two sides, or in Gott's preferred orientation, the Northern and Southern Hemispheres, which conveniently allows the equator to run around the edge. Either way, this is a map with no boundary cuts. To measure distances from one side to the other, you can use string or measuring tape reaching from one side of the disk to the other, he suggested.

"If you're an ant, you can crawl from one side of this 'phonograph record' to the other," Gott said. "We have continuity over the equator. African and South America are draped over the edge, like a sheet over a clothesline, but they're continuous."

This double-sided map has smaller distance errors than any single-sided flat map—the previous record-holder being a 2007 map by Gott with Charles Mugnolo, a 2005 Princeton alumnus. In fact, this map is remarkable in having an upper boundary on distance errors: It is impossible for distances to be off by more than ± 22.2%. By comparison, in the Mercator and Winkel Tripel projections, as well as others, distance errors become enormous approaching the poles and essentially infinite from the left to the right margins (which are far apart on the map but directly adjacent on the globe). In addition, areas at the edge are only 1.57 times larger than at the center.

That's a lot of problems solved to produce a reasonably simple 2D representation of the world that both minimizes distortion and remains useful as a map. Even better, their innovative concept may be patentable:

To the best of their knowledge, no one has ever made double-sided maps for accuracy like this before. A 1993 compendium of nearly 200 map projections dating back 2,000 years did not include any, nor did they find any similar patents.

"Our map is actually more like the globe than other flat maps," Gott said. "To see all of the globe, you have to rotate it; to see all of our new map, you simply have to flip it over."

At this writing, a U.S. patent application for the invention of the world disk map has yet to be filed. The Inventions in Everything team will periodically check to see if that status changes.

### References

David M. Goldberg et al. Flexion and Skewness in Map Projections of the Earth, Cartographica: The International Journal for Geographic Information and Geovisualization (2007). DOI: 10.3138/carto.42.4.297

Envelope Polyhedra: arxiv.org/abs/1908.05395

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