Janney: "What the hell is that??"
Fallow: "It's where you've been living this whole time"

— The West Wing, Season 2 Episode 16 [youtube]

Icosahedral World Map

This map is a representation of the Earth's surface that, when folded up, forms an icosahedron: a 20-sided 3D shape whose faces are all equal equilateral triangles. In addition, it minimizes the land areas split up when unfolded.

This project was inspired by the Dymaxion map [Wikipedia], created in 1954 by American architect Buckminster Fuller and Japanese cartographer Shoji Sadao. It was greatly aided by Unfolding the Earth: Myriahedral Projections, a paper by Professor van Wijk of the University of Eindhoven which presents the key algorithm used.

While I've tried my best to explain the ideas as clearly as I can and to produce the right visuals to accompany the explanations, I'm no professional when it comes to communicating complex mathematical ideas. So if there's anything you don't understand or that's not clear, that's on me.

Visualizing the Computer Program

I wrote a computer program using the method presented in Unfolding the World to find the optimal arrangement. Optimal in this context means as little land area split apart by the boundaries of the map as possible. The following videos demonstrate its operation.

First, the icosahedron is rotated about the Z axis and the best map is found for each angle.

Same for the X axis...

...and the Y axis.

Combinations of all 3 axes must also be considered.

After examining more than 46 million combinations (360 degrees around each axis), here are the results sorted from worst to best. The final frame is the configuration that splits up the least land areas (sadly, it was not possible to avoid it entirely) and is the configuration presented at the top of the page.

Symmetry

I was curious if the symmetry of the icosahedron could be used to reduce the search space. Fewer angles need to be searched if I could show that some of them are redundant.

For example, these rotations of 0°, 72°, and 144° about an axis through a vertex yield the same result ("same" meaning indistinguishable from the initial position). The initial positions of the vertices are marked by red points.

In fact there are 60 ways to rotate an icosahedron such that it remains unchanged — this is the icosahedral symmetry group.

That means for a given rotation we can get 59 others for free. Here's that idea represented as rotation vectors: vectors whose direction indicates the axis of rotation, and whose magnitude represents the angle of the rotation (0 would be no rotation, magnitude 180 would be a 180 degree rotation). The expanding sphere in the first plot shows the initial set of samples. In second plot those samples have the 60 members of the icosahedral symmetry group applied and plotted, so other regions of the space fill up at the same time. These other regions show where the equivalent states area. The last plot shows those samples converted back to Euler angle space. From here we can see that only a small volume needs to be explored to cover all the possibilities.

Going in reverse: the total state space can reduced by each member of the icosahedral symmetry group to find the minimum region.

Unfortunately I've reached the limit of my mathematical ability. I can't think of an equation that would precisely describe this reduced volume that would help reduce the number of combinations examined. Moreover, even if an exact definition of the reduced volume were possible it does not necessarily reduce the number of required samples, it may simply mean that the full state space is equivalent to sampling the reduced volume very densely, still with a similar number of samples. It seems that we cannot do any better than a brute-force search.

Motivation

The puzzling world of map projections has fascinated humans for at least 2,500 years. ... Although the subject is inherently mathematical and very complex in some instances, it also is highly visual and readily calls forth human reactions ranging from near disbelief to unquestioned acceptance as truth, from confusion to a clear understanding of information.

— An Album of Map Projections, Snyder 1994

Maps are a flat representation of the (supposedly) not-flat Earth's surface. As such they must make tradeoffs: something has to be squished, stretched, or cut apart in order to flatten the surface, much like peeling an orange. A map projection is a specific procedure to convert the Earth's surface into a flat map, usually defined by a set of mathematical equations.

Consider the Mercator projection. It is likely the most viewed map projection worldwide by virtue of being used by Google Maps [source].

Note how Greenland and Africa appear comparable in size. In reality, Africa is about 14x larger than Greenland by land area. Here's what they look like in Google Earth, where the Earth is viewed as a sphere instead of a flat surface.

This is a result of the mathematics behind the Mercator projection, and is a fairly well-known shortcoming of this map projection. Features toward the top and bottom of the map get really stretched.

A perhaps less well-known issue: it's not possible to see the North and South poles in the Mercator projection. The stretching that gets worse toward the north and south of the map increases to infinity at the poles, meaning that no matter how big one makes the map they will never be visible. For this reason, Google Maps limits the map to between 85N and 85S (give it a try: search for "90N 0W" or "90S 0W" (the North and South pole coordinates) in Google Maps, the pin will not be visible)

This is not to say that the Mercator projection is useless. It was initially meant for maritime navigation, as the map was designed with the property that travelling at a costant compass bearing would trace a straight line on the map [source]. Meaning that if you wanted to figure out in what direction to sail to get from point A to point B, you'd just plop down a ruler between the two points, and the angle would tell you the direction on the compass to follow. Not every map projection has this property (the Dymaxion map certainly doesn't) and it greatly simplifies crossing oceans, all at the relatively low cost of some areas near the poles where hardly anyone goes anyway.

We don't use paper maps to cross oceans anymore, but the Mercator projection is probably known to more people today (by visual recognition if not by name) than its inventor ever could have imagined and dictates so much of what we think a map "should" look like. North and south at the top and bottom, east-west lines are parallel, Greenland being the same size as Africa, Antarctica being an amorphous white mass.

None of this need be the case - you've already seen an example demonstrating an alternative at the top of this page - and yet the distortions inherent in the Mercator projection have become so transparent to us that they have become part of our reality, and we forget what made the map such a useful tool in the first place.

I think what makes the Dymaxion map so striking is that it shatters almost every assumption that Mercator has burned into our collective subconscious expectation of a world map: north is not at the top, south is not at the bottom, going right does not mean going east, Antarctica has a well-defined shape (and is quite small!), Greenland is tiny, and perhaps above all, it is not a rectangle!

The Dymaxion is not without its disadvantages: it's hard to tell which edges connect to which; it doesn't efficiently use rectangular displays; and though our expectation of cardinal directions in maps is arbitrary, the fact remains that it is a widely-understood convention, one that the Dymaxion discards completely. The very features that differentiate it from the Mercator projection are the features that make it hard to use, precisely because it is different from the norm. I'd hazard that to be the reason why it does not see any significant use today.

Still, I think there's value in seeing how things could be done differently, even if they may not necessarily be better than the status quo.

Technical Notes

The search algorithm was implemented in Python, as were all the visuals. Important packages include numpy and scipy for matrix operations and transformations, and pyvista for 3D visualization.

Datasets for the Earth's surface, in both raster and vector form, came from naturalearthdata.com.