world maps with constant-scale natural boundaries

It takes a book to explain this project. I’ll post an excerpt later.

I’ll also post published abstracts, conference posters and other “csnb” (constant-scale natural boundary world) maps.

For now, enjoy the first world map with constant-scale natural boundary, courtesy of the provocation of jim hagan, who revived my dormant interest in making world maps by pointing me at the great pyramid of Egypt. More later on that, too.

If you center up the south pole in a world map — or if you hold a globe at the same view — the continents of the southern hemisphere will snap to an orthogonal grid (with the Pacific Ocean holding down the fourth quadrant), and the northern hemisphere continents will snap to a similar orthogonal grid a quarter-turn (out of phase) to the first.

In other words, the disposition of continents fills up the sides of a square in one hemisphere, and the corners of the square in the opposite hemisphere.

In the century before last, Charles Sanders Peirce exploited this disposition of land to make his quincuncial (five-square) map of the world. You can find it — like I did after I made this map in 1990-93 — in Snyder and Voxland’s Album of Map Projections, and here: Peirce quincuncial projection (these map examples are centered on the north pole; for the opposite point of view, centered on the south pole, cut the quincuncial into four squares and reverse the outside points). If I had found Peirce’s map earlier, we’d probably never have world maps with constant-scale natural boundaries, because Peirce’s map shows all I was going for with Map #1: answering jim’s question about what could possibly tie the great pyramid to an Antarctic-centered view of the world.

Answer: orthogonal symmetry, a quirky kind of orthogonal symmetry, one with a clipped or dogeared corner, a kite with a tail, but here let me highlight the serendipity of discovery. Shy Peirce’s map, I used techniques of perspective and anamorphic sculpture to make a world map that stretches not at the edge, like conventional maps do, but in the middle. “Constant-scale” and “natural boundary” begets maps with unusual characteristics and novel properties.

Turns out this is very handy cartography for modern problems like mapping asteroids. Here’s a link Phobos arts and crafts to a well-written report about a csnb map of an irregularly shaped moon of Mars. The map folds up to a model of the moon.

The art historian Erwin Panofsky, in his book Albrecht Durer, called this “prototopology,” which means merely that the map, when properly folded, resembles the object. Now — finally! — we have a map a heron can use to get to Brazil. Not that there are herons on Phobos, but that was the fanciful problem I’d been given as a child of six, my apprentice problem inducting me into the guild of architects, the problem that kidnapped my attention throughout grammar school, the problem that sat dormant until hagan came around with his pyramid and global disaster books.

Panofsky’s colleague at the Institute for Advanced Study, Marston Morse, the mathematician from Maine, was fond of a quote from Kepler about stuff like this. He ended his Authors Note for his essay Mathematics and the Arts, with it:

“The paths by which we arrive at our insights into celestial matters seem to me almost as worthy of wonder as those matters themselves.”

Nice sentiment, hm?