The large red arrowhead at the map’s bottom, just a bit left of center, is where — so they say, if there are such things — great waves occur; the Indian Ocean surface current trying its best to squeeze its way around the southern tip of Africa and getting pinched now and then by the three currents to the south of it going the other direction.

When the edge of a world map is a natural boundary, the shape of the map has physical meaning.

World maps with natural boundaries were discovered in the mid-twentieth century by a fascinating old coot named Athelstan Spilhaus, a polymath from South Africa who, according to his obituary, spent World War II in Burma and southwest China as a meteorological sciences adviser to not-yet-chairman Mao, spent the 50′s at Area 51 sending up experimental, extremely high altitude weather balloons and about whom, because of a popular Sunday funnies science-strip Spilhaus wrote, President Kennedy said, “The only science I ever learned I learned from Athelstan Spilhaus.”

Working ingeniously with scissors and tape, Spilhaus exploited conventional projection formulas, extending them here, cropping them there, along shorelines and tectonic features to make “world maps with natural boundaries,” the title — finally, the right words together! — of his last published paper, with cartographic luminary John Parr Snyder, of 1993. The trouble, if there was one, was that he and Snyder exploited conventional projections, which, extended beyond their usual limits, became grossly distorted at their peripheries. So, even though the edges of their maps had physical meaning, it was like looking at a map of the world through a fun-house mirror. For Spilhaus and Snyder, the problem became 1) of selecting a particular projection (from hundreds available) which would minimize the fun-house effect for a given projection, and 2) of accepting the limitation of boundary extent that the chosen projection allowed. So, they could (as they did in their 1993 paper) conceive of a world map edged by continental divides, and comment on the “obvious utility” of such a map — it would organize the world into its constituent watersheds, the basic units of geomorphology — but they could not figure out how to make it.

Here is the map they would like to have made.

Because the geometry for plotting points for csnb world maps is not a conventional (two parameter, x and y) projection system — csnb only projects the line of interruption (a single parameter, x) at a constant scale — the two problems that bedeviled Spilhaus and Snyder, extreme peripheral distortion and limited boundary selection, are moot. And — here’s the bonus, the beauty of the csnb approach to making world maps — when a world map’s natural boundary is drawn at a constant scale, the map’s shape has geometrically accurate physical meaning, like looking at your reflection in a flat mirror.

Land is white, water is blue, edge of map is continental (and sub-continental) divides, and the habitual perimeters of primary ocean currents (imagine holding your finger over the end of the water hose).
What is nice here is that at a single glance you see Earth’s complete surface in proper proportion, the world’s ocean and the lands that drain into it, all in proper shape and size.

From the point of view of a raindrop, falling on the map’s edge, everything is downhill (or swept along the current). From the point of view of the sailor, coming into port, walk inland as far as you want, but beware if you go over the top of the hill. And:

• The overall proportion of land to water is accurate
• The relative proportion of each ocean to the others is accurate
• The proportion (and disposition) of land draining into each ocean is accurate
• The proportions, shape and size of the particular watersheds, relative to each other, are accurate.

• Continents are color-coded.
• Straits (Bering and the various Indonesian) are shown with curvy hyperlines
• Canals (Suez and Panama) are shown with straight hyperlines (Suez, a sea level canal, has a radius corner; Panama, a locked canal, has a hard corner).

Because no segment is very large in relation to the total surface, the map folds to a pretty good globe.

If you think about the words “constant” and “scale,” it’s obvious that any csnb world map, even when its segments are very large, has to fold up, at least to some sort of volume. More about that in later posts, where I’ll cover the “children” (more compact adaptions, prunings of the boundary tree) of this map.

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?