Pragmatic Quincuncial Cartography

By Sean Carroll | September 13, 2005 4:15 pm

Matt McIrvin, on a quest to figure out when the USA was displaced from the center of the world (at least where map-makers are concerned), points to a fascinating map projection site put together by Carlos Furuti. It goes through all the different ways people have thought of to project a spherical Earth onto a flat map, doing their darndest to preserve nice features like shapes and sizes. Only after looking at all these different attempts does it really hit you how distorting most world maps are, if only because the nice features of one will draw attention to the glaring shortcomings of some other one. Round spheres are really quite geometrically different from flat planes — who knew?

My favorite projection is the Quincuncial Projection shown below. It is “conformal” (angle-preserving) almost everywhere, except at the four points where the Equator takes a dramatic right turn. These are also where the size distortions are the most dramatic; fortunately, we can stick these points in the middle of various oceans, where nobody is the wiser. The other obvious problem is that Antarctica is sliced into four little bits. But Antarcticans aren’t a crucial constituency, so we can learn to live with it.

Peirce Map

The reason this is my favorite, besides the fact that it’s both fairly accurate and intrinsically cool, is that this projection was invented by Charles Sanders Peirce, someone known much more for his philosophy than for his cartographical skillz. (And “Peirce” is pronounced like “purse,” just so you don’t come off as a poseur when you drop his name in conversation.) Peirce was the orginator of pragmatism as well as semiotics, and was labelled by Bertrand Russell as “certainly the greatest American thinker ever.” His manuscripts, if Wikipedia is to be believed (hey, why doesn’t Wikipedia support trackbacks?), run to over 10,000 pages. And here he is inventing new ways to map the world.

All of which is simply to say: if Charles Sanders Peirce were alive today, he would definitely have a blog.

  • Clifford

    My favourite is by far the Peters projection, which attempts to preseve relative area, if I recall. It gives you a dramatic understanding of how huge continents like Africa are compared to Western Europe and North America, despite what you see on the more traditional projections. That’s the world map I have on my wall. I hope they have that on the site too?


  • Clifford

    Hmmm, looking at the site, I guess it must be this projection to which I referred above (I learned it as the Peters projection, and that’s what it says on my map…). They give the striking example of Greenland and the African continent.


  • citrine

    Cool map!

    Now if we can only find a way to get the majority of kids (and some adults, too) in the USA to find their country on *some* world map.

    This is slightly off topic, but I can’t resist bringing it up. I grew up in a S.Asian nation and came here as an undergrad. I have gotten the following questions/ comments from people I’ve met in the USA:

    * “I didn’t know there were pianos in that part of the world” (after hearing me play a Mozart sonata). This was from a Physics instructor.

    * “Are there brown skinned Christians?”

    * “Are there eggs in your country?”

  • Risa

    Love it.

  • Simon DeDeo

    There is a long poem about CSP by Susan Howe; when she came to give a reading, she also showed some of his weird “logical diagrams” which kind of look like Feynman diagrams on cocaine.

    CSP got into lots of trouble for being more into free love than was standard for the time (I think cohabiting without marriage?)

  • ljs

    TOTALLY agree with citrine on US kids being so ignorant in geography. How about what is the capital of their own state for starts!!

  • Chaz

    Pretty funny, citrine… it reminds me… someone once asked a friend of mine whether Jews celebrate Halloween.

  • Anonymous

    CSP got into lots of trouble for being more into free love than was standard for the time (I think cohabiting without marriage?)

    This and many other interesting things are in Louis Menand’s book The Metaphysical Club. I recommend it highly.

  • Suz

    hey Sean,
    your two MIT talks were totally written in my calendar, but alas, I’ve been mega stressed with my own talk to give (mandatory super-group meetings; not nearly as exciting as your talks) and other work. Aaargh! I hope you enjoyed your time at MIT.

  • Wolfgang

    If you click on my name you will find two links to “universally corrected” maps of our globe. (hint: Australians see the world differently…)

  • Matt McIrvin

    As the site says, while the goal of using an equal-area projection is entirely laudable, the Peters projection is a bit of a sham. Peters ignored centuries of existing work on equal-area map projections, and independently rediscovered an old projection by Gall that is not used much because it induces so much shape distortion; he then successfully managed to sell it to educators and activists by using politically loaded rhetoric, implying that he’d discovered the only egalitarian projection.

    The Mercator projection, which makes Greenland bigger than Africa, was Peters’ great strawman, and at the time, it really was used occasionally for wall world maps in education, though it was already fading from fashion. The Mercator is terrible for that purpose. But there are lots of projections that are better for world maps than the Mercator. For that matter, Peirce’s quincuncial projection doesn’t distort relative sizes that much, though it’s not equal-area.

    And now for some cranky unsupported speculation of my own: I think the Peters map’s sales pitch works in part because we are social bipeds, and associate *height* with “bigness”. While it represents areas accurately, the Gall-Peters makes tropical countries look relatively bigger even than they are, by stretching them vertically so they look tall. That gives a sort of shock of the new: look how huge Africa is! But if you turn the map sideways, suddenly Eurasia looks huge again.

  • Matt McIrvin

    …By the way, the Mercator projection has lately made a resurgence at Google Maps! For all its faults, it’s actually about the best you can do for that specific purpose, since it’s the only map that both (1) is conformal (that is, locally shape- and angle-preserving, so that magnified views of it won’t look squashed), and (2) always has north pointing the same direction.

    To do better, you have to abandon flatland and go to Google Earth.

  • Clifford

    Matt, the issue here is not who thought of it first, or whether Peters was a charlatan or not…. The issue is the idea of emphasizing the area over other political concerns. There will always be distortions, right? You have to make some sacrifices. So whether you call it Peters or Gall, or Rumpelstiltskin’s projection, there are some merits to choosing to emphasise area over other things. I for one like to be reminded that North America and Western Europe are not quite as huge and important as they think they are….



  • Sean

    Well, the fact that Peters gained notoriety for his maps by emphasizing political concerns and ignoring centuries of cartographical research is an issue, and an interesting one. There are lots of equal-area projections out there, going back quite some time.

    My personal preference is not for strict equal-area projections, since they distort shapes so badly. Better to compromise between shape/size issues, as Peirce’s map does pretty well.

    As Wolfgang points out, putting the South Pole at the top makes an interesting political statement. Although you still have to defend your equatorial-centrism.

  • Clifford

    I guess I misspoke and/or was not clear. Sorry. I meant that it was not the issue that I was emphasizing in my previous comments concerning liking equal area (despite its shortcomings). I did not mean to imply that “rediscovery” of an idea is not an issue in general.


  • Matt McIrvin

    Equal-area projections are good for countering Northern Hemisphere ethnocentrism, but there are better ones out there than the Peters, such as the Mollweide or this particularly nice flat polar quartic one, or the interrupted equal-area projections. They’re often used for statistical maps where equal area is particularly important.

  • Clark Goble

    People I know who are Peirce scholars don’t like Menand’s book. I think it was less free love than wanting a divorce in a society that didn’t condone such things. It also led to pretty much the end of his formal academic career.

    But he definitely is an amazing philosopher. There’s a scholarly site for papers on him that has many of his main papers.

  • Alejandro Rivero

    AKAIK, every islamic map has the south in the top. I am in doubt about islamic globes. Are they inverted when shown in museums? A related issue is constelation naming.

  • Greg Kuperberg

    Conformal maps of the Earth are a great introduction to complex analysis. If you identify the Earth with the Riemann sphere, then the Mercator map is exp(i*z), while the quincuncial projection is a Weierstrass elliptic function. Or you could view it as a 2-to-1 conformal projection from a torus to a sphere with four ramified points. I imagine that it is relevant to one-loop calculations in string theory in that guise.

    It would be fun to use a modular form to make an Escher-like repeating conformal map of the earth in the hyperbolic plane. Escher himself veered towards similar possibilities. In addition to the famous hyperbolic tiling, Circle Limit IV or “Angels and Devils”, he used the same angels and devils to tile the sphere, as a wood carving. The real starting point is a smaller sphere tiled by one angel and one devil, with both of the Escher works as branched conformal lifts. (Indeed if you look carefully, the angels and devils are both just decorated triangles, rendering angels and devils equivalent to each other by reflection. So you could start with a disk with three marked points for the two wingtips and the joined feet of either creature.)

  • Plato

    Conformal maps of the Earth are a great introduction to complex analysis. If you identify the Earth with the Riemann sphere, then the Mercator map is exp(i*z), while the quincuncial projection is a Weierstrass elliptic function. Or you could view it as a 2-to-1 conformal projection from a torus to a sphere with four ramified points. I imagine that it is relevant to one-loop calculations in string theory in that guise.

    While I like to see this movement, I wanted to develope it into a modern view.

    Not only this, but how perception is dependant when seeing tessellations. Sort of like using a line of a shadow or light, depending on your point of view to support the line?:)

    What mathematics would describe this?

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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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