The 25 March 2024 xkcd honours Greenland’s place as a measure of cartographic distortion. It’s also, unexpectedly, a riff on the idea of the 1:1 scale map (cf. Borges), especially if you consult the comic’s alt text: “The Mercator projection drastically distorts the size of almost every area of land except a small ring around the North and South Poles.”

Previously: xkcd: The Greenland Special.

I’ve been struggling for months now on how to deal with Arno Peters and his world map. Every time I turn to the subject, I just get bogged down by the complexity of the scattered and multifaceted literature, by the insanity of much of Peters’ map work, and by the different responses to his work. […] After at least three tries to say something new, and floundering each time, I am presenting this blog entry simply as an attempt to organize the information about Peters in a way that makes sense to me, which is to say historically. Think of it as a long bibliographical essay based on what I have to hand (so not comprehensive, especially in the German-language literature). I’m not sure that it says anything necessarily new or significant. So please join me as I go through a series of cuts at Peters and his map work.

Daniel Huffman’s map projection trading cards are making a comeback. “While my colleagues and I did our best to let everyone know about these cards, some people inevitably missed out during the first print run. I’ve had many people contact me asking and hoping to get their hands on a pack or two. So, I am bringing them back for one final print run via Kickstarter,” writes Huffman. “I hope you’ll share this widely, so that we don’t miss anyone this time around, as this is almost certainly the last time these cards will be printed.”

Previously: Map Projection Trading Cards.

The latest in Randall Munroe’s Bad Map Projection series on xkcd is perhaps his most evil yet: it turns all longitudes positive—i.e., it turns west longitude into east longitude, putting Quebec somewhere in Kazakhstan and the Panama Canal off Sri Lanka.

Fresh off of producing a (now sold-out) line of map projection trading cards, Daniel Huffman has produced a 16×24-inch poster showing the surprisingly close and entangled relationships between the various map projections.

I first learned about a couple of these connections several years ago. I don’t quite remember how or where, but I found out that the Mercator projection was equivalent to a Lambert Conformal Conic with the standard parallels set opposite each other across the Equator. And that if you moved both those parallels up to a pole, you got a Stereographic. My mind was suitably blown, and I saved it as a fun fact to share with people. This year, while working on The Projection Collection, I spent a lot of time on daan Strebe’s site looking up details, and I often saw his notes (usually derived from Snyder/Voxland) about how projections were related to each other. I started to realize there were a lot of these connections out there, and I thought it might be fun to diagram them in some way.

The diagram is digital-only (PDF) and donationware.

Daniel Huffman’s map projection trading cards are now a thing you can order. Daniel, earlier this month:

A couple months back, I floated an idea for making some fun trading cards based on map projections. I’m very happy to report that several dozen of you responded and contributed designs to help make the set happen. I’ve been spending several weeks on managing everyone and working through logistics, and I’m pleased to now be able to offer a pre-order of The Projection Collection.

The cards can be pre-ordered here, with delivery later this year (or pickup at NACIS). Each pack has 16 cards, with complete sets not available by design—these are meant to be trading cards in the classic sense. Pre-orders will close on July 6, so you have until then.

Uncharacteristically for xkcd’s Bad Map Projection series, the Madagascator is actually totally legitimate as a projection. Not that it’s any less mischievous, mind.

Update, 3 May: Turns out there was more to this xkcd cartoon. See Mercator: Extreme.

Randall Munroe’s map projection humour is increasingly on point, as last Friday’s xkcd demonstrates. (The mouseover text is even better: “There are two rules on this ship: Never gaze back into the projection abyss, and never touch the red button labeled DYMAXION.”)

Previously: xkcd: The Greenland Special; xkcd: All South Americas; Blame the Mercator Projection; xkcd’s Time Zone Map; xkcd’s Liquid Resize Map Projection.

Previously: Map Projection Playground; Comparing Map Projections.

Previously: The Spilhaus Projection Comes to ArcGIS Pro; Everything’s Coming Up Spilhaus; About the Spilhaus Projection.

It’s a Spalding ball exclusive to Urban Outfitters, it costs $29, and it hurts my brain. [r/Maps]

At some point, xkcd cartoonist Randall Munroe is going to put out a book focusing on his map-related cartoons, isn’t he. The latest in his “Bad Map Projection” series (previously: All South Americas, Time Zones, Liquid Resize) is The Greenland Special, an equal-area projection except for Greenland, which uses Mercator. And I thought he was messing with us before.

But that’s not to say that I haven’t picked up some knowledge: I’ve turned my longstanding interest in fantasy maps into a few published articles (with more still in the works or in press), so I will concede the point on that front. But in general what I do have is exposure. Over the past 18 years I have seen just about everything to do with maps, and so I know a little bit about just about everything. Not enough to be employed at any map-related job, but 18 years of paying attention, of synthesizing everything I’ve seen and read, has afforded me some perspective.

Enough to call out obvious horseshit when I see it.

Also, because I’m not a cartographer, because I don’t have that background or training, because my expertise is a hundred miles wide but a millimetre thick, I’d be extraordinarily reluctant to tell cartographers that what they’ve been doing for the past few centuries has been completely wrong, and that I’ve come up with something better that no one has ever thought of before—only for the something better to be utterly old and familiar to those who know what they’re doing.

I don’t have that kind of chutzpah.

Arno Peters did, though. In 1974 the German historian presented the Peters World Map (a retread of an 1855 equal-area projection by James Gall) as the antidote to a Mercator projection that emphasized temperate regions over the tropics: the Global West over the Global South. In doing so Peters was fighting a battle that, Mark Monmonier has argued, was mostly already won by the 1970s. The Mercator had long been seen as unsuitable for world maps, with wall maps and atlases already having moved on to the Goode’s homolosine, Mollweide and Van der Grinten projections, among others, by the mid-20th century.

Even so, cartographers generally hated the Peters map because it was foundationally ignorant: Peters was dabbling in map projections without understanding their history. He and his adherents invented a false dichotomy—Peters vs. Mercator—and marketed the projection to credulous audiences (e.g. Boston schools as recently as four years ago) as a solution to a problem that in truth was neither unsolved nor really a problem.

Guess what? It’s happening again.

In 2007 a pair of physicists, David Goldberg and J. Richard Gott, published an article in Cartographica in which they proposed a way of measuring and scoring map projections by six kinds of distortion—area, shape, distance, boundary cuts, flexion and skewness (the latter referring to bending and lopsidedness). According to their system, the Winkel Tripel projection, currently in use at National Geographic, had the best score: 4.563. (The lower the score, the better: a globe’s score is zero. The Mercator’s score is 8.296.)

Last month, in an unpublished paper uploaded to Arxiv, Goldberg and Gott, along with Robert Vanderbei, tried to come up with something better than the Winkel Tripel, and arrived at a pair of azimuthal equidistant projections centred on each pole and extending to the equator; the twist, as they see it, is to make the map double sided.

“To the best of their knowledge,” says the piece from Princeton’s communications office, “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.”

Except that maps showing the world in two hemispheres date back at least as far as the 16th century (there’s one on my wall) and polar azimuthal projections aren’t exactly new either: they’re splitting hairs awfully fine to make that claim. Regardless, their double-sided map gets a score of 0.881 on their Goldberg-Gott scale.

Matthew Edney was not impressed; as you might expect, he did not hold back. “I am utterly and thoroughly gobsmacked,” he wrote last month in a piece that marvels at the claims made in the PR piece.

Underpinning all this sheer stupidity and naivety are some serious points about what these astrophysicists understand maps to be. It is not that they are ignorant of the mathematical principles; two have published a paper in a map journal on their measures of map distortion (Goldberg and Gott 2007; also Gott, Mugnolo, and Colley 2007). But it seems that from their highly mathematicized perch they have realized that world maps are actually useful for imaging and visualizing the world. But they want the maps to also be as accurate as possible, according to their own idiosyncratic criteria. […]

Ultimately, once one has stripped away the immense amount of PR guff and hyperbole, there’s little to recommend this as a “new” and “different”—other than the proposal to paste the two halves together. And I’m pretty sure I’ve seen an eighteenth- or nineteenth-century hand-held fan with hemispheres drawn on either side …

They have really only reinvented the wheel.

In a separate piece Edney has some questions about the parameters used in Goldberg and Gott’s scoring system, a couple of which he finds dubious, along with the way they’re tallied up.

Someone coming in from outside the field to “solve” the map projection problem sounds an awful lot like Arno Peters all over again—especially since their little paper has picked up a certain amount of media attention—see coverage from The Verge and the New York Times—that trumpets the possibility that this projection “fixes” flat maps, or the distorted view of the world that we get from flat maps.

Hoo boy. That’s a hell of a claim—one we’ve seen before. Except that Peters purported to solve map projections’ problem of representation. These professors sound like they’re going after the impossible Holy Grail of map projections: to find the most perfect, least distorted map, the One True Map that will leave all others in its dust, the One Projection that can be used all the time and in every circumstance. (The number of questions I’ve seen on Quora, for example, that insist upon this impossibility—that there is such a thing as an “accurate” map projection—is striking.)

Only they’ve defined “most accurate” as “fewest distortions according to our own criteria.” In doing so they’ve reduced map projections to a simple math problem that ignores centuries of map history, to say nothing of real-world uses. In other words: a spherical cow—a model reduced to the point of oversimplification so that it works.

Because you’d be hard pressed to find an actual use case for their double-sided map of the world. As Edney points out, only half of the world is viewable at once: “The point of world maps is to show the whole earth, but one can’t see the entire earth, so the most accurate world map doesn’t show the whole earth” (which somehow doesn’t incur a penalty in their schema). And heaven forbid you should want your world map to show all of Africa, or South America, or Indonesia, at the same time.

Thing is—and regular readers will know I’m preaching to the choir here—there is no such thing as the perfect map projection. Only the right projection for the job at hand. The antidote to a given map projection’s distortion is a different map projection whose distortion is less problematic for what you’re trying to map. Some distortions you can live with in order to preserve fidelity elsewhere. I’m fond of the Equal Earth projection, but I wouldn’t use it if I wanted to emphasize the polar regions.

It’s a practical choice, in other words, one that I’ve come to understand, from my 18 years of paying attention to this, as a basic challenge of map design. One that you can’t simply come in and solve with math. It’s tempting to see this as a variant of the Dunning-Kruger effect—or at least the popular understanding of it: an inability to recognize your lack of ability—that in my experience seems to afflict an awful lot of physicists and engineers. Reduce the problem until it’s solvable, then solve it.

First, assume a spherical cow.

Interesting as an exercise, but not particularly useful.

Kai’s Comparing Map Projections mashes up two code blocks by Mike Bostock: Map Projections Distortions is a visualization of the types of distortion inherent to each projection; Projection Transitions morphs between projections. Combining the two is a neat trick. Refreshing to see the usual two combatants excluded. [Maps Mania]

Saturday Morning Breakfast Cereal’s take on the Mercator projection is … not what you’d expect. The punch line is similar to Christopher Rowe’s short story, “Another Word for Map Is Faith”: if you can’t make the map conform to the territory, make the territory conform to the map. Since we’re dealing with the Mercator projection, this requires some … escalation.

As John Nelson promised last September, with the release of ArcGIS Pro 2.5, the popular and viral Spilhaus projection is now available in ArcGIS Pro.

Meanwhile, Mark Gargul writes in response to Ken’s critique of his cartogram of the election results. Mark describes himself as an amateur and readily admits that other cartograms are “clearly more aesthetically pleasing. On the other hand, I was going for something different with my cartogram—specifically, to try to preserve riding-adjacency as much as possible.”

@kennethfield
… On the other hand, what I was going for was preserving, to the extent possible, riding adjacency. If Markham-Stouffville shared a border with Markham-Unionville on a real map, I wanted that border on the cartogram. Hence, the ugliness.

— Mark Gargul (@GargulMark) October 26, 2019

The other thing Mark was going for in his cartogram was to indicate the urban-rural split: metropolitan areas are given a black border: it’s easy to see which ridings are in Montreal or Toronto; seats that are partially urban and partially rural straddle those borders.

I'll summarize for you guys what I let Ken know in more detail: I wasn't going for pretty, but I was going for illustrating the rural-urban split, which doesn't come across well in the other cartograms or maps I have seen

— Mark Gargul (@GargulMark) October 26, 2019

So it’s doing several things at once that may not be immediately apparent.

Previously: A Cartogram of Canada’s Election Results; More Canadian Election Maps.

Update on mapping #CanadaElection2019 by me & @williamscraigm. Data errors fixed. Value-by-Alpha Unique Values, Prop Symbol (both modified by pop density), Dot density (winners), Dot density (all parties >1%). 1 dot=100 votes. Screen grabs. #NotMercator pic.twitter.com/MJbIRRGqsw

— Kenneth Field (@kennethfield) October 23, 2019

I hit “Publish” too soon last night. Kenneth Field and Craig Williams put together a series of maps showing the Canadian election results in a number of different ways: we have a value-by-alpha map, a proportional symbol map, and two kinds of dot density maps: one showing the winners, one showing all votes per constituency. (One dot equals 100 votes; the dots are spread evenly across constituencies, even when people aren’t. You can’t have everything.) And it’s on the Lambert, not the Mercator.

Speaking of the Mercator. Maps Mania’s roundup of Canadian election results maps notes that the Canadian media’s interactive maps (e.g. CBC, Global, Globe and Mail) invariably resorted to Web Mercator, largely because of the mapping platform used. (In-house infographics team? Don’t be ridiculous.) Web Mercator is singularly bad for Canadian election maps, because Nunavut: it’s the largest electoral district by area (1.9 million km²) and the smallest by population (31,906). It’s enough of a distortion on the Lambert: Mercator makes it worse.

As for cartograms, Ken hated the one I posted last night; Keir points to Luke Andrews’s Electoral Cartogram of Canada, which is a bit nicer, and uses only one hexagon per riding instead of seven. Keir also points to this animation that shifts between a geographical map and a cartogram. It’s hard to recognize Canada in cartograms, because it’s difficult for us to grasp just how many people live in southern Ontario.

Previously: A Cartogram of Canada’s Election Results.

John Nelson reports that the Spilhaus projection will be supported in the next version of ArcGIS—version 2.5, to be released in a few months. This odd projection, which centres Antarctica on a world map showing the oceans as a single, uninterrupted body of water; went viral last year. Requests for ArcGIS support soon followed. Thing is, ArcGIS support requires the math behind the projection: figuring out that math took some sleuthing. The Spilhaus is, it turns out, basically an oblique aspect of the Adams World in a Square II projection.

Previously: About the Spilhaus Projection.

Here’s a gallery of all 68 map projections supported in ArcGIS 10.7.1 and ArcGIS Pro 2.4, from Bojan Šavrič and Melita Kennedy. Some interesting projections included here, some of which are only suitable for a specific region (such as the complex New Zealand map grid). No Gall-Peters, interestingly.

Chris Whong uses a clementine as a substitute for Tissot’s indicatrix: “I found myself eating a clementine this morning, and thought it would be interesting to slice up the orange peel on an 8×8 grid to visualize how much of the earth’s surface is represented in WebMercator tiles at zoom level 3. This is kind of an inverse of the Tissot’s Indicatrix above, showing chunks of the spheroid’s surface over the projected tiles that represent them in web maps.”

Alberto Cairo’s short piece arguing that the Mercator projection isn’t a monstrosity doesn’t cover particularly new ground: the Mercator was created for a specific purpose (bearing-based navigation) and is a good choice for small-scale maps, but it has no business on a world map. But it’s probably worth reiterating, since I still see over-the-top condemnations of the projection on colonialist grounds, channeling Arno Peters (which, you know: not new).

On Sunday Tom Patterson announced that the Equal Earth Physical Map is now available for download in JPEG, Illustrator and GeoTIF formats. Unlike its political counterpart, no territorial boundaries appear on this map (though cities do). Not having borders doesn’t mean that Tom and his collaborators won’t get into trouble with the names of natural features, though: I note they use Sea of Japan rather than East Sea, for example (see above). But, importantly, they’ve released the map into the public domain: if you don’t like their labels, or their choice of cities or colours or textures, you can make changes to the map and put out your own version.

Previously: Equal Earth Gets a Wall Map.

Last Friday’s xkcd suggests that the Mercator projection’s reputation can be used to convince anyone of any false geographical fact.

Not that I’d suggest you do that, mind. No.

It was announced today at NACIS that the Equal Earth projection is now available as a wall map—which is a necessary thing if it’s going to go toe-to-toe with the Peters map. The political wall map is only available as a download (three versions, centred on Africa and Europe, the Americas, or the Pacific): the 19,250 × 10,150-pixel, 350 dpi file results in a 1.4 × 0.74 m (55″ × 29″) print—assuming you have access to a large-format plotter. Not everyone does, so it’s only a matter of time, I suspect, before they have prints available for sale.

The map shows countries and territories in surprising detail (it includes Clipperton, for example); and while it does show disputed regions as such, its choices of boundaries and nomenclature won’t make it many fans in South Korea or India.

Previously: The Equal Earth Projection; Equal Earth Updates; More on Equal Earth.

This image went a bit viral earlier this week. Some context. It’s from an August 2015 blog post at Le Cartographe, in which Alexandre Nicolas discussed (and rendered, above) a projection produced in 1942 by South African oceanographer Athelstan Spilhaus. In Spilhaus’s oceanic projection, centred on Antarctica, the world’s oceans form a single, uninterrupted body of water. Which, you know, it is. The continents form the edges of the map; there is … some … spatial distortion. As Alexandre wrote in 2015, “This projection is rarely used and it’s a real shame!”

Previously: The Penguin Projection (speaking of Antarctica-centred projections).

The Gall-Peters projection is a second-rate projection with first-rate public relations; cartographers’ responses to the projection that focused on its cartographic shortcomings ended up missing the point. Something different is happening with the Equal Earth projection, which was announced last month as a response to Gall-Peters: an equal-area projection with “eye appeal.” It’s getting media traction: the latest news outlet to take notice is Newsweek. So, finally, there’s an alternative that can be competitive on the PR front, without having to mumble something about all projections being compromises until the eyes glaze over.

It’s turning up in GIS packages, too: in D3, in G.Projector and in proj4. There’s even a t-shirt.

Previously: The Equal Earth Projection; Equal Earth Updates.