Introducing the Danseiji Projections

This post outlines work that is described more thoroughly in the article “Minimum-error world map projections defined by polydimensional meshes”, which has been published in the International Journal of Cartography. If you want the full technical details and can’t afford to get around the paywall, you can read the preprint here:

For centuries, cartographers have wrestled with the challenges of making accurate maps. They were limited by not only the mathematical impossibility of accurately portraying a curved surface on a plane, but also by their computational capacities. In Mercator’s day, creating a single conformal cylindrical map required performing a Riemann sum for every single latitude. Even today, some of the most mathematically complex map projections cling to simple geometric shapes, like the straight parallels of Tobler or the tetrahedrons of AuthaGraph. Modern cartographers work to minimize distortion and constrain it to the least important regions of the globe—usually the oceans—within the constraints of these basic formulas through a combination of trial-and-error, combining and varying existing projections, and optimization by advanced calculus.

And yet, this is the twenty-first century. The Mayan apocalypse has come and gone! There are over seven generations of Pokemon! Cartographers need not bow to these computational constraints any longer! The time has come to shake free of these arbitrary lines of latitude and longitude, to throw off these circles and hyperellipses and power functions, to deny control over our maps to these unchallenged assumptions that physical area is the most important quantity of a country, that Antarctica is unimportant and negligible, and that the International Dateline is the end of the world! The time has come for the Danseiji projections.

The Danseiji (/dɑnˈseɪdʒi/; from “弾性 地”) projections are the optimal map projections, in the most literal sense of the word. Each one is represented by a fine mesh of vertices that represent an elastic spherical surface. That surface is allowed to come to rest, thus reducing the distortions of size and shape throughout the mesh. The core of the Danseiji projection’s strength comes from the fact that it has no equations. It is free to be anything or anyone, regardless of how ungeometric or irregular that may be. This family of maps can be easily adapted and applied to any purpose. Comparing areas? Showing statistics? Brainwashing children? There’s a Danseiji projection for you!

If you want to learn more about the technical details, you should check out the preprint, currently being reviewed for publication by the International Journal of Cartography. Here, I’m going to talk about each of these new projections more subjectively and how I think they can fit into our existing cartographic ecosystem.

Note that while I will be using Kavrayskiy distortion criteria to compare map projections throughout this article, I won’t put too much emphasis on them or dwell on specific numbers. This is because, while they provide an easy way to quantitatively group map projections by their distortion, I found while working on this that the projections with the lowest Kavrayskiy distortion criteria are kind of terrible. These criteria are good enough to give us an idea of what projections are similar to each other, buth flawed enough that I don’t think we should spend too much time trying to squeeze every last millibel out of Airy–Kavrayskiy. Again, there are more details in the paper. Now, onto the map projections!

The Danseiji N projection

The Danseiji N projection is the most basic of the bunch. Symmetrical across both axes and interrupted everywhere but the antimeridian, it is similar in Kavrayskiy criteria to Robinson, Winkel Tripel, or Wagner’s compromises. Robinson and Winkel Tripel are both quite popular in atlases, posters, and statistical maps, implying that there is a place for a projection like this in our modern world.

Notably, unlike pretty much any other map projection with similar distortion characteristics, Danseiji N shows the poles as points, not as lines. I consider this aesthetically important, as I generally think that points on the globe ought to be represented as points on a map. I tolerate pole lines them in some of my favorite conventional projections because it’s just really hard to express formulaically the kind of extreme (but finite) stretching at the poles that’s needed to keep polar distortion away from the extreme longitudes. Danseiji N has no such challenge, and thus could potentially replace Robinson and Winkel Tripel as go-to general-use world map projections.

The Danseiji I projection

But of course, equal-area projections are in far more demand nowadays. Look at Gall–Peters supporters, who usually claim to be so because they think Gall–Peters is the only area-preserving map out there. The Danseiji I projection is not perfectly equal-area, but its areal deviations are imperceptible to the eye. Its angular distortion, by Kavrayskiy criteria, is comparable to that of other modern equal-area projections like Equal Earth, Tobler hyperelliptical, and Strebe 1995.

Note that this time, there are pointed-polar comparisons to be made. Beyond Tobler’s hyperelliptical, there are the older and far more common Hammer, Mollweide, and Sinusoidal. Equal area projections are always in fashion, and there have always been many of them. Because of this, the value proposition of Danseiji I is less clear than that of Danseiji II. It comes down to a matter of superficial preference.

In my personal opinion, there are no good lenticular* equal-area projections; I think that Hammer and Strebe ’95 both stretch out the extreme longitudes too much. Danseiji I is lenticular, but only subtly so. The parallels are very straight in the center and only bend as much as they need to on the sides. At the same time, though, I think that Eckert IV is perfectly serviceable as a pseudocylindrical equal-area projection, and Equal Earth is a fantastic projection when you take familiarity into account. And I don’t think that area-equality is as important as a lot of people say it is. So while I might use this projection in my own works because I think that it’s cool, I don’t expect it to displace the myriad of existing equal-area projections any time soon.

The Danseiji II projection

The Danseiji II projection is similar to both N and I, but with different weighting. It places about as much emphasis on angular distortion as it does on areal distortion. As you can see, the outer portions of the map are noticeably inflated, but not so egregiously as some other compromise projections such as Van der Grinten or Gall stereographic. Its Kavrayskiy criteria are closest to the Kavrayskiy VII projection, but not that close.

Danseiji II fills a much larger gap in map projection space than N and I do. I can see it being quite appealing to those who care about angular distortion, but are wary of the severe size imbalances so often derided in Van der Grinten and conformal projections. It would be a good projection for large posters, where angular distortion is more important as a general rule.

The Danseiji III projection

This is where it starts getting interesting. The Danseiji III projection is hard to classify. It introduces an interruption into the ocean to improve the way the landmasses look, which makes it kind of like Goode’s homolosine or Cahill–Keyes. But it uses about eight times less interruption than Goode does and six times less than Keyes, depending on how you count, and most of the oceans are as intact as they ever are, so they’re actually quite dissimilar. It has been tuned to push distortion out of the landmasses and into the less important oceans, which is how Dan Strebe described his process in creating Strebe ’95. But it does this much more directly than Strebe did; he said that symmetry was a design factor in that map projection whereas it clearly isn’t in this one. Danseiji III is an entirely new kind of map projection.

Unlike N, I, and II, Danseiji III cares about land more than sea. As a result, it goes out of its way to bend its graticule around the continents. You can see from the Tissot indicatrix that the northern Atlantic gets compressed, the Pacific gets stretched, and the Indian gets skewed, but all of the landmasses (including many of the larger islands) look more or less fine. It does this with only one small interruption, and without drastically changing the positioning or orientation of the continents. Therefore, while it has no direct competitors in the current ecosystem, I think that it has great potential for all kinds of use cases. Its fairness to all latitudes is great for political maps, its accurate depiction of all countries’ shapes and sizes is a must for educational maps, and its familiar layout and terrestrial favoritism are perfectly appropriate for statistical maps (though I think Danseiji V would also be good at that). It would be rubbish for nautical navigation, since the oceans are so distorted, but why would you use anything other than Mercator or a globe for navigation, anyway? Overall, I think this is the most widely applicable Danseiji projection.

The Danseiji IV projection

The Danseiji IV projection is my favorite of the bunch. It discards all assumptions of conventional map projections and lays the continents out in the orientation that the computer decided was optimal. It is largely useless for practical purposes, as it is needlessly difficult for a layperson to locate and identify specific countries, but it shows the relationships between the continents (especially North America and Eurasia) more faithfully than any lenticular projection could.

This is far from the first map projection to use alternative layouts like this. It’s not unlike Fuller’s Dymaxion projection, and in a more abstract sense, other polyhedral projections like Waterman and AuthaGraph. However, its organic shape allows it to use less interruption and take up less space than Dymaxion or Waterman, and have less distortion than AuthaGraph. These advantages give it great applicability to all cases where Dymaxion is used today—that is, artwork and maps of human migration.

The Danseiji V projection

The Danseiji V projection is more conventional than IV, but perhaps more controversial. It warrants some explanation.

Consider the problem of telling a wobsite owner which countries they get the most traffic from. The simplest solution is a table, listing each country and the number of hits the wobsite has gotten from it. There are hundreds of countries in the world, though, and even a WordPress blogger who only writes about cartography and flat earth theory would quickly require a table 70 rows long. Graphs are a great way to make tabular information more palatable, and the best choice for datums like this would normally be a pie or bar chart. Since the categories are countries, though, there’s a shortcut that one can exploit to make the information easier to read: people already associate a position and shape with each country, so use the x and y axes to indicate category and let the color axis indicate number.

It might look something like this. Note that the vast areal distortions of the Mercator projection don’t really impede the ability of the map to communicate its information; they just make Greenland, Canada, and Russia take up an unnecessary amount of vertical space. Also note that most of this map (~65% ignoring margins) is whitespace: the oceans. They convey no information, as they represent no category.

Thus the motivation for Danseiji V. It intentionally introduces areal distortion to make the oceans smaller and the continents closer together, thus saving space at the expense of geographic accuracy that is usually irrelevant, anyway. It also places a little bit of added emphasis on shape, since the only really important thing in a map like this is that the countries all have familiar shapes and positions.

And overall, I think it does a great job. Africa and Siberia are a little bent, and all of the islands are noticeably shrunk, but overall the terrestrial distortion is minimal, and the whole map takes up way less space. As with III, I don’t know of any existing map projections that are at all similar to this, so I can’t really compare it to prior art, but I think that this is easily the best choice for any statistical cartography.

The Danseiji VI projection

Finally, we reach the Danseiji VI projection. This is definitely the weirdest one. It was inspired by a YouTube comment I saw on a trailer for the ODT documentary Arno Peters: Radical Map, Remarkable Man.

The Peters Projection is absolute Garbage. Europe has a much higher density of Cities and Countries than Africa so it’s better to depict it bigger than it is for more precision. Also I’m sorry to tell you, but Africa IS less important for World history, that’s an Objective fact. Seems like facts are Racist.


The argument is, of course, ill-informed. It seems to defend the Mercator projection as if its inflation is proportional to density of cities, density of countries, and/or importance in world history rather than simply vicinity to the poles. But it got me thinking: what if a map did intentionally inflate regions with more geographic features? From another angle, if Gall–Peters supporters are really concerned with fairness, shouldn’t they support a map that allocates space to countries relative to their populations rather than their acreages? Mark Monmonier argued as much in Rhumb Lines and Map Wars, and I agree.

Such maps exist, of course, and they’re called cartograms. They are rarely used for actual geographic information, though. The extreme angular distortion and discontinuity they introduce typically makes them completely unusable for anything but making a point about the global distribution of whatever they’re portraying. Also, for some reason, global cartograms never seem to show statistics at anything finer than the national scale, which is a real shame when the biggest countries on the map are vast, heterogeneous realms like China, India, and the US.

So I decided to try making a compromise between a regular map projection and an area cartogram: Danseiji VI. Like V, it intentionally introduces areal distortion. However, rather than dilating the continents uniformly, it dilates regions of high population density. I used the finest data I could find so that it would reallocate space within countries as well as between them.

The result is certainly interesting. The reproportioning is noticeable, but almost all countries are still recognizable by their shape. It looks terrible with a graticule, but delightfully jarring without one. There are aspects of it that I dislike, though. For one thing, I had to blur the population density to avoid the mesh artifacts that you get when the scale varies too much between adjacent elements. In doing so, I inadvertently vastly reduced the scale difference between the most and least densely-populated areas, so you don’t get to see Russia or Canada shrink as dramatically as on a real cartogram. There are also certain more specific things that bug me about it. For instance, I’m not sure why the size ratio between France and Germany is exaggerated so much when Germany is actually more populous than France. It’s possible that this one in particular would benefit from more tuning. For now, though, I present it along with the rest of my work in the hopes that someone might find it useful.


The Danseiji projections are objectively the best map projections, given a very particular definition of “best”. While I think that a great many of the map projections we use today are arbitrary and fail to meet their full potential, I hardly think that Danseiji is going to replace every other map projection tomorrow. However, I also think that these projections have great potential in many use cases, which I outlined above.

If you want to make some Danseiji maps for yourself, check out the data files at Mendeley Data. Alternatively, if you don’t want to write your own code, you can use my Map Designer program, available for free on GitHub.

* While the term “lenticular” is not universally accepted, I use it here to mean any map projection that is symmetric about the x and y axes and has parallels that curve poleward.

† Their username is actually just their full name, but they don’t have a strong internet presence, so crediting them in this context seemed mean.

2 thoughts on “Introducing the Danseiji Projections

  1. I was so disappointed that I couldn’t find a very accurate map anywhere on the internet in this day and age of technology but here it is… I am going to print the Danseiji IV projection and stick it on my wall. Great job man.


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