"Your side projects always seem to involve the Hard Way" (Tom Carden, November 14th, 2007)
(Update, March 1: Check out a new revision of this map with a calmer, more predictable drag behavior)
Late last year, I posted about the "faumaxion" world map, a slightly modified version of Buckminster Fuller's famous Dymaxion World Map. I've finally put the finishing touches on this project, implementing it as a command-line script for composing static images (similar to Modest Maps compose.py) and a Flash slippy map.
There are a bunch of interesting things going on here.
The panning action is different from a typical Google Maps interface. With the mercator projection centered on (0N, 0E) that all the popular providers use, North is always vertical. Here, the faces of the icosahedron wrap around the sphere and meet at non-right angles. As you pan the map, the focal point marked by the small circle is kept North-oriented. A less jarring way to design this would be have the drag action work just like a normal slippy map, and animate the reorientation to North at the end of a move. I'm curious, though, how this version is perceived, and whether it's too infuriating to use.
I definitely think it does a good job of illustrating how the interrupted projection works.
Unlike your typical mercator or albers projection, the map is divided into twenty separate gnomonic projections, each framed in one triangle from an icosahedron. Buckminster Fuller designed his original projection so that the edges of the triangles fell on water as much as possible, dividing the globe neatly into chunks of inhabited land - it's a humanistic map, designed to focus on human views of the world. I'm calling this one "faumaxion" because it doesn't strictly follow Fuller's model - he didn't use the gnomonic projection, and I don't do any additional cuts near Australia to keep it whole (sorry, George).
Typically, the imagery we handle in Modest Maps is continuous, with aerial photography or road maps covering large areas that are broken down into smaller and smaller squares. The math for handling rectilinear tiles is fairly straightforward, and is covered by Modest Maps' geometry classes: Location, Point, and Coordinate. In the faumaxion case, the tiles are equilateral triangles at a range of zoom levels.
The tiles are all being served from Amazon S3. They're regular square JPEGs in the Flash version, masked behind triangles in the display. For the Python version, I'm using 24-bit transparent PNGs with the bits outside the triangle already cut out. You'll notice that at the higher zoom levels, some tiles are missing or screwed up. Sorry.
A strategy that seems to work for interactively arranging icosahedron faces here is based on an article from Scott Bilas of Gas Powered Games, The Continuous World Of Dungeon Siege. I make no claims about this interactive toy being anything like a complete 3D game, but I borrowed Scott's idea of maintaining a central reference point ("there is no world space") and performing whatever linear transformations are necessary to arrange a world around that point on every frame. Faumaxion's central reference point is the circle in the middle - it's the first face to be drawn, and every other face is continually arranged around it using a variation of the "ghost finger" hack I described in "gefingerpoken," a recent post on multi-touch interfaces.
My hope is that in combination with some excellent work that Tom Carden's been doing to port Modest Maps to Processing, the transformation-based placement of faces here will eventually migrate into the mainstream ActionScript 3.0 version of Modest Maps, making it possible to display a wider range of intermediate zoom levels and generally make the TileGrid a little less crazy.
Although I didn't go so far as to add geographic markers to this map, it's possible to do so, and I threw the tiny latitude/longitude display into the upper-left hand corner to show it. For all its oddities, the Dymaxion World Map is wonderfully suited to showing global concerns: Fuller created arrangements focused on different parts of the globe to demonstrate the sense of historical political and military moves, such as Japan's grab for empire in the 1930s and 40s, or British domination of the south seas in the early 19th century. Other uses might include visualization of transcontinental flights that otherwise look like wasteful loops when plotted on a standard mercator projection. I'm looking at you, noted Web 2.0 travel-sharing website.
This is fantastic, Mike! Really fascinating and fun to scroll around... I just finished reading "Guns, Germs and Steel", and this would be a great way to visualize some of the crazy population movements he describes, like the Austronesian colonization of Madagascar... The only thing I don't get is: why does "north" change so suddenly sometimes when you slip across certain boundaries between triangles? Shouldn't the orientation of north be just as continuous as the contents of the map?
Thanks Cassidy! I think the reason for the jump is that the orientation is local-north rather than global-north. A straight vertical line from the center point won't necessarily hit the pole in the map, but it will match a local compass reading at that point. Because the projection changes as you move from face to face, you get small discontinuities in the parallels and meridians. Here's an example, two images created with the same parameters, one of which shows Blue Marble imagery, while the other shows a latitude/longitude grid. You can see folds or breaks between each pair of triangular faces: http://teczno.com/faumaxion/blue-marble.jpg http://teczno.com/faumaxion/grid.png
Oh, okay, I get it now. That makes sense! It seems like a "global north" would also behave discontinuously, what with the north pole jumping around every time the triangles snap into a new position. But I wonder if it'd be possible to create some kind of alternate "faux north" that behaves more smoothly, at the expense of being slightly out of kilter with local north? I can't think of a simple equation for this off the top of my head, but you could probably fake it by rendering RGB vector values into each triangle and filtering them across the boundaries, or something. (Too bad the icosahedron isn't symmetrical about the poles-- if it were, you could get away with mapping faux north for just two triangles, and then mirroring it to the rest!)
Lots of map projections, like the British national grid that I grew up with, have a concept of 'grid north' and 'true north' which deviate slightly depending on location. So that seems like a tolerable compromise. In this case though, I think the whole thing needs a sprinkle of multi-touch/rotational magic. Not all of the great stories to be told on the globe run left-to-right :)
This is very impressive, though as you mention, a bit aggravating and squirrelly in use. I think the solution of reorienting after a drag (perhaps with minor animation, to help visualize the reorientation) is probably a good idea. Still, I'm a bit surprised nobody has tried to do this with the Dymaxion Map before (at least not to my knowledge). It's really a great way to explore the concept.
Thanks for the feedback - I've just posted a new version of this that has the orient-after-drag behavior, and it's much calmer: http://teczno.com/faumaxion-II/
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