Help me make planet dice.

March 19, 2012 11:12 AM

Is there an easy way to map a sphere onto a dodecahedron? I'm playing with the idea of generating "planet dice."

I've been thinking of ways to make a projection of a planetary map onto a die for the purposes of using it as a pencil and paper RPG prop. It seems to me that a d6/cube would be too few surfaces and not quite satisfyingly round and a d20/icosahedron would result in small hard-to-read surfaces. Also, a lot of gaming stuff already comes optimized for hex maps.

I actually know for sure this is possible, because I've found a page with map projections. (And Thinkgeek did a icosahedron (though it does not look rollable.)

That's about exactly what I want to do, but I'm trying to work out how I would do something like that myself. From the standpoint of the Earth, I might like to experiment with rotation to see the best arrangement of vertices against interesting surface areas, or even give Pangaea a shot. If that worked out, then doing the moon or fictional planets would a next step.

Is there any good existing mapping software for something like this or am I better off trying to do my own math? Are there good open-source maps of, say, Traveler planets, that could be used for practice?

posted by Karmakaze to sports, hobbies, & recreation (11 answers total)

Also, a lot of gaming stuff already comes optimized for hex maps.

a dodecahedron has pentagons.
posted by empath at 11:26 AM on March 19, 2012

I recently was looking at a Google Sketchup plugin called Waybe that "unfolds" a 3D model into something you could print and cut and glue together.

It seems to me that an icosahedron would mesh easier with a hex map than a dodecahedron. 3-sides fits into 6-sides easier than 5-sides, right?
posted by RobotHero at 11:27 AM on March 19, 2012

This guy did it and mentioned he developed his own algorithm. Maybe it's worth asking him about it?
posted by circular at 11:37 AM on March 19, 2012

Doh, sorry, missed your link.
posted by circular at 11:38 AM on March 19, 2012

Dymaxion projection on an icosahedron.
posted by Jawn at 11:39 AM on March 19, 2012 [4 favorites]

I think for fictional planets, this doesn't matter - you would just draw directly onto an unfolded dodecahedron template, and figure out which edges met where, so you could preserve continuities.
posted by carter at 11:53 AM on March 19, 2012

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%)

I'm not totally sure this means that the best dodecahedral mapping of a sphere would be less distorting than the best icosahedral mapping, but I think it does, because the average distance from sphere to dodecahedron would be less than to the icosahedron.
posted by jamjam at 1:11 PM on March 19, 2012

Is there an easy way to map a sphere onto a dodecahedron?

Easy theoretically, or easy practically? Because, yeah, theoretically, it's easy: put the vertices of the dodecahedron onto the circumscribing sphere (which shares the same center), and then poke rays from the center, through the dodecahedron, and then onto the sphere. There thus is a one-to-one correspondence between points on the sphere and points on the dodecahedron.

This apparently is what your web page calls gnomonic projection. (I would usually call it central projection, but I don't work with cartography.) It's possible that they are projecting the sphere onto a dodecahedron where the sphere is inscribed in the dodecahedron, instead of circumscribing it.
posted by leahwrenn at 2:13 PM on March 19, 2012

A Photoshop plug-in I wrote does exactly what you describe if you pick 'dodecahedron' for the output.
posted by naturetron at 12:08 AM on March 20, 2012

You can look at a easy/cheap japanese software called Pepakura. I use it almost daily to make more animal like sculptures, but you can make/print any object you want, specifying exactly where you want your fold lines and cut lines etc. You could start by looking on turbosquid for someone who's modeled the exact shape your looking for first, then plop that into pepakura...
alternartively, would this be good enough? It comes in white with black lines too, and is very beautiful in person... can't vouch for rollablity...
posted by PardonMyFrench at 12:08 AM on March 20, 2012

I still haven't quite gotten my head around doing the projections, but I did find a page with good templates for printing and folding paper polyghedra:
hbmeyer.de/flechten/indexeng.htm

Based on that, I'm actually tempted to go with an icosahedron - I tried the pseudoglobe pattern there and got a much better result than any of the patterns that involved gluing little tabs under the faces of the paper sides.
posted by Karmakaze at 10:24 AM on May 21, 2012