of maps and Platonic solids


I love maps.  One of my favorite blogs is Strange Maps, and I try to visit it only when I have some time to kill.  I have a USGS quad of the Mt. Tam/Muir Woods area in my office.  At home, I have Axis Cartography's San Francisco map hanging in a prominent location.  Give me an atlas, and I will be occupied for hours, poring over the maps inside. 

I think most mathematicians (geometers and topologists in particular) have an appreciation for maps.  There's a lot of interesting mathematics going on in them that most people never really think about.  The main issue is one of projection--the Earth is round and maps are flat.  Anyone who has tried to wrap a ball at Christmas knows that there's no way to do the job smoothly.  Any projection from the curved surface of the Earth to a plane will necessarily create some distortion; how much and which kind is a function of the projection chosen.  Atlases get around this problem by cutting the surface into chunks and drawing the pieces.  At close range, the Earth looks flat (hence the old popular belief), and it is, approximately.  That allows cartographers to create accurate maps of cities and states with little distortion of the shapes.

But this isn't a post about the mathematics of map projections.  The university's library book store is closing this week and everything is on sale.  I wandered in to browse and discovered flat file drawers full of USGS quads.  Most of them were of pieces of Florida (naturally), but I also found some from Washington and Montana.  I grabbed 40 without even looking at them; at 10 cents apiece I couldn't resist.  What will I do with them?  I don't know.  Some of them are rather beautiful and even deserve to be framed.  In particular, I found a few orthophotographic maps of portions of the Gulf Coast and some farmland that had been plowed for crops.  They look more like pieces of abstract art than maps.

In my haste, I managed to grab multiple copies of some maps.  What to do with them?  Fold origami, of course.  The photo above shows two dodecahedra that I made from cutting one map into 24 3" by 4" rectangles, folding them into pentagonal units, and then assembling them into the polyhedra.  They're sitting on my 13" laptop, so you can get a sense of the finished size (about 3.5 inches in diameter).  I folded both of these in about an hour total; it's a lot of fun.  

I'm teaching my one-credit origami class again this term and I was trying to think of a big project to do as a class instead of the level 3 Menger sponge we did last time.  I had the following crazy idea.  Each person creates a tessellated rectangle 4.5 ft by 6 ft.  I showed the students how to fold a basic unit to do this from a 6" square of paper; creating the large rectangle will take 432 such units.  We could then take these large rectangles, fold them into pentagonal units, and put them together to make very large dodecahedra.  Large here means about 5 feet tall (I think).  Will it even work?  Who knows?  We're trying a scale model first with 18" by 24" rectangles.  I'll keep you posted.

By the way, Platonic solids are very interesting in their own right.  There are only 5--tetrahedron, cube, octahedron, dodecahedron, and icosahedron.  Euclid proved this in his Elements, but my favorite proof uses Euler's formula from topology.  My students will learn all this as I teach them to fold all five of these solids--the dodecahedron as above and the others using business cards. 

So you see, math can be fun.