A Twelve-Part Puzzle Based on
the (4, 2) Goldberg Polyhedron
"Tectonic Plates"

George W. Hart

This puzzle is a 4.5-inch diameter spherical jig-saw based on the (4, 2)-Goldberg polyhedron. In other words, these parts snap together to make a cool looking sphere. Each of the twelve pieces above contains exactly one pentagon; the remaining faces are hexagons..

Here is is assembled.  The parts interlock nicely to hold together, then can be disassembled.  It has 282 faces, of which 170 are hexagons and 12 are pentagons.  Can you find the three easily visible pentagons in this photo and see how they are related?

To get a sense of how it works, this picture shows eleven parts together and the last one ready to be inserted. But no particular part needs to be last; they can be assembled in any order.

In the above image, the six pentagons in this hemisphere have been colored red.  (Six more pentagons are on the back hemisphere.)  The blue line indicates how the pentagons are related. Start in any pentagon, facing any one of five directions, and take four steps on hexagons. Then turn right 60 degrees and take two more steps. You always land on another pentagon. That is the meaning of the "4" and the "2" in calling this the (4, 2)-Goldberg polyhedron. The mathematician Michael Goldberg worked out the theory of this family of polyhedra in the 1930s. A nice example of how pure math ideas often are later found to have useful applications is that since the 1980's these forms are familiar to chemists as "Bucky balls".

If you have access to a 3D printing machine, you can make your own copy of this puzzle from the twelve stl files which are available here.  It is not too hard to assemble the pieces correctly if you remember to check the (4, 2) property as you go along.  That will eliminate most of the incorrect matings which might initially look good.  If you get totally stuck, the six images above show how the twelve parts go together.

Above is a second instance of the same puzzle.  I dyed this one yellow and I think that helps make the regions clearer, so when it is sitting on a shelf, people can see it is a puzzle.  It gives a nice sense of tectonic plates fitting together, so I call this a Tectonic Plates Puzzle. But geologists will no doubt point out that the geometry of these regions is nothing like the geometry of real tectonic plates.

In this image, the (4,2) puzzle is assembled and sitting inside a slightly larger (5,3)-Goldberg puzzle which I made previously. Pictures of that puzzle and the stl file for fabricating it are available here.  As I get time, I plan to continue this series of different concentric Goldberg puzzles nesting inside each other.

For more about Goldberg polyhedra and other interesting uses for them, see my paper "Goldberg Variations," to appear in Shaping Space.