A Sculpture/Puzzle

by George W. Hart

Dept. Computer Science
Stony Brook University

G4G6 exchange item
Atlanta, Georgia

March 2004

This puzzle consists of twenty identical flat pieces, each with the shape shown on the reverse of this page. Photocopy that template (enlarged) on to twenty sheets of card stock, cut on the edges, and assemble the pieces to form the structure above. When properly configured, the parts can be taped together just at their tips, without any interior contact. With paper, it is relatively easy to form the configuration if you bend the parts to get the arms past each other when necessary. For a tougher challenge, cut twenty copies of the shape out of rigid plastic, plywood, or metal. Then the arms are often blocked on the wrong side of each other compared to where you would like them to be. However it can be assembled; below left is a wooden model with twenty laser-cut parts glued together.


Snarl. Assembled wooden model, 12 inch              Selective laser sintering, nylon, 1 inch

The small attached version of Snarl, shown above right, is a solid freeform fabrication model of the assembled form. (I find it suggestive of the dried seedpod of a sweetgum tree.) It is made from fine nylon powder by computer-controlled laser-melting of cross sections, i.e., selective laser sintering (SLS). At this one-inch scale, there are slight irregularities, because its thin struts are at the technology’s limit. An .stl file for reproducing this on any SLS machine is available at    

Snarl Template.  Make twenty copies on stiff card stock, rigid plastic, plywood, or metal.

Mathematically, the form derives from the uniform compound of five regular octahedra, a well known polyhedral compound shown below left, which has pairs of concentric coplanar equilateral triangles in twenty icosahedrally arranged planes.  The figure above lies within two equilateral triangles as shown below right. It is a subset of the plane, carefully designed to not intersect the nineteen copies of itself in the other icosahedral planes, yet to meet other copies of itself along the short outside edges.