# Soccer Stars A total of 450 binder clips are assembled into 32 stars and joined like the faces of a soccer ball.

Designed and constructed in March, 2012. Thanks to Daniel, Nick, and Alex for help with construction. A closeup shot showing the details of individual 5-pointed and 6-pointed stars. A different sculpture made with a similar technique. This Binder Clip Snub Dodecahedron is made from twelve 5-pointed stars and has 150 binder clips. The stars follow a slightly different design from those in Soccer Stars.

Twelve 5-pointed stars and twenty 6-pointed stars are joined together according to the faces of a soccer ball (truncated icosahedron). The result is a sturdy and surprisingly heavy sphere built from 450 binder clips.

The polyhedron formed by these stars is a snub truncated icosahedron (illustrated below), which has 12 pentagons, 20 hexagons, and 240 triangles. In general, any polyhedron can be snubbed by replacing each face with a star as was done here. The most famous examples are the snub cube (equivalent to the snub octahedron), the snub dodecahedron (equivalent to snub icosahedron), and the snub tetrahedron (which is just an icosahedron). Because of the generality of the snubbing procedure, many variations of this Soccer Stars model can be built. For example, this Binder Clip Snub Dodecahedron uses twelve modified 5-pointed stars and has a total of 150 binder clips.

The polyhedron underlying Soccer Stars—the snub truncated icosahedron—cannot be built from regular polygons: if all faces were regular, than each vertex with one hexagon and four triangles would be perfectly flat! This polyhedron is shown in the image below in canonical form, meaning each edge is tangent to a sphere, each face is flat, and the points where the edges touch the sphere have center of mass at the sphere's center. A marvelous theorem says that every polyhedron has a unique such representation! (This is a strengthening of the Koebe-Andreev-Thurston circle packing theorem.) For the canonical snub truncated icosahedron pictured below, the pentagons are regular, and the hexagons are equilateral but not equiangular. Furthermore, there are four different shapes of triangles—two isosceles and two oblique. Can you tell which is which? The Snub Truncated Icosahedron (right) is constructed by separating the faces of a Truncated Icosahedron (left), attaching triangles to form "stars" around each face, and joining them as shown. The Snub Truncated Icosahedron is shown in canonical form.