# Clipped Corners

An icosidodecahedron made from 30 binder clips.

Designed and constructed in April, 2011.

Thirty binder clips are joined into an icosidodecahedron, with one clip per corner and one handle per edge. The icosidodecahedron, a quasi-regular polyhedron constructed from 12 pentagons and 20 triangles, can be made from a dodecahedron by "clipping off" each corner at the midpoints of its edges. It can likewise be constructed by "clipping" the corners of an icosahedron instead of a dodecahedron. This corner-clipping process is called rectification, and this example illustrates the more general fact that rectifying a polyhedron produces the same result as rectifying its dual.

In the method used here to join binder clips, each clip has four handles connected to it, where two opposite handles "belong" to the clip and the other two come from adjacent clips. Which polyhedra can be constructed in this way? Not accounting for physical difficulties, any polyhedron where all vertices have exactly four edges should be possible: for such a polyhedron, it is always possible to assign handles to clips correctly (prove this!). There are many polyhedra whose vertices have degree four, and enticing possibilities include the rhombicuboctahedron (or the quasirhombicuboctahedron!) and the rhombicosidodecahedron. My friend Andrea Hawksley succeeded in building a cuboctahedron, which is also known as a rectified cube or rectified octahedron. In fact, any rectified polyhedron has degree four vertices everywhere and is thus theoretically constructible in this fashion. My sculpture Zodiac is exactly a "rectified soccer ball" ("rectified truncated icosahedron") made with a variant of this corner joint, with some clips removed for "stellar" effect.