Paperclip Snub Dodecahedron

A snub dodecahedron made from 120 paperclips.

Designed and constructed in July, 2011.

A closeup view of a vertex.

A view down a 5-fold symmetry axis.

A view down a 3-fold symmetry axis.

A larger construction using the same technique, made with 360 paperclips in the shape of a snub truncated icosahedron.

Constructed in December, 2013.

A snub dodecahedron made from 120 paperclips. I designed this in response to George Hart's paperclip snub dodecahedron challenge, imposing a few additional constraints of my own: the paperclips should support themselves without glue, solder, etc., and the pieces should be recognizable as paperclips, with as little modification as possible. In this design, half of the paperclips have been bent at the middle, and the other half are intact and outline the central triangles.

Because paperclips have a "long" and a "short" half, not all edge lengths of this snub dodecahedron are the same. If the long and short halves of the paperclip have lengths \(1\) and \(e < 1\) respectively, then there are 12 regular pentagons and 30 equilateral triangles with side length \(1\), and there are 60 triangles with side lengths \(\{1,1,e\}\). A true snub dodecahedron corresponds to \(e=1\) so all faces are regular. But varying the value of \(e\) produces a continuum of polyhedra that unfurl from an icosidodecahedron at \(e=0\) to a rhombicosidodecahedron at \(e=\sqrt{2}\), as in the animation below. In fact, many paperclip brands have a long-to-short ratio close to \(\sqrt{2}\), so making this model with the roles of the long and short ends swapped would result in a shape strongly resembling a rhombicosidodecahedron with diagonals across the square faces.

When the edge between two yellow triangles has length 1, the polyhedron is a perfect snub dodecahedron.

As the length \(e\) changes, the polyhedron morphs between an icosidodecahedron at \(e=0\), a snub dodecahedron at \(e=1\), and a rhombicosidodecahedron at \(e=\sqrt{2}\).