Cute

Thirty hair ties in dodecahedral pattern with decorative knots at the vertices.

Designed and initially constructed in August 2010; this instance constructed in April 2012.

A closeup of one of the vertex knots. This knot cannot locally come undone, but because of this, only 19 of the 20 vertices can have this pattern.

The last, "cheating" knot, which can be taken apart just by pushing the ends through.

This Cute sculpture is carefully knotted from 30 loops (hair ties)—one per edge of a dodecahedron. Somewhat surprisingly, the structure can support its own weight despite the springiness of the hair ties. The three edges meeting at each vertex of the dodecahedron are tied together in a decorative-knot-like fashion, and the knots shown in the image above are "locally unsplittable," meaning if we zoom in on a single vertex and try to untangle the three ends in that small region, we will not be able to separate them. (Technically speaking, these vertex knots are non-split tangles.)

The fact that these loops have no loose ends greatly complicates the tying process. In fact, I have proved a fun, knot-theoretic theorem to accompany this sculpture: If all 20 vertices were tied with this (or any other) "locally unsplittable tangle," then the resulting link would be unsplittable—it would be impossible to separate the 30 strands from each other. In particular, it would be impossible to tie this structure without cutting the hair ties! (The property of unsplittability is also discussed in Pair o' Boxes)

I chose to overcome this obstruction by allowing one "cheat" vertex: at one of the 20 vertices (shown here), I used a different knotting that can be locally undone. Even with this allowance, the structure is still quite challenging to tie together!

I am currently writing about the theorem mentioned above and many extensions thereof in an article, Brunnian Links and Impossible Knotwork on Graphs (in preparation). I'll put a link here when it is finished.