# Intensegrity Intensegrity is made from 30 lollipop sticks and 30 hair ties (no glue).

Designed and constructed in August, 2010.

A swarm of colorful hair ties twists and dances through a framework of lollipop sticks; the structure is held in a delicate balance by their mutual tug-of-war. The entire structure is quite small—7cm in diameter, or about the size of a tennis ball. This made assembly very challenging, but certainly worthwhile.

The sticks naturally assume a shape reminiscent of five interlocking tetrahedra, so the figure has chiral dodecahedral symmetry—at least to the color blind. Accounting for color, there are no symmetries at all! Indeed, if each stick has a hair tie of color $$a$$ across its center and two hair ties of colors $$b$$ and $$c$$ looped around its sides, then each of the $$5\cdot\binom{4}{2}=30$$ possible color combinations $$(a,\{b,c\})$$ appears exactly once. For example, the frontmost stick in the figure above is the unique rod with $$a=\text{orange}$$ and $$\{b,c\}=\{\text{red},\text{blue}\}$$. Symmetries of the dodecahedron induce even permutations of hair tie colors, and this provides a combinatorial identification of the Dodecahedral group with alternating group $$A_5$$.

Despite its name, this figure does not qualify as a tensegrity, which requires all forces to be purely tensional or compressive. Indeed, both the cables (hair ties) and bars (sticks) experience "bending" forces that are not purely tensional or compressive, such as where a cable pulls on the middle of a bar, or where two bars touch and push on each other. Also, the tensegrity model would not account for hair tie and stick thicknesses, which seem relevant here. These issues greatly complicate questions like the following: Is this symmetric configuration a stable equilibrium in the absence of friction? The sticks do have an unfortunate tendancy to slip slightly out of place when handled, but this could possibly be attributed to variation in hair tie strength.