# Ethan Bolker

Professor of Mathematics Emeritus

UMass Boston

Newton, Massachusetts

I've built mathematical models ever since I read Hugo Steinhaus's "Mathematical Snapshots" more than 60 years ago. Over time I graduated from crude cardboard platonic polyhedra and soma cubes to some constructions friends have graciously called art.

K(5,5)

40 x 40 x 40 cm

Wood and brass

2017

Counting suggests that the complete bipartite graph K(5,5) might be rigid even though it has no triangles. Ben Roth and I proved that in "When is a bipartite graph a rigid framework?" (Pacific Journal of Mathematics, 90, 1981). The graph has one more edge than the number of degrees of freedom, so can be realized as a tensegrity structure. In this realization four of the five vertices from each set form a tetrahedron with the fifth vertex near its center. The tetrahedra overlap so that the center vertex of each is inside the other. Eight struts join each center vertex to the four vertices of the other tetrahedron. The other 17 edges are cables.