Unstable Polyhedral Structures

Summary

Editors' Note: Michael Goldberg was trained as an electrical engineer at the University of Pennsylvania and did graduate work at George Washington University. He then worked for 40 years as a naval engineer. Since 1922 his name has appeared as a problem solver in the American Mathematical Monthly (and somewhat later in Mathematics Magazine). He wrote extensively on polyhedra, dissection problems, packing problems and linkage mechanisms. A striking paper of his on the isoperimetric problem for polyhedra (a problem still not completely solved) appeared in 1935 in the Tôhoku Mathematical Journal, 40 (1935), 226–36. His first article in Mathematics Magazine appeared in 1942.

For more problems of the type in this paper see Jack E. Graver's Counting on Frameworks/Mathematics To Aid the Design of Rigid Structures (MAA, 2001).

It was shown by Cauchy [1] and Dehn [2] that a convex polyhedron made of rigid plates which are hinged at their edges is a rigid structure. However, if the structure is not convex, but still simply connected, there are several possibilities. It may be any of the following cases:

1. (a) rigid,

2. (b) infinitesimally movable (shaky),

3. (c) two or more stable forms (multi-stable),

4. (d) a continuously movable linkage.

Shaky polyhedra

The regular icosahedron of twenty triangular faces is convex and rigid. If six pairs of faces with an edge in common are replaced by other pairs of isosceles faces with their edges in common at right angles to the original common edge to make a non-convex icosahedron, we may obtain the orthogonal icosahedron of Jessen [3], shown in Figure 1.