Volume-Increasing Inextensional Deformations of Platonic Polyhedra

It is known that the volume of a convex polyhedron can be increased by suitable isometric deformation of its surface resulting in a non-convex shape. Deformation patterns and the associated enclosed volumes of the Platonic polyhedra were theoretically and numerically investigated by a few authors in the past. In this paper, a generic diamond-shaped folding pattern for all Platonic polyhedra is presented, optimised to achieve the maximum enclosed volumes. The numerically obtained volume increases (44.70%, 25.12%, 13.86%, 10.61%, and 4.36% for the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively) improve the existing results (44.00%, 24.62%, 13.58%, 9.72%, and 4.27%, respectively). Quasi-rigid inflatable membrane representations of such deformed polyhedra imply a significant change of structural shape due to initial inflation and subsequent compressive stresses transverse to the crease lines.