Chiral Icosahedral Hinge Elastegrity’s Geometry of Motion

G4G13

Chiral Icosahedral Hinge Elastegrity’s Geometry of Motion

Eleftherios Pavlides

Art

The Chiral Icosahedral Hinge Elastegrity resulted from a Bauhaus paper folding exercise, that asks material and structure to dictate form. The key new object obtained in 1982 involved cutting slits into folded pieces of paper and weaving them into 8 irregular isosceles tetrahedra, attached along 24 edges, to 12 right triangles, that in pairs form elastic hinges, creating an icosahedral shape held together by elastic forces. The chiral icosahedral hinge elastegrity has noteworthy physical and geometric properties.

At G4G12, shape-shifting through further folding of the hinge elastegrity was presented. It led to a number of familiar geometric objects, as well as some new ones. It can flatten into a multiply covered square, morph into shapes with the vertices of each of the Platonic shapes, model the hypercube, as well as morph into new figures with the vertices of figures of congruent faces that are not regular polygonal regions. With the help of co-presenter professor Thomas Banchoff, we generalized a unique new monododecahedron that had been obtained through folding, into a family of monododecahedra using analytic geometry.

For G4G13 the proposal is to present the Geometry Of Motion of the Chiral Icosahedral Elastegrity. At rest the icosahedral hinge elastegrity has 6 openings framed by the 12 hinged triangles, that form gates that open and close, with a set of orthogonal axes going through their center and the structure’s center. As the structure contracts into an octahedron, the gates close, and the tetrahedra pivot so that the 3 orthogonal axes extend thought the vertices of the octahedron. The gates open maximally when the structure expands back into a regular icosahedron. As the structure gyrates expanding into a cuboctahedron the slits close becoming 6 diagonals of the 6 squares of the cuboctahedron.

The asymmetrical tetrahedra move along a second set of 4 axes that gyrate around the center of the structure. When two tetrahedra are pressed together, along any of the 4 axes, all 12 hinges are activated simultaneously, contracting the 8 asymmetrical tetrahedra chirally, spinning isometrically in unison along the 4 rotating axes, moving towards the center of the structure, into an octahedron. When any two asymmetrical tetrahedra are pulled away along one of the 4 axes, the hinges activate the entire structure extending the 8 tetrahedra spinning in unison with reverse chirality, along the 4 axes that rotate in reverse direction, pivoting around the structure’s center, into a cubeoctahedron.

When external forces are removed, elastic forces in the hinges return the structure into its original regular icosahedral shape.

G13-095