Modeling knotology surfaces with inside-out Bassetti hinges

The square pillow (the knotology basket directed by the digon graph) realized with inside-out Bassetti rubber band hinges. The bold black edge coincides with the diagonal hinge or crease.

The preceding post showed how knotology surfaces can be assembled from congruent isosceles right triangles. Because every hypotenuse edge in a knotology surface must pair up with another hypotenuse edge, knotology surfaces can also be seen as being composed of squares that are hinged, creased, or folded along a designated diagonal. This is probably the preferred way to model these surfaces because the number of pieces needed is reduced by half.

Square tiles with appropriately color-marked corners (as illustrated in the images here) will be called bi-squares, because they force construction of a square-faced, bipartite crack-graph—the graph made evident by separating the square tiles slightly—when the bi-squares are assembled with their corner colors matching.

A cardstock bi-square that has been cut and notched for attachment with Bassetti rubber band hinges. The overall size is 3" x 3". The creased internal square is 2-3/16" x 2-3/16". The notches were cut straight across with scissors after the flaps were folded flat. This size works well with the widely available Rainbow Loom elastic bands.

The same card stock bi-square as above with the ears folded flat.

In 1959, architect Frederick F. Bassetti invented a good way to hinge polygonal cards together with elastic bands to form polyhedral shapes. The main drawback to making a model with Bassetti hinging is that the outside of the finished model wears protruding flaps and rubber bands. I find that it is possible, though a bit more time consuming, to work Bassetti hinges inside-out so that the flaps and elastic bands are hidden inside the model.

The bi-squares illustrated here were made from 3" x 3" pieces of 65-lb (176 g/m2) card stock, that was creased along the edges of a 2-3/16" square diamond and also along one of its diagonals. After the "ears" were folded flat, the corners were truncated with straight scissor cuts leaving clean-cornered rectangular notches that hold Rainbow Loom elastic bands fairly securely. Working carefully, one elastic band engagement at a time, and with the help of a small gauge (2-1/4 mm) crochet hook, I find it is possible to assemble the Bassetti hinges inside out. In the end, there is just a peek of elastic band showing at the corners. It also helps to pre-curl the ears a bit so that they can pass around behind the elastic band.

View from the inside of a bi-square with two pre-curled ears. 

Stage in the closure of a particularly difficult inside-out Bassetti hinge.

Using a narrow gauge crochet hook to properly set the elastic band in an inside-out Bassetti hinge.

Close-up view of the inside-out Bassetti hinges in the smallest knotology surface: the triangular pillow (the surface directed by the line segment graph.)