In February, 2008, Walt van Ballegooijen (in the Netherlands) sent me photos of his lovely model inspired by Anton Hanegraaf's 7-piece hinged dissection of a 2x1x1 rectangular block to a cube. (I had described Anton's dissection on pages 198-199 of Hinged Dissections and also on pages 236-238 of Dissections: Plane & Fancy.)
Walt derived his dissection by taking eight copies of Anton's dissection, producing mirror images in each of the three dimensions, and gluing appropriate pieces together. The resulting linkage has 24 pieces. Below we see the 2x1x1 rectangular block on the left, the cube on the right, and below them an intermediate block of dimensions r2 x r x 1, where r is the cube root of 2.
Below the linkage is opened to show off all eight cyclic hingings that are present. (Anton's 7-piece hinged dissection had just two cyclic hingings.) Note that the plastic flashlight in the middle is used to prop the linkage open.
There are a surprising number of intermediate positions for the linkage. The one on the left below is halfway between the 2x1x1 rectangular block and the block of dimensions r2 x r x 1. The intermediate position on the right below carries some of the pieces further towards their final position in the cube.
This model is all the more impressive when you realize that it is over 10 kilograms (22 pounds)! Walt has produced 2 videos of the linkage in action, which are displayed on my video gallery.
Photos supplied through the courtesy of Walt van Ballegooijen,
who retains all rights to them.
All else is copyright 2008, Greg N. Frederickson.
Last updated February 13, 2008.