Transforming cube/octahedron linkage

This is a linkage that transforms shape. Two of its states are a cube and an octahedron, passing through a cuboctahedron state. It also has other shapes that it can fall into, which makes it a bit of a puzzle to work out how to make it go to the state you want.

To make this you will need:

• 24 laser cut links (I used 3mm MDF. The pdf file will cut six links at once.)

• 24 3D printed connectors

• 60 M3 bolts (at least 10mm)

• 60 M3 nyloc nuts

First, connect the links together in pairs with 12 bolts and nuts to make 12 "X" shapes, hinged at their centers. Each X will be oriented with its bolt pointing towards the center of the cube or octahedron, so there is an inner and an outer link in each X.  Each end of each link connects (with another bolt and nut) to one end of a connector. The connectors have an inner and an outer end: connect the inner end to an outer link and the outer end to an inner link.

Beware: it is very easy to get confused and connect the wrong end of a connector to a link, so it goes further up or further down than it should. There are only supposed to be two "heights" of the links above the center of the polyhedron.

When you're done, the links and connectors should form into four hexagons. This is easiest to see in the "midway" photograph.

The nyloc nuts are necessary to allow the bolts to act as axles. Without the nylon inserts either the nuts will work themselves loose or you will have to tighten them to the extent that the hinges will not turn.

This structure was independently (and three months before me!) discovered by Yuan Liao, Gökhan Kiper, and Sudarshan Krishnan. See their paper "Reconfigurable polyhedral mechanisms using scissor-like elements with cantellation transformation between dual geometries" in Mechanism and Machine Theory, Volume 215 (2025) 106187 (https://doi.org/10.1016/j.mechmachtheory.2025.106187).

Sabetta Matsumoto and I also made a version of the design with fabric panels to show the faces of the cube and the octahedron. For this version, use the connectors with thread holes. The triangle_template and square_template files can be printed and cut out to use as templates to cut out the fabric panels. These have a centimeter or so of margin that can be folded over a ribbon (for strength) and stitched.

Thanks to Edmund Harriss for taking some of these photos (the version without fabric), and thanks to the FabLab at Sorbonne University where I prototyped this model.