Yoshimoto Cube #1, Double Star Flexicube, Shensei Mystery, are some of the names for this geometric wonder. At one stage of folding, the outside form is a cube which opens to reveal a stellated rhombic dodecahedron filling the inter volume of the cube. This can be removed, manipulated to form a cube and the original cube folded into another dodecahedron which fits back inside. There are many online references available.

The assembled Yoshimoto cube #1 consists of 8 smaller cubes and can show 6 separate images on the overall cube facets. Unfolding and refolding leads to another cube with an additional 6 outside facets, so 12 images can be used for viewing. Additional artwork can be put on the dodecahedron surfaces with triangular facets which nest inside the cube.

If you have Snapmarks on your Glowforge, they greatly simplify scoring and cutting the many pieces needed for construction. I’ve tried two construction routes. The first makes a pyramid unit having a square base with an isosceles triangle with apex angle ArcCos(1/Sqrt[3]) . Three more identical triangles are joined one to another to complete the unit template with tabs for gluing.

Three of these units are glued together to form a half-cube. There are 8 full cubes needed, making at total of 48 unit pyramids. These must be connected by hinges to form the overall cube with specific degrees of freedom for manipulation.

The second construction route uses a template which forms a whole half-cube, thereby requiring only 16 pieces. https://www.youtube.com/watch?v=nM3lm37OCrQ&feature=share

gives a tutorial on folding using the second construction route. It also shows the hinge locations…

The first construction route seems to give better nesting for me.

A basic problem is that the nesting of the completed pieces works theoretically for zero thickness material. Unit squares about one inch on a side have not worked well for me. Going to 1.5 inch unit side helps. I have also tried making the hinges out of flexible material (i.e. silkspan paper) in hopes that the hinge spacing could adapt to being larger/smaller as needed.

An elastic hinge material would be ideal, if it could be found. Any suggestions?