Turn Two Cubes into a Rhombic Dodecahedron II

This model illustrates an interesting connection between a cube and a rhombic dodecahedron (a twelve-sided polygon that has rhombi for faces). You can essentially turn a cube inside out and drape it over another cube to form a rhombic dodecahedron.

Steps

1. Cut out six copies of the small pattern pieces.
   
2. Fold and glue them to create six square pyramids.
   
3. Tape the square bases of the pyramids together so they form a shape of a cross. Leave a gap of 1 - 2mm (1/16 of an inch) between the pyramids depending on the thickness of the paper you are using. The gap helps the pyramids fold up to form a cube.
   
4. Cut out one copy of the net of a cube, fold it up, and glue it.
   
5. Turn the cube made of pyramids "inside out" and drape it over the cube so each pyramid base rests on the face of the solid cube. The resulting solid is a rhombic dodecahedron.
   

Notes

• You can think of a rhombic dodecahedron as a stellation of the cube. Each pyramid in this model extends the face of a cube.
  
• The acute angles in the rhombi that make up the dodecahedron have a measure that is the inverse cosine of 1/3. This is also the angle formed at the intersection of two body diagonals of a cube so it shows up in the pyramids.
  

References

• (Loeb 1991)