Turn Two Octahedra into a Rhombic Dodecahedron

This model illustrates an interesting connection between an octahedron and a rhombic dodecahedron (a twelve-sided polygon that has rhombi for faces). If you place triangular pyramids on each face of an octahedron, you create a rhombic dodecahedron. What's more, four of the triangular pyramids can be put together to make a tetrahedron.

Materials

• Card stock
  
• Glue
  
• Re-stickable (not permanent) glue stick
  

Steps

1. Cut out eight copies of the piece on the top half of the pattern.
   
2. Glue them together to create eight triangular pyramids.
   
3. Apply the re-stickable glue stick to the faces of the pyramids.
   
4. (if you don't have re-stickable glue stick) Tape the bases together as shown. Leave a gap of 1 - 2mm (1/16 of an inch) between the pyramids depending on the thickness of the paper you are using. You can fold these pyramids up to form two connected tetrahedra.
   
5. Cut out one copy of the piece at the bottom of the pattern.
   
6. Fold and glue it to make an octahedron.
   
7. Attach the pyramids to the octahedron. (If you taped the pyramids you can turn the tetrahedra made of pyramids "inside out" and drape it over the octahedron so each pyramid base rests on the face of the octahedron.) This forms a rhombic dodecahedron.
   

Notes

• You can think of a rhombic dodecahedron as a stellation of the cube. Or conversely, if you connect the long diagonals of each rhombus in a rhombic dodecahedron, you form an octahedron.
  

References

• (Loeb 1991)