The Rhombic Dodecahedron

1. Build a rhombic dodecahedron using Zometool. Start by building a cube with blue struts for edges. Then build a square-based pyramid outward on each square face using yellow struts.

2. Convince your neighbour that two blue-yellow-yellow isosceles triangles that share a blue edge are in fact coplanar.

3. The short diagonals of each rhombic face coincide with the edges of what polyhedron?

4. The long diagonals of each rhombic face coincide with the edges of what polyhedron?

5. What is the ratio of the lengths of the diagonals of each rhombic face?

6. What are the dihedral angles? These are the angles between two adjacent faces.

7. What is the volume of the rhombic dodecahedron using as a unit of volume the polyhedron you found in Question 3?

8. What is the volume of the rhombic dodecahedron using as a unit of volume the polyhedron you found in Question 4?

9. Will the rhombic dodecahedron fill space? If yes, convince your neighbour. If no, what other polyhedron is needed to fill the gaps?

Philip Mallinson

PCMI July 2001