Exercises 5

1. Recall that one makes the dual of a polyhedron by putting a vertex at the centre of each face and joining vertices by edges if the corresponding faces meet in an edge.
   What are the duals of the rhombic dodecahedron and of the truncated octahedron ?

   Solution to question 1
   
   
   
   

2. Identify the dual of the stella octangula and hence find its direct and full symmetry groups.
   If we regard the stella octangula as the union of two tetrahedra T ∪ J(T) and colour T white and J(T) black, what are the symmetry groups then ?
   Describe how you would colour the stella octangula to get a figure F with S_d(F) = D₄ and S(F) = D₄ × < J > .

   Solution to question 2
   
   
   

3. A figure F consists of a triangular prism with three square faces and two equiangular triangular faces.

   Describe the direct symmetry group of F and the full symmetry group of F.
   
   
   Two copies of F are put together in two different ways to make figures F₁ and F₂ as shown on the right.

   Describe the direct and full symmetry groups of F₁ and F₂.

   Solution to question 3
   
   
   

4. Show that the group of rotations of a regular prism P_n = P × I ⊆ R³ with P a regular polygon and I the unit interval, is the dihedral group D_n.
   What is its full symmetry group ?

   A square antiprism (with two parallel square faces and eight isosceles triangles for the others) is as shown on the right:
   Find its direct and full symmetry groups.

   Solution to question 4