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. The direct symmetry group of the cube is isomorphic to the symmetric group S₄ . Put markings on the faces of the cube so that the direct symmetry group of the marked cube is:
   (a) C₄     (b) D₂ (generated by rotations in 3 dimensions)     (c) A₄

   The full symmetry group of the tetrahedron is isomorphic to S₄ . Give examples of symmetries corresponding to 2-cycles. How many of them are there?
   Give an example of a symmetry which corresponds to a 4-cycle and identify it as a rotatory reflection with a certain axis. How many such symmetries are there?

   Solution to question 3

   
   

4. In the group A₅ of direct symmetries of the dodecahedron, identify symmetries corresponding to the three different (non-trivial) conjugacy classes of permutations:
   (a) 5-cycles     (b) 3-cycles     (c) products of two 2-cycles.
   In each case identify the axes of the rotations and verify that you have enough axes to account for all the permutations of that shape.
   
   
   Solution to question 4

5. Democritus of Abdera (in about 400BC) knew that the volume of a pyramid or a cone satisfies V = ¹/₃ Bh where B is the area of the base and h is the height. Use this to calculate the volume of a regular tetrahedron with edge-length 2 and of a regular octahedron with edge-length 2.
   If one cuts off a tetrahedron with edge-length 1 from each corner of a tetrahedron with edge-length 2 what is left? Use this to verify your earlier calculation.

   Solution to question 5