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- 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*? - 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*) =*D*_{4}and*S*(*F*) =*D*_{4}× <*J*> . - The direct symmetry group of the cube is isomorphic to the symmetric group
*S*_{4}. Put markings on the faces of the cube so that the direct symmetry group of the marked cube is:

(a)*C*_{4}(b)*D*_{2}(generated by rotations in 3 dimensions) (c)*A*_{4}The

*full*symmetry group of the tetrahedron is isomorphic to*S*_{4}. 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?

- In the group
*A*_{5}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 *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.

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