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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*> . - 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*F*_{1}and*F*_{2}as shown on the right.Describe the direct and full symmetry groups of

*F*_{1}and*F*_{2}. - Show that the group of rotations of a
*regular prism**P*_{n}=*P*×*I*⊆**R**^{3}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.

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