Course MT3818 Topics in Geometry

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## 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 ?
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 TJ(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 Sd(F) = D4 and S(F) = D4 × < J > .
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.

4. Show that the group of rotations of a regular prism Pn = P × IR3 with P a regular polygon and I the unit interval, is the dihedral group Dn.
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.

SOLUTIONS TO WHOLE SET
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JOC March 2003