1 Attachment(s)

Finding symmetries of a triangular prism

Attachment 27686

The above prism has three identical rectangular faces and equilateral triangles at the top and base. The positions of the faces of the prism have been numbered so that we may represent the elements of the group G of all symmetries of the prism as permutations of the set {1,2,3,4,5}.

Write down all the symmetries of the prism in cycle form as permutations of the set {1,2,3,4,5} and describe each symmetry geometrically.

Re: Finding symmetries of a triangular prism

Quote:

Originally Posted by

**Jojo55** Attachment 27686 The above prism has three identical rectangular faces and equilateral triangles at the top and base. The positions of the faces of the prism have been numbered so that we may represent the elements of the group G of all symmetries of the prism as permutations of the set {1,2,3,4,5}. Write down all the symmetries of the prism in cycle form as permutations of the set {1,2,3,4,5} and describe each symmetry geometrically.

Here's just one: $\displaystyle 2 \rightarrow 3, 3 \rightarrow 5, 5 \rightarrow 2, 1 \rightarrow 1, 4 \rightarrow 4$

i.e. counter-clockwise rotation on z-axis (2 3 5)(1)(4) = (2 3 5)

Can you find the others?

Re: Finding symmetries of a triangular prism

Quote:

Originally Posted by

**majamin** Here's just one: $\displaystyle 2 \rightarrow 3, 3 \rightarrow 5, 5 \rightarrow 2, 1 \rightarrow 1, 4 \rightarrow 4$

i.e. counter-clockwise rotation on z-axis (2 3 5)(1)(4) = (2 3 5)

Can you find the others?

Thanks.

I've got eight with that one included.

The identity

1 counter- clockwise rotation on z-axis = (235)

2 counter- clockwise rotations on z-axis = (253)

Rotation by pi (where 1 and 4 changes places) = (14)(23)

Reflections I get

(25) cuts through the middle of 3, 1 and 4

(35) cuts through the middle of 2, 1 and 4

(23) cuts through the middle of 5, 1 and 4

(14) is a reflection in the horizontal axis

Re: Finding symmetries of a triangular prism

Very naughty this is an Open University T.M.A. assessment question-Don't get caught!