Course MT4521 Geometry and topology

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## Exercises 3

1. Find a condition for the product of two rotations in I(R2) to be a translation.
Prove that the product of two reflections in lines in R2 is a rotation if the lines meet and a translation otherwise.
When is the product of two glide reflections a rotation ?

2. Prove that the composite of a rotation about a point a and reflection RL in a line L is a reflection if and only if a lies on the line L and a glide reflection otherwise.
[Hint: Write the rotation as a product of reflection in a line parallel to L and another reflection.]

3. Let Ha be a half-turn (rotation by π) about a point a in the plane. Prove that any translation can be written as a product of half-turns.

Find and prove a condition for the composite Ha RL to be a reflection.

Prove that the composite Ha Hb Hc is a half-turn about the point a - b + c .

4. Prove that the group I(R) of symmetries of a line is isomorphic to the group of matrices { | a, bR, b = ±1 } under the usual matrix multiplication.
[Hint: Show that this group maps the set of points of the form to itself.]
Generalise this to find a group of matrices isomorphic to I(R2).

5. If AB is a (directed) line in R2, let GAB be the glide reflection along AB.
Prove that "gliding around a rectangle" gives the identity.
i.e. If ABCD is a rectangle, then GDA GCD GBC GAB = identity.
Show that gliding around a general quadrilateral in the plane does not in general give the identity and find a condition that ensures that gliding around it does give the identity.

6. If h, k are any symmetries, prove that hkh-1k-1 is a direct symmetry (a rotation or translation). If h and k are both direct, what can you say about hkh-1k-1 ?
Show that any direct symmetry can be written as hkh-1k-1.

7. Write down the 2 × 2 matrix which represents an anticlockwise rotation of 90° about the origin in R2 (w.r.t. the standard basis). Hence write down 3 × 3 matrices representing rotations of 90° about the x and y axes in R3.
Find the axis of the rotation got by combining these last two rotations.

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JOC February 2010