Course MT4521 Geometry and topology

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## Exercises 2

1. Prove that an element of the orthogonal group O(3) acting on R3 is either:
1. a rotation about some axis a,
2. a reflection in some plane,
3. a rotation about an axis a followed by a reflection in a plane perpendicular to a.

Which of these three possibilities are the following ?
1. Successive reflections in two planes meeting in a line b,
2. The transformation x -x,
3. Successive rotations by π/2 about three mutually perpendicular intersecting axes,
4. Successive reflections in three mutually perpendicular planes.

Solution to question 1

2. Show that any rotation acting on the plane R2 can be written as a product of two reflections.
Hence prove that any element of O(3) can be written as a product of at most three reflections in planes in R3.

3. When we write an element of the group I(Rn) of isometries of Rn as a product T L with T a translation and L an orthogonal map, show that T and L are uniquely determined.
Show that one can also write an isometry as L' T' with L' an orthogonal map and T' a translation. 4. In the diagram on the right ABCD and XYZT are two equal squares. Find how many symmetries of R2 map one of these squares into the other and describe all such symmetries.

Solution to question 4 5. The diagram on the right is part of the logo of a pharmaceutical company and consists of two congruent rhombi.
State how many isometries of the plane map the left-hand rhombus into the other one and justify your answer.
Make a sketch to show (roughly) the centres of the rotations and the lines of the glides which map the rhombi to one another.
6. Let ABC and A'B'C' be two congruent plane triangles.
If ABC and A'B'C' both go clockwise then prove that the perpendicular bisectors of AA', BB' and CC' either meet at a point or are parallel.
If ABC goes clockwise and A'B'C' goes anticlockwise then prove that the midpoints of AA', BB' and CC' lie on a line.
[Hint: look at an isometry taking ABC to A'B'C'.]

SOLUTIONS TO WHOLE SET
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JOC February 2010