Course MT3818 Topics in Geometry

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## Isometries in 3 dimensions

As in the last example, we have direct and opposite symmetries.

1. Direct symmetries

These are of the form f = Ta L with L SO(3).
They can be of three kinds:

1. A rotation (about any line in R3),
2. A translation,
3. A screw translation (or screw): a rotation about some line followed by a translation parallel to that line.

Proof
If L is the identity, then f is a translation, otherwise the following result handles things.

Lemma
Let L be a rotation about the line containing the vector b. Then Ta L is a screw unless a and b are perpendicular.

Proof
Write a = b + C with C perpendicular to b. Then f = Tb (TC L).
Now TC L is rotation about some axis parallel to b since we are reduced to the two-dimensional situation in the plane perpendicular to b.
Hence f is a screw translation unless = 0 in which case it is a rotation.

2. Opposite symmetries

These are of the form f = Ta L with L O(3) - SO(3).
They can be of three kinds:

1. A reflection (about any plane in R3),
2. A glide reflection: reflection in a plane P followed by a translation parallel to P,
3. A rotatory reflection: reflection in a plane P followed by a rotation about an axis perpendicular to P.

Proof
An element of O(3) - SO(3) is either reflection a plane or a rotatory reflection. The first of these two possibilities is handled by:

Lemma
Let RP be reflection in the plane P. Then Ta RP is reflection in a plane if a is perpendicular to P and is a glide reflection otherwise.

Proof
Let b be a vector perpendicular to P. Write a = B + C with C a vector parallel to P.
Then f = Ta RP = TC (Tb RP) and Tb RP is reflection in a plane parallel to P since this is essentially the two-dimensional situation considered ealier.
So if the vector C = 0 then f is a reflection and otherwise it is a glide reflection.

The second possibility is handled by:

Lemma
Let L be a rotatory reflection. Then Ta L is also a rotatory reflection.

Proof
Let L = RP Rotb where the vector b is perpendicular to the plane P. Write a = b + C with C P.
Then f = Ta L = (Tb RP) (TC Rotb) since TC commutes with RP.
Since b is perpendicular to P, the first bracket is reflection in a plane parallel to P.
Since C is perpendicular to b, the second bracket is rotation about an axis parallel to b.
Hence this is a rotatory reflection as required.

This completes the classification of the opposite symmetries.

Remarks

1. We get the following summary about isometries of R3.
 Fixed point Direct symmetry Opposite symmetry None Translation or Screw Glide reflection dim 0 Rotatory reflection dim 1 Rotation dim 2 Reflection dim 3 Identity

2. One can also classify the different kinds of symmetry by looking at how they can be written as products of reflections in planes. Note that a product of an even number of reflections is a direct symmetry, a product of an odd number is opposite.
1. Reflection in a plane

2. A products of two reflections is a translation if the planes are parallel and a rotation (about the line where the planes meet) otherwise.

3. A product of three reflections is a rotatory reflection if the three planes meet in a point. If two of the planes are parallel and meet the third or if the three planes are parallel, then we get a glide reflection.

4. To get a screw translation we need a product of four reflections.

3. In fact any symmetry of Rn can be written as a product of at most n + 1 reflections in (n - 1)-dimensional hyperplanes.

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JOC February 2003