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

**Direct symmetries**These are of the form

*f*=*T*_{a}∘*L*with*L*∈*SO*(3).

They can be of three kinds:

- A
*rotation*(about any line in**R**^{3}),

- A
*translation*,

- 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 T_{a}∘ L is a screw unless**a**and**b**are perpendicular.**Proof**

Write=*a**λ*+*b***C**with**C**perpendicular to. Then*b**f*=*T*_{λb}∘ (*T*_{C}∘*L*).

Now*T*_{C}∘*L*is rotation about some axis parallel tosince we are reduced to the two-dimensional situation in the plane perpendicular to*b*.*b*

Hence*f*is a screw translation unless*λ*= 0 in which case it is a rotation.

- A
**Opposite symmetries**These are of the form

*f*=*T*_{a}∘*L*with*L*∈*O*(3) -*SO*(3).

They can be of three kinds:

- A
*reflection*(about any plane in**R**^{3}),

- A
*glide reflection*: reflection in a plane*P*followed by a translation parallel to*P*,

- 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 R*_{P}be reflection in the plane P. Then T_{a}∘ R_{P}is reflection in a plane if**a**is perpendicular to P and is a glide reflection otherwise.**Proof**

Letbe a vector perpendicular to*b**P*. Write=*a**λ**B*+**C**with**C**a vector parallel to*P*.

Then*f*=*T*_{a}∘*R*_{P}=*T*_{C}∘ (*T*_{λb}∘*R*_{P}) and*T*_{λb}∘*R*_{P}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 T*_{a}∘ L is also a rotatory reflection.**Proof**

Let*L*=*R*_{P}∘*Rot*_{b}where the vectoris perpendicular to the plane*b**P*. Write=*a**λ*+*b***C**with**C**∈*P*.

Then*f*=*T*_{a}∘*L*= (*T*_{λb}∘*R*_{P}) ∘ (*T*_{C}∘*Rot*_{b}) since*T*_{C}commutes with*R*_{P}.

Since*λ*is perpendicular to*b**P*, the first bracket is reflection in a plane parallel to*P*.

Since**C**is perpendicular to, the second bracket is rotation about an axis parallel to*b*.*b*

Hence this is a rotatory reflection as required.

This completes the classification of the opposite symmetries.

- A

- We get the following summary about isometries of
**R**^{3}.

**Fixed point****Direct symmetry****Opposite symmetry**

None Translation *or*ScrewGlide reflection

dim 0 Rotatory reflection

dim 1 Rotation

dim 2 Reflection

dim 3 Identity

- 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.

- Reflection in a plane
- A products of two reflections is a translation if the planes are parallel and a rotation (about the line where the planes meet) otherwise.
- 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.
- To get a screw translation we need a product of four reflections.

- Reflection in a plane
- In fact any symmetry of
**R**^{n}can be written as a product of at most*n*+ 1 reflections in (*n*- 1)-dimensional hyperplanes.

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