Why is "glide symmetry" its own type?

Question

Artin's Algebra pages 155 & 156 list the types of symmetry of a plane figure as:

1. Reflective
2. Rotational
3. Translational
4. Glide

He then goes on to say "Figures such as wallpaper patterns may have two talk about other figures having combinations of independent" symmetries. EDIT: He says "... having combinations of independent translational symmetries". See Joriki's answer.

Why doesn't glide symmetry count as having two independent (reflection + translation) symmetries? If we're going to count combinations, why not have rotation + translation etc. as their own types too?

Answer 1

The book says "Figures such as wallpaper patterns may have two independent translational symmetries".

The example for that sentence is a figure with two independent translational symmetries in two different directions. Each of those symmetries is a symmetry in its own right.

In the case of glide symmetry, there isn't a reflection symmetry in its own right; only the combination of a reflection and a translation leaves the figure invariant.

In your last example, rotation + translation, there's nothing new, since that can be expressed as a rotation about a shifted axis.

Answer 2

It's possible to have glide symmetry without having reflection symmetry. I think what's meant by two independent symmetries is that a pattern can simultaneously have reflection symmetry and translation symmetry, such as an infinite strip of Cs:

CCCCCCCCCCCCCCCCCCCCCCCCCCCCC

This has reflection symmetry over a horizontal line and translation symmetry by a horizontal vector of magnitude an integral multiple of the width of the C character.

A pattern like the one below, however, has glide symmetry and translation symmetry, but not reflection symmetry.

bpbpbpbpbpbpbpbpbpbpbpbpb

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edit: Given the various discussions in the comments, I think it might be useful to point out that all non-identity plane isometries can be expressed as a reflection, a composite of two reflections, or a composite of three reflections. An isometry that is a single reflection is just a reflection; if a figure is invariant under such a transformation, we say it has reflection symmetry. An isometry that is a composite of two reflections over parallel lines is a translation; if a figure is invariant under such a transformation, we say it has translation symmetry. An isometry that is a composite of two reflections over intersecting lines is a rotation; if a figure is invariant under such a transformation, we say it has rotational symmetry. An isometry that is a composite of three reflections (and cannot be expressed as a single reflection) is a glide reflection; if a figure is invariant under such a transformation, we say it has glide-reflection symmetry. Any composition of more than three reflections can be expressed in terms of fewer reflections.

Answer 3

Perhaps some more examples will be useful.

Look at Artin, fig. 1.4 on p. 156, depicting a stylized stem with leaves on alternating sides. This figure has glide symmetry, because a reflection in the horizontal line (the "stem") followed by a horizontal translation (through the distance between two consecutive leaves) carries it back to itself. Note that it does not have any reflective symmetry, i.e. there is no line over which reflection will carry the figure back to itself. The point is that, although the glide is a composition of a reflection and a translation, neither of these motions by itself carries the figure back to itself. After the reflection alone, the leaves are on the wrong sides of the stem; same with the translation alone.

Notice the same is true of Isaac's ...bpbpbpbp... example. A reflection in the horizontal axis turns everything upside down. Thus it is not a symmetry of the figure because it changes the figure. (If you blink while I perform the reflection, you will still know something is different.) However a glide, i.e. the composition of this reflection with a horizontal translation the length of one letter, will carry each b to the next p and vice versa. This is a symmetry because it carries the figure back to itself. (If you blink while I perform the glide, you will not notice that the figure has changed.)

One more (beautiful) example: M.C.Escher's Horsemen: http://www.tessellations.org/eschergallery18.shtml There is a glide (reflection over a vertical axis followed by a vertical translation) that carries the red horsemen to the white ones and vice versa. However, there is no reflection that by itself carries horsemen to horsemen.

As for why there isn't a separate name for a translation composed with a rotation, joriki's answer is the reason: (rotation + translation) is a rotation about a different point. To illustrate, consider rotating the plane 180 degrees about the origin and then translating it 2 units to the right. The rotation sends $(x,y)$ to $(-x,-y)$ and the translation sends this to $(2-x,-y)$. Notice that the composed map $(x,y) \mapsto (2-x,-y)$ is the 180 degree rotation about $(1,0)$.