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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?

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Because it doesn't have two independent symmetries. – Qiaochu Yuan Jul 11 '11 at 2:13
@Qiaochu: Could you elaborate? p. 157 defines glide as "obtained by reflecting about a line l and then translating by a nonzero vector a parallel to l". This sounds like "two independent symmetries"? – Xodarap Jul 11 '11 at 2:19
No, it sounds like one symmetry depending on two operations. The reflection is not an independent symmetry; just performing the reflection doesn't leave the figure invariant. – joriki Jul 11 '11 at 2:27
It's good style, when you edit the question such that (a part of) a previously correct answer becomes incorrect, to indicate that edit, since otherwise it makes it look as if the answer was wrong. (In the present case, you removed the quote, and now my sentence about changing the context of the quote no longer made sense, but would have remained if I hadn't happened to notice your edit.) – joriki Jul 11 '11 at 2:29
@Joriki: thanks, have tried to update the question. – Xodarap Jul 11 '11 at 2:35
up vote 5 down vote accepted

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.

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Is this correct: all objects that have rotation + translation symmetry also have both rotational and translational symmetry. But not all objects with glide symmetry have reflection + translational symmetry. This is why "glide" counts as its own but translation + rotation does not. – Xodarap Jul 11 '11 at 2:43
No, that's not correct. Rotation + translation = rotation, so every system with rotation also has rotation + translation, but it doesn't necessarily have translation. The reason why rotation + translation isn't a type of its own is that it's just rotation. Same with rotation + reflection = reflection. The only combination that isn't equal to one of its ingredients is translation + reflection. – joriki Jul 11 '11 at 4:49

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:


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.


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.

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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: 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 $(1,0)$.

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