### Group Theory 3: Expanding on Symmetry

Yesterday, I talked about the basic intuitive idea of what symmetry means. Today, I'm going to try to take it a bit further. I will eventually get to the point of explaining how groups express this; but this stuff really fried my brain the first time I saw it, so I'm trying to take small steps, with as much intuition as I can muster at each step.

Intuitively, when we talk about symmetry, we think about reflection - like we could pick out a line, or a plane, and switch things around the sides of it. In normal english usage, that's all that symmetry means: reflective symmetry around a line or plane. The standard real number multiplication symmetry fits that nicely: it's reflective symmetry around the line "x=y".

In math, and in group theory in particular (since group theory is what defines symmetry), symmetry means something more than that. Mathematically, symmetry means something more like "immunity to transformation" - where the exact transformation could be almost anything. Immunity to transformation means that after the transformation is applied, there is no way to distinguish between before and after the transformation.

The familiar symmetry, reflective symmetry, means that if you mirror-reflect what's on each side of the line, and switch it to the other side of the line, you'll end up with something indistinguishable from what you started with.

There are other kinds of symmetry, which mean that you can do some other kind of transformation without having any distinguishable effect.

Translational symmetry means that something can be moved in some way without having any distinguishable effect; rotational symmetry means that something can be rotated in some way without having and distinguishable effect.

For example, if you have an infinite sheet of graph paper, and you move it one square in any direction, you can't tell that you moved it: it is symmetric with respect to translation by integral unit translations.

If you have a pentagon, you can reflect it five along five different lines:

You can also rotate it; it has rotational symmetry around the angles produced by the five reflective symmetry lines:

Finally, a particularly cool example: this Escher image has six reflective symmetries; 3 rotational symmetries; translational symmetries in 8 directions,

and a 3-way reflective symmetry (meaning you can divide it into three sections and mirror-swap each section separately). Instead of marring such a beautiful image, I'll leave it alone and let you puzzle those out.

Intuitively, when we talk about symmetry, we think about reflection - like we could pick out a line, or a plane, and switch things around the sides of it. In normal english usage, that's all that symmetry means: reflective symmetry around a line or plane. The standard real number multiplication symmetry fits that nicely: it's reflective symmetry around the line "x=y".

In math, and in group theory in particular (since group theory is what defines symmetry), symmetry means something more than that. Mathematically, symmetry means something more like "immunity to transformation" - where the exact transformation could be almost anything. Immunity to transformation means that after the transformation is applied, there is no way to distinguish between before and after the transformation.

The familiar symmetry, reflective symmetry, means that if you mirror-reflect what's on each side of the line, and switch it to the other side of the line, you'll end up with something indistinguishable from what you started with.

There are other kinds of symmetry, which mean that you can do some other kind of transformation without having any distinguishable effect.

Translational symmetry means that something can be moved in some way without having any distinguishable effect; rotational symmetry means that something can be rotated in some way without having and distinguishable effect.

For example, if you have an infinite sheet of graph paper, and you move it one square in any direction, you can't tell that you moved it: it is symmetric with respect to translation by integral unit translations.

If you have a pentagon, you can reflect it five along five different lines:

You can also rotate it; it has rotational symmetry around the angles produced by the five reflective symmetry lines:

Finally, a particularly cool example: this Escher image has six reflective symmetries; 3 rotational symmetries; translational symmetries in 8 directions,

and a 3-way reflective symmetry (meaning you can divide it into three sections and mirror-swap each section separately). Instead of marring such a beautiful image, I'll leave it alone and let you puzzle those out.

## 6 Comments:

In talking about the Escher drawing, you say it has "8 transformational symmetries". I'm assuming you mean "translational symmetries".... and I'm assuming we're imagining this to tile the plane.... I think there are a lot more than 8 directions for translational symmetry though....

By Anonymous, at 12:03 PM

Thanks, that was a type, which I've corrected.

True, there are more translational symmetries; but they're combinations of the eight basic ones. There are translational symmetries in the vertical direction; horizontal, and two diagonals. The other translation symmetries are all combinations of those.

By MarkCC, at 12:19 PM

Yes, I can type :-). That was supposed to say "that was a typo". I believe that would be called a meta-typo.

Someone please shoot me. :-)

By MarkCC, at 12:21 PM

I'm still thinking you only need 4 of those 8 though. You can make the "up" translation from a "northeast, northeast, west" combination. And, "west, southwest" is equivalent to "southeast".

I'm thinking { east, west, northeast, southwest } covers it.

By Patrick Stein, at 4:06 PM

patrick:

I think you're close, but not quite right; I don't think that "west,southwest" is equivalent to "east". The point of the symmetry is that you can move the graph in a direction without changing anything; west,southwest may do something similar to east, but it's not the same.

But your basic point, that the 8 symmetries I point out are not actually the minimal set of symmetries of the Escher tesselation is absolutely right.

By MarkCC, at 8:17 PM

Yep... my mistake... I meant "east, southwest" is equivalent to "southeast".

By Patrick Stein, at 12:07 AM

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