What symmetry is — the ladder
-
What a symmetry actually is
Not a property of a shape but a motion that leaves it alone. Once symmetry is a verb rather than an adjective, everything else in the subject follows — including why there can only ever be seventeen wallpapers.
-
The four motions of the plane
Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.
-
Why it is a group and not a list
The symmetries of a pattern cannot be chosen independently. Do two of them in succession and the result is forced to be a third, which is why there is no eighteenth wallpaper for anybody to invent.
-
The orbit is the pattern
A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out.
-
The same symmetry, somewhere else
Two mirrors in a pattern can be the same symmetry or two different ones, and looking will not settle it. Conjugation is the operation that decides, and it turns an intuition about sameness into arithmetic.