In my readings of Mirman (1995), "Group Theory: An Intuitive Approach", on p.35 he asks me to consider a so-called "water group" that has 4 transformations. I'll list them for completeness, but I'm only concerned about two of them.
- The identity element
- Reflection in the plane element
- Rotation by $\pi/2$ about a line through "O" perpendicular to the line joining the two "H"s
- Reflection through the line described in 3.
Now for some context. This book is primarily a book about group theory, but the author likes to use examples from physics in order to provide examples that yield a concrete picture of what the formalism can and can't describe.
Leading up to the example of the water group, I calculated the Cayley table for the symmetry group $S_3$; the dihedral group $D_3$; and then the table for an isosceles triangle as an example of symmetry breaking from the equilateral triangle in the prior example. All of that I get.
What I don't understand is how a quarter-turn rotation leaves the figure invariant in the water example. I tried it from two different angles:
using cylindrical coordinates, there is only one possibility: namely a 1/4 of a full turn in the plane of the figure which yields a picture that is different than what you started with.
or, using spherical coordinates, you can perform the rotation out of the plane but that leaves you with a top-down view and the three "atoms" following in a straight line which isn't what you started with, either.
So, I'll ask again: is there a way to do a 1/4 turn of an isosceles triangle that leaves the original shape unchanged?