[Math] Rotational Symmetry for Rangoli Designs

Question

I am viewing these Rangoli Patterns

And it says

Which one has no line symmetry but matches four times as it turns
around?

But to me, they all have line symmetry, i.e. you fold it in half, and it is identical.

Please clarify

Answer 1

The details matter. Only one of them has line symmetry—the one on the far right. The other three all contain elements that are not mirror symmetric. The one on the far left, for example, contains eight scythe-shaped elements that turn clockwise from the points where they meet the body of the figure. A mirror reflection would result in these elements turning counterclockwise instead. That one does, however have eight-fold rotational symmetry (and therefore also has four-fold rotational symmetry).

The one on the top fails to have four-fold rotational symmetry because of the square in the center containing the diagonal strip pattern. In fact, it fails even to have two-fold ($180^\circ$) rotational symmetry because the black stripes wind up in the in the positions of the white stripes and vice versa after a $180^\circ$ turn. Therefore I believe the text preceding the figures is incorrect where it says "All of them have rotation (turning) symmetry. So you can spin them around and on the way around they match up with own starting position at least twice."

The remaining figure—the one on the bottom— has four-fold, but not eight-fold, rotational symmetry.