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Rotations Are Not Single Round Here

Age 14 to 16 Challenge Level:

If you haven't seen the problem 'A Roll of Patterned Paper' you may want to try that first.

In that problem I'd been finding designs for streamers - this problem follows on from that.

I noticed this about the streamers that had rotation symmetry :

In the ones I tried, if there was one centre of rotation there always seems to be another centre that worked as well.

Two centres of rotation

Your challenge :

Can you find a design with only one centre of rotation?

Or if you thought that was in fact impossible, could you say why that's true ?

Taking it further :

I noticed that if there are mirror lines across the strip, and if there is also a horizontal mirror line, then the strip always seems to have rotation symmetry.

Can that be explained?