One fine spring morning, at some point, I was simply tearing a few sheets of paper towel off a fresh roll. As the sheets came off and the roll rotated about the vertical axle of the holder (Figure 1), I lazily pondered the geometry lurking in the situation. I realised that sometimes I would let the roll rotate as the unwound part moved straight until tearing, while at other times, I might hold the roll in place and unwind the sheets around the roll. More often, it would be some spontaneous combination of the two modes of unwinding.