With a roll of two ply toilet paper, when you tear off the sheets the perforations line up. However, if you take the top layer and wind it back around the roll and then try to tear off the sheets the perforations do not line up at all.
Mr Moglet agues that :
*The extent to which the outside edge of the top layer sheet will exactly line up with the outside edge of the sheet in the layer below is a function of the ratio of the length of the individual sheet to the circumference of the roll itself.
If the circumference is exactly divisible by the sheet length then there will be no overlap and the edges will exactly correspond. If not, then the overlap will correspond to the extent to which the length of the individual sheets required to wind back around the roll exceeds or falls short of the actual circumference of the roll.
For example if the circumference of the roll is 15cm and the sheet size is 5cm then there will be no overlap. If, however the circumference of the roll is 13 cm, then the overlap will be 2cm.*
I say it has nothing to do with the length of the individual sheets. I postulate that if the toilet paper roll was made from two continuous pieces, and rather than perforated it is simply cut off evenly and then wound back around the roll, the two cut ends would still not line up.
This is… almost right. He’s not taking into account that it works for both sheets of toilet paper, not just one. Take the roll and mark where it ends, and unroll it exactly one revolution. It doesn’t line up with the perforation for both sheets.
The easy way to solve it is to find out how toilet paper is actually made. The way toilet paper is actually made is it’s unrolled from two huge master rolls onto the cardboard tube, and stamped with perforations and texture by a machine as it goes on. When the proper amount is on the roll, it’s cut off, glued to the roll, then the roll itself is sliced into 5" widths (or however wide toilet paper rolls are). Why two ply? Marketing. People feel like they’re getting more use out of two ply and it’s more robust. Why not just make [del]10 louder[/del] one-ply thicker? Because then it would feel like wiping with a paper towel, and many people don’t like that. Yes people are weird. Paper towels actually were invented by a manufacturing mishap, in which the tissue thin paper was made way too thick. Rather than waste it, Arthur Scott marketed them as “paper towels” had the rolls cut into 13" lengths, and marketed them as a sanitary, medical necessity.
Well, yes, but that does not make him wrong. He is trying to be quantitative about it and figure the size of the mismatch, working in units of sheet length. You are just saying that regardless of sheet length they won’t line up. He is trying to say by how much.
In no circumstances will the end of the wound-back sheet line up with the end of the other sheet, but if the circumference is equal to an integer multiple of the sheet length then the end of the sheet you did not unroll will line up with one of the perforations on the unwound sheet, and you can tear off the extra sheet or sheets of one-ply and be back to having a roll where the perforations line up.