What would it look like if you marked each perforation on a roll of toilet paper and rolled it all back up? A striped pattern? A spiral pattern? I wondered about that in college, and wrote a program to answer the question - here's it's latest incarnation:
Of course, the pattern depends on the sheet thickness and the sheet length. Try clicking different places on the screen - the farther your mouse is to the right, the thicker the sheet, the farther down, the longer the sheet. I was surprised at what a difference small changes in these values had on the overall pattern. My fractal sense is tingling - is yours?
At one point I found an article about this type of pattern (an Archimedean spiral marked at fixed intervals), but have since lost track of it - can you help me out?