Instead, I found an interesting math puzzle after I was done painting. I had taped all of the woodwork (quite a lot including trim and railings) and needed to take it all down. I could have just made a huge messy ball, but, boring! So instead, I made up a game:
- Start with a piece of tape or long strip of paper
- Make a few folds at whatever angle you'd like
- Continue wrapping by making a fold to align with the edge (like this)
- You lose if your strip ends up with a vertex in the middle (i.e. no edge to fold along)
I started with a 45-45-90 triangle. Of course, it wasn't exactly 45-45-90. All three sides of the triangle ended up much wider than the strip, but the pattern continued nicely. I quit when I got bored, I never lost.
Then I tried to make an equilateral triangle. This was quite satisfying. It lasted as long as I wanted to play and didn't get too much wider than the tape.
I tried to make triangles with other angles. Those resulted with the messy shapes on the far left of both the middle and bottom rows:
My rectangle is completely uninspired. I tried to make a different rectangle by changing the rules - I folded so an edge would line up next to a previous edge. It started making diagonals across a quadrilateral of sorts. But I got distracted when the tape got stuck to the wall and messed this one up. (not pictured, big mess)
I'm most proud of the pentagon! I continued wrapping for a long time and yet two sides stayed exactly the width of the tape. I know that a regular pentagon is possible too because I've made those paper strip stars.
What polygons work perfectly? (I'm defining perfect as: the side length stays the same as you wrap once the original polygon is formed)How quickly does a small error magnify? Is this chaos theory or slow and steady? Does a 59-61-60 degree triangle react very differently from at 60-60-60 triangle?
Are there any shapes that start out working and then you eventually lose?