January 20, 2014

Folding Polygons

This weekend I painted my stairs. Most people think the math involved in painting is calculating the volume of paint required. I will grant you that the area of the walls in my stairwell is more interesting than your average wall and includes some cool composite shapes, but really, I only buy paint by the gallon. So knowing the exact area of the walls I needed to paint was not necessary.

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?

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