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Fold a Strip of Paper into Approximate n-ths

You can easily fold a paper in half, into fourths and so on, but how do you fold a piece of paper into 5ths, or 7ths? Here is a demonstration of the Fujimoto's approximation, a method that will allow you to fold a strip of paper in any odd units. For this example, we'll fold a strip into fifths.

Steps

1. Make a guess where you think one fifth of the paper is and make a pinch there to mark the spot. Open the paper again.
   
2. Fold the far end to that pinch mark and make a mark there. (This mark will be approximately at the 3/5ths mark.)
   
3. Fold the far end to the latest pinch mark and make another pinch mark. (4/5th)
   
4. Fold the beginning of the strip to the latest pinch mark and make a final pinch mark. (2/5ths)
   
5. Fold the beginning to the strip to the latest mark and this time crease the paper. This is a very accurate 1/5th mark.
   
6. Fold other fifths from this length by performing an accordion fold.
   

Notes

• Don't let the word approximation throw you off. The fold will be probably as good as you can get with an "exact" method given the inaccuracies in folding.
  
• How does this work? If you think of your initial guess as being at a distance exactly 1/5th plus a small error, the next fold ends up folding the error in half! As you repeat the folding, the error is reduced by a factor of two each time.
  
• For another "self-correcting" method of folding see Fold Pentagons from a Strip of Paper.
  
• The book by Thomas Hull which introduced me to this fold is a great book for teachers of mathematics at an advanced high school or college level. It introduces a simple activity and then explores the rich mathematics behind it.
  

References

• (Hull 2006)

Comments

  © Laszlo Bardos