Expert: Paul Klarreich - 11/29/2007

Question
Cut a standard 8.5 x 11 piece of paper into a strip of length 8.5 inches.  The width is irrelevant.  Fold the left end of the paper toward the right side, then fold the right side back towards the left so that the crease is made exactly where the original fold stopped.  Then, fold the left side back across the right so the crease is made exactly where the right side stopped.  You should theoretically be able to repeat this folding and make infinite folds of this paper.  What is the distance between the first fold and the right hand side of the paper so that you can make infinite folds?

Answer
Questioner:   Max
Category:  Calculus
Private:  No
 
Subject:  folding a rectangular piece of paper
Question:  Cut a standard 8.5 x 11 piece of paper into a strip of length 8.5 inches.  The width is irrelevant.  Fold the left end of the paper toward the right side, then fold the right side back towards the left so that the crease is made exactly where the original fold stopped.  Then, fold the left side back across the right so the crease is made exactly where the right side stopped.  You should theoretically be able to repeat this folding and make infinite folds of this paper.  What is the distance between the first fold and the right hand side of the paper so that you can make infinite folds?
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Hi, Max,

This looks like one of those 'Golden Section' things.
I think it has to work like this:  At every stage, the length to be folded over must be the same fraction of the entire current length.

At stage 1, let the current length be one 'hand'.  [My hand is 8.5 inches - check yours.]

At stage 1, let x = the portion folded over:

   x                   1 - x
--------------+-----------------------

   x           1 - 2x
=============+----------
<--folded-- ><- single->

At stage 2, the Current Length is 1 - x.  The Portion Folded Over is 1 - 2x.

But we must have

PFO2   PFO1
---- = -----
CL2    CL1

This gives:

1 - 2x     x
------ = ------
1 - x      1

Multiply out:

1 - 2x = x - x^2
x^2 - 3x + 1 = 0

Solve the quadratic:
[use only the minus solution.]
    3 - sqrt(9 - 4)
x = ----------------
          2

    3 - sqrt(5)
x = ------------
          2

x = 0.382, about.

Fold over 0.382 hands. (You work out the inches.)