Exploring the Crease Length Problem

The classic crease length problem: fold the lower-left corner of a sheet of paper inches wide and inches high, with , to a point on the right edge, and crease. Which fold gives the shortest crease? The problem becomes interesting when you consider folds with "flaps", not just folds with creases that end on the left and bottom edges.
The "textbook" solution assumes creases end only on the left and bottom edges, and the "textbook" answer is the crease whose lower endpoint is three-quarters of the way to the right corner. That is the correct answer for this case, but only when . it is never the shortest crease. The shortest crease is either the vertical crease, when , or the crease for the fold to the opposite corner, when , where (the golden ratio). Both these creases are shortest when .
This Demonstration lets you explore these and other results, such as the creases corresponding to local extrema of the crease length function and the order of the lengths of these special creases.
Choose "paper" to play with folding to for various and . Note the significance of the circles.
Choose "crease length" and "2D study" to study the crease length function for folds of to as a function of for fixed . Click "constrain to right edge" to make and see the crease length function for .
With , vary to observe how the crease length function depends on . Note the relative extrema, corresponding to five critical creases (vertical, upper-left corner, traditional, 45°, and opposite corner). Note the relative minimum for the traditional critical crease is never the minimum.
With , set . The crease is the 45° crease. This crease has a relative maximum length. Vary and observe when it has the maximum length and when the upper-left corner crease has the maximum length. They have the same maximum length when . Note that then the vertical crease and the opposite corner crease also have the same minimum length.
Choose "crease length" and "3D study" to study the crease length surface as a function of .
Choose "fly" to study in detail how the crease length function changes as and are varied. The shape of the graph depends on . The graph looks like /\/\ (like a flying bird), with local extrema , , , , and for the five critical folds, and the bird appears to fly as is varied. Click the button for to allow to vary. Click the button for to set . These are the "special" for which . The order of the is .
When the local maximum is equal to the local maximum and the crease length function is increasing at . When , it is increasing on and decreasing on .
Choose "order of critical creases" to see representations of paper with the five critical creases, in order by crease length. Click and vary to see the permutations change, or click to see their order when . Observe the folds giving minimum and maximum crease lengths are different when and when .
The crease length orders between the special are
if ,
if ,
if ,
if ,
if ,
if .
Interestingly, if , then and the textbook answer for creases ending on both the left and bottom edges is also "correct" for creases ending on the bottom edge. Examples are (for Stewart) and (for Gardner). But for or paper, though, , and the opposite corner crease is shortest.
Choose "critical table" to see formulas or values for , crease endpoints , and crease lengths for the five critical creases.
Choose "surprise" to see how different the results can be for paper of almost the same shape, such as and , because .


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My step-grandson William Jons, studying for a UMTYMP (University of Minnesota Talented Youth Mathematics Program) exam, asked me about problem 65 in section 4.7 of Stewart. I am grateful to him for stirring my interest in using Mathematica to investigate the problem.
Stewart includes a figure with the crease ending on the left and bottom edges, and adds, "In other words, how would you choose (the distance along the bottom from the left corner) to minimize (the length of the crease)". The student is discouraged from considering folds with flaps. Gardner has a figure with a flap and the crease ending on the top and bottom edges, but his solution ignores this case. Luckily, he used 8×11 paper and not 8×10 or 8.5×11.
The Demonstration "Optimize the Length of the Crease of a Folded Piece of Paper" considers folds with creases that end on the two short sides, but not creases which end on the two long sides. This misses the opposite corner fold, which has the shortest crease for 8.5×11.75 paper.
After discovering what I thought were amazing things, including the connection with the golden ratio, I found Ellermeyer had already discovered most of the same things.
J. Stewart, Calculus: Early Transcendentals, 5th ed., Belmont, CA: Brooks/Cole, 2007.
M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles and Diversions, New York: Simon and Schuster, 1961.
S. Ellermeyer, "A Closer Look at the Crease Length Problem," Mathematics Magazine, 81(2), 2008 pp. 138–145.
R. B. Kirchner, "The Crease Length Problem Revisited," Spring Meeting, MAA North Central Section, 25 April, 2009; CDF file for the presentation is here: https://dl.dropbox.com/u/17725131/CreaseLengthProblem/CreaseLengthProblemRevisitedPlayer.nbp
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