The crease length problem: fold the lower-left corner of a sheet of paper  inches wide and  inches high, with  , to the right edge and crease. What fold gives the shortest crease? The problem becomes interesting when you consider folds with "flaps", not just folds that intersect the left and bottom edges. The "textbook" solution is to fold so the lower endpoint of the crease is three quarters of the way to the right corner. When  , this crease is shortest among creases that intersect the left and bottom edges. But it is never the shortest crease. The folds giving the shortest and longest creases depend in a remarkable way on the ratio  . When  , where  (the golden ratio), there are two distinct shortest creases and two distinct longest creases. There are other special  , when local extrema have the same value. Choose "paper" to play with folding  to  , noting the significance of the circles, or choose "crease length" to study crease length. Vary  and  to change the shape of the paper. The point P can be restricted to the right edge of the paper. Note how the graph of crease length for P along the right edge changes with  , for example, how the locations of the minimum and maximum change. The 3D graph can be rotated for better viewing. Folding  to  ,  , the graph of crease length as a function of  looks like a flying bird, /\/\. There are five critical folds, with local extrema  ,  ,  ,  , and  . The order of these local extrema changes as  is varied. In fact, there are seven "special  ",  such that if  , then  . The critical lengths have a different permutation for  in each interval:  . Choose "critical table" and select "formulas" or "values" to study formulas or numerical values for P, the crease endpoints  , and crease length  at the five critical folds. Choose "fly" to fly the crease length graph by selecting "  " and varying  or  . See how the order of the local extrema changes at the special  . It is interesting to set  to 8 or 8.5 and vary  . Select one of the  to see the graph when  . The table to the right shows the special values of  for a given  . Observe that one of  and  is always less than  , the "textbook" minimum. Choose "order of critical creases" to see critical folds in order of crease length. Study by varying  (with  selected) or by selecting one of the  . The min and max crease lengths are either  and  (when  ) or  and  (when  ). The crease length orders between the special  are  if  ;  if  ;  if  ;  if  ;  if  ;  if  . Interestingly, if  , then  and the textbook answer (for folds intersecting the left and bottom edges) is also "correct" for folds that only intersect the bottom edge. Examples are  (for Stewart) and  (for Gardner). But for  or  paper,  , and both  and  are less than  . Choose "surprise" to see how different the results can be for paper of almost the same shape, such as  and  , where  .
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'm grateful to him for stirring my interest in using Mathematica to investigate the problem. Stewart includes a figure with the crease intersecting 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 intersecting 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. After discovering what I thought were amazing things, including a connection with the golden ratio, I found Ellermeyer had already discovered some 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.
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