December 18, 2016

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 ... .